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第1行： − − − '''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again. − − Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again. − − ==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑 − − ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）词条名Compartmental models in epidemiology翻译为复杂传染病是否妥当 − − − The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]]. − − The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927. − − − − The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze. − − The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze. − − − Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population. − − Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population. 第37行：
第12行： − − − − The '''SIR model'''is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments: − − The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments: − − :'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment. − − S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment. − − :'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals. − − I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals. − − :'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant". − − R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant". − − − − This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]]. − − This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella. − − − These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/> − − These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control. − − − − As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment. − − As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment. − − − − − Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e. − − Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e. − − − − − − For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N<sup>2</sup>'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N<sup>2</sup>'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants). − − For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N<sup>2</sup>, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N<sup>2</sup> is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants). − − − Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]). − − Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution). − − − − For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected. − − For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected. 第137行：
第51行： − − − − The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Beckley"/> − − The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations: 第156行：
第64行： − − − − − − where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three. − − where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three. − − − − This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]]. − − This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross. − − − This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from: − − This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from: 第195行：
第84行： − − − − − expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables. − − expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables. − − − − Secondly, we note that the dynamics of the infectious class depends on the following ratio: − − Secondly, we note that the dynamics of the infectious class depends on the following ratio: 第216行：
第92行： − − − − the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math> − − the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math> − − − − By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get − − By dividing the first differential equation by the third, separating the variables and integrating we get 第237行：
第101行： − − − − where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects. − − where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects. − − Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and − − Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and − − <math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has − − <math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has − − − − − − − − − − − − − − − − (note that the infectious compartment empties in this limit). − − (note that the infectious compartment empties in this limit). − This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]], namely − − This transcendental equation has a solution in terms of the Lambert function, namely − − − − − − − This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects. − − This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects. − − − The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows: − − The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows: 第313行：
第128行： − − − − it yields that if: − − it yields that if: 第328行：
第137行： − − − − then: − − then: 第341行：
第144行： − − − − i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if − − i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if 第354行：
第151行： − − − − then − − then 第367行：
第158行： − − − − i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important. − − i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important. 第379行：
第164行： − − − − Note that in the above model the function: − − Note that in the above model the function: 第391行：
第170行： − − − − − models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>): − − models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>): 第404行：
第176行： − − − − − Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process. − − Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process. 第417行：
第182行： − − − In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization − − In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization − − − − :<math>\mathcal{S}(u)= S(0)u </math> − − :<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math> − − − − for − − for − − − − − with initial conditions − − with initial conditions 第460行：
第201行： − − − − where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>. − − where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>. − − − An equivalent analytical solution found by Miller yields − − An equivalent analytical solution found by Miller yields 第487行：
第217行： − − − − − − Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>. − − Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>. − − − Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" /> − − Effectively the same result can be found in the original work by Kermack and McKendrick. − − − − These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related. − − These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related. 第517行：
第228行： − − − Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is: − − Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is: 第535行：
第241行： − − − − for which the disease-free equilibrium (DFE) is: − − for which the disease-free equilibrium (DFE) is: 第548行：
第248行： − − − − − − In this case, we can derive a [[basic reproduction number]]: − − In this case, we can derive a basic reproduction number: 第564行：
第256行： − − − which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that: − − which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that: − − :<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math> − − − − The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population. − − The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population. 第599行：
第278行： − − Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again. − − Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again. 第612行：
第287行： − 【图5：SIS compartmental modelSIS传染病模型】+ − − − − We have the model: − − We have the model: 第631行：
第300行： − − − − Note that denoting with ''N'' the total population it holds that: − − Note that denoting with N the total population it holds that: 第644行：
第307行： − − − − It follows that: − − It follows that: 第657行：
第314行： − − − i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>: − − i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>: 第675行：
第327行： − − − It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),such that the basic reproduction rate is greater than unity. The solution is given as − − It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as 第686行：
第333行： − − − where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>. − − where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>. − − − − As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to: − − As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to: 第707行：
第343行： − − − − In the long run, in this model, all individuals will become infected. − − In the long run, in this model, all individuals will become infected. 第729行：
第359行： − − Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>). 第738行：
第366行： − The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations: 第751行：
第378行： − − where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively. − − where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively. 第762行：
第385行： − − − For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model. − − For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model. 第794行：
第412行： − − − Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C. − − Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C. 第813行：
第426行： − − − − For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed). − − For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed). 第830行：
第437行： − − − Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model: − − Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model: 第849行：
第451行： − − − − We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable. − − We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable. − − − For this model, the basic reproduction number is: − − For this model, the basic reproduction number is: 第870行：
第461行： − − − Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions − − Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions 第881行：
第467行： − − − − it holds that: − − it holds that: 第897行：
第477行： − − − − In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients: − − In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients: 第915行：
第489行： − − − − is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane). − − is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane). 第928行：
第496行： − − − The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end. − − The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end. 第940行：
第503行： − − In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model: − − In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model: 第962行：
第521行： − − − For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model. − − For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model. 第987行：
第541行： − − − − An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended. − − An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended. 第1,005行：
第553行： − − − It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic. − − It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic. − − − In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate − − In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate 第1,025行：
第563行： − − − with period T equal to one year. − − with period T equal to one year. − − − Thus, our model becomes − − Thus, our model becomes 第1,050行：
第578行： − − − − (the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if: − − (the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if: 第1,063行：
第585行： − − − − whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input. − − whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input. − This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic. − − This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic. − − − − − − − The SIR model can be modified to model vaccination − The SIR model can be modified to model vaccination 第1,103行：
第608行： − − In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>: − − In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>: 第1,120行：
第621行： − − − − where <math>V</math> is the class of vaccinated subjects. It is immediate to show that: − − where <math>V</math> is the class of vaccinated subjects. It is immediate to show that: 第1,132行：
第627行： − − − − − thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that: − − thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that: 第1,147行：
第635行： − − − − In other words, if − − In other words, if 第1,158行：
第640行： − − the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease. − − the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease. 第1,169行：
第647行： − − − Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects: − − Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects: 第1,179行：
第652行： − − In such a case the eradication condition becomes: − − In such a case the eradication condition becomes: 第1,188行：
第657行： − − i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline. − − i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline. 第1,199行：
第664行： − − − In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows: − − In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows: 第1,215行：
第675行： − − − − − leading to the following eradication condition: − − leading to the following eradication condition: 第1,233行：
第686行： − − − This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects: − − This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects: 第1,249行：
第697行： − − − − − It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by: − − It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by: 第1,262行：
第703行： − − − − − and that the eradication condition is: − − and that the eradication condition is: 第1,280行：
第714行： − − − − Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that: − − Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that: 第1,301行：
第729行： − − − − (where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s: − − (where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations: − − − − :<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math> − − − :<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math> 第1,322行：
第737行： − − where: − − where: 第1,332行：
第743行： − − − − − is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages. − − is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages. − − − Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed: − − Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed: 第1,353行：
第752行： − − − − − but that are nonlocal for the density of susceptible newborns: − − but that are nonlocal for the density of susceptible newborns: 第1,367行：
第759行： − − − where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults. − − where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults. − − Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains: − − Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains: 第1,388行：
第771行： − − − In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold: − − In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold: 第1,401行：
第779行： − − − and the demographic equilibrium is − − and the demographic equilibrium is 第1,413行：
第786行： − − − − automatically ensuring the existence of the disease-free solution: − − automatically ensuring the existence of the disease-free solution: 第1,427行：
第794行： + − − A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator. − − A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator. − − 基本再生数可以通过求解适当的函数算子的谱半径来计算。 + − == Other considerations within compartmental epidemic models == − 分区传染病模型中的其他考虑事项 − − − In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment. − − In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment. 第1,454行：
第810行： − − − − Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /> − − Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission. 第1,466行：
第816行： − − − − Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" /> − − Other occurrences which may need to be considered when modeling an epidemic include things such as the following: − *Non-homogeneous mixing − 不同类型的疾病混合 − + − 动态变化的传染率 − + − 空间上的不均匀分布 − + − 由大型寄生虫引起的疾病 + − ==Deterministic versus stochastic epidemic models== − 确定性和随机流行病模型 − It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref> − It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.+ − + − − − To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications. − − To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications. − − − One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts. − − One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.
无编辑摘要
'''<font color="#ff8000">仓室模型Compartmental models</font>'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。
'''<font color="#ff8000">仓室模型Compartmental models</font>'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。
这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>
这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>
这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。
这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。
模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。
模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。
==SIR模型==
==SIR模型==
'''<font color="#ff8000">SIR 模型</font>'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> 是最简单的仓室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:
'''<font color="#ff8000">SIR 模型</font>'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> 是最简单的仓室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:
S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。
S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。
I：染病者。这类人群已经感染疾病，并且有能力感染易感者。
I：染病者。这类人群已经感染疾病，并且有能力感染易感者。
R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。
R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。
这个模型可以合理地预测<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref>传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。
这个模型可以合理地预测<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref>传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。
[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]
[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]
这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变） ，我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。
这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变） ，我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。
从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。
从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。
[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]
[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]
人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：
人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：
[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]
[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]
===传染率===
===传染率===
为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''<font color="#ff8000">质量作用定律the law of mass action</font>'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。
为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''<font color="#ff8000">质量作用定律the law of mass action</font>'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。
在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''<font color="#ff8000">爱尔郎分布Erlang distribution</font>'''<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>)。
在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''<font color="#ff8000">爱尔郎分布Erlang distribution</font>'''<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>)。
对于没有染病者康复的特殊情况（<math>γ=0 </math>） ，SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。
对于没有染病者康复的特殊情况（<math>γ=0 </math>） ，SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。
===缺少生命动力学的SIR模型===
===缺少生命动力学的SIR模型===
流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''<font color="#ff8000">仓室模型comparenmental models</font>'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref>
流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''<font color="#ff8000">仓室模型comparenmental models</font>'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref>
</math>
</math>
其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。
其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。
这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。
这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。
这个系统是非线性的，但是可以推导出隐式的'''<font color="#ff8000">解析解analytic solution</font>'''。首先:
这个系统是非线性的，但是可以推导出隐式的'''<font color="#ff8000">解析解analytic solution</font>'''。首先:
:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>
:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>
用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。
用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。
其次，我们注意到不同种类人群的动态取决于以下比例:
其次，我们注意到不同种类人群的动态取决于以下比例:
:<math> R_0 = \frac{\beta}{\gamma},</math>
:<math> R_0 = \frac{\beta}{\gamma},</math>
所谓的'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> 那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>
所谓的'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> 那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>
通过将第一个'''<font color="#ff8000">微分方程differential equation</font>'''除以第三个微分方程，分离变量并进行积分，我们得到
通过将第一个'''<font color="#ff8000">微分方程differential equation</font>'''除以第三个微分方程，分离变量并进行积分，我们得到
:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>
:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>
其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。
其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。
易感人群的初始比例<math>S_0=S(0)/N</math>，以及
易感人群的初始比例<math>S_0=S(0)/N</math>，以及
在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有
在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有
:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>
:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>
(请注意，在这个极限时，没有染病者)。
(请注意，在这个极限时，没有染病者)。
这个超越方程有一个Lambert函数解，<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref>即
这个超越方程有一个Lambert函数解，<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref>即
:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>
:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>
这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。
这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。
'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:
'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:
:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>
:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>
如果:
如果:
:<math> R_{0} \cdot S(0) > N,</math>
:<math> R_{0} \cdot S(0) > N,</math>
那么：
那么：
:<math> \frac{dI}{dt}(0) >0 ,</math>
:<math> \frac{dI}{dt}(0) >0 ,</math>
也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果
也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果
:<math> R_{0} \cdot S(0) < N,</math>
:<math> R_{0} \cdot S(0) < N,</math>
那么
那么
:<math> \frac{dI}{dt}(0) <0 ,</math>
:<math> \frac{dI}{dt}(0) <0 ,</math>
也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''和最初的易感者人群数量都极其重要。
也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''和最初的易感者人群数量都极其重要。
====感染力====
====感染力====
注意，在上面的模型中，函数:
注意，在上面的模型中，函数:
:<math> F = \beta I,</math>
:<math> F = \beta I,</math>
建立了从易感者到染病者的传染率模型，这被称为'''<font color="#ff8000">感染力the force of infection</font>'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''<font color="#ff8000">感染力the force of infection</font>'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :
建立了从易感者到染病者的传染率模型，这被称为'''<font color="#ff8000">感染力the force of infection</font>'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''<font color="#ff8000">感染力the force of infection</font>'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :
:<math> F = \beta \frac{I}{N} .</math>
:<math> F = \beta \frac{I}{N} .</math>
Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref>和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。
Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref>和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。
====SIR 模型的精确解析解====
====SIR 模型的精确解析解====
2014年，Harko 和合作者推导出了 SIR 模型的精确'''<font color="#ff8000">解析解analytical solution</font>'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化
2014年，Harko 和合作者推导出了 SIR 模型的精确'''<font color="#ff8000">解析解analytical solution</font>'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化
:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
为了
为了
:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>
:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>
有初始条件
有初始条件
:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>
:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>
其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''<font color="#ff8000">超越方程transcendental equation</font>''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。
其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''<font color="#ff8000">超越方程transcendental equation</font>''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。
等价的'''<font color="#ff8000">解析解analytical solution</font>'''由米勒<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref>发现
等价的'''<font color="#ff8000">解析解analytical solution</font>'''由米勒<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref>发现
</math>
</math>
在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。
在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。
实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。
实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。
注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。
注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。
===具有生命动力学和稳定人口的SIR模型===
===具有生命动力学和稳定人口的SIR模型===
考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:
考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:
</math>
</math>
:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>
:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>
:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>
:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>
这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:
这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。
EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。
[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]
[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]
有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。
有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。
SIS compartmental model
SIS compartmental model
[图5：SIS compartmental modelSIS传染病模型]
我们有以下模型:
我们有以下模型:
</math>
</math>
注意，用N表示人群总数:
注意，用N表示人群总数:
:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.
:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.
由此可见:
由此可见:
:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,
:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,
也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:
也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:
这个模型可以找到一个'''<font color="#ff8000">解析解analytical solution</font>'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''<font color="#ff8000">平均场方程the mean-field euations</font>''') ，<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> 使基本再生率大于单位数。给出了解如下:
这个模型可以找到一个'''<font color="#ff8000">解析解analytical solution</font>'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''<font color="#ff8000">平均场方程the mean-field euations</font>''') ，<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> 使基本再生率大于单位数。给出了解如下:
:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.
:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.
其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。
其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。
作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:
作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:
\frac{dI}{dt} \propto I\cdot (N-I).
\frac{dI}{dt} \propto I\cdot (N-I).
</math>
</math>
从长远来看，在这种模式下，所有的个体都会被感染。
从长远来看，在这种模式下，所有的个体都会被感染。
[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]
[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]
[图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。]
[图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。]
The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:
The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:
易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:
易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:
</math>
</math>
这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>
这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>
===MSIR模型===
===MSIR模型===
对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。
对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。
===病原携带状态===
===病原携带状态===
一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。
一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。
===SEIR模型===
===SEIR模型===
对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。
对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。
[图10：SEIR compartmental modelSEIR传染病模型]
[图10：SEIR compartmental modelSEIR传染病模型]
假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:
假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:
</math>
</math>
我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。
我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。
对于这种模式，'''<font color="#ff8000">基本再生数basic reproduction number</font>'''是:
对于这种模式，'''<font color="#ff8000">基本再生数basic reproduction number</font>'''是:
:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>
:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>
类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件
类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件
:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>
:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>
它认为:
它认为:
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:
当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:
</math>
</math>
是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。
是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。
=== SEIS模型 ===
=== SEIS模型 ===
SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。
SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。
:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>
:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>
在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:
在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:
=== MSEIR模型 ===
=== MSEIR模型 ===
考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。
考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。
=== MSEIRS模型 ===
=== MSEIRS模型 ===
MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。
MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。
===可变接触率===
===可变接触率===
众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。
众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。
此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力
此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力
:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>
:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>
周期T等于一年。
周期T等于一年。
因此，我们的模型变成了
因此，我们的模型变成了
</math>
</math>
(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:
(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:
:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>
:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>
然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。
然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。
这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。
这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。
==疫苗接种模型==
==疫苗接种模型==
SIR 模型可以经过修改，对疫苗接种进行建模<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}</ref>
SIR 模型可以经过修改，对疫苗接种进行建模<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}</ref>
===新生儿疫苗接种===
===新生儿疫苗接种===
当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:
当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:
</math>
</math>
其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:
其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:
:<math> \lim_{t \to +\infty} V(t)= N P,</math>
:<math> \lim_{t \to +\infty} V(t)= N P,</math>
因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:
因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:
:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>
:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>
也就是说，如果
也就是说，如果
:<math> P < P^{*}= 1-\frac{1}{R_0} </math>
:<math> P < P^{*}= 1-\frac{1}{R_0} </math>
疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。
疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。
===疫苗接种与信息===
===疫苗接种与信息===
现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:
现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:
:<math> P=P(I), \quad P'(I)>0.</math>
:<math> P=P(I), \quad P'(I)>0.</math>
在这种情况下，传染病根除的条件是:
在这种情况下，传染病根除的条件是:
:<math> P(0) \ge P^{*},</math>
:<math> P(0) \ge P^{*},</math>
也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。
也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。
===非新生儿疫苗接种===
===非新生儿疫苗接种===
如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:
如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:
\end{align}
\end{align}
</math>
</math>
推导出以下的根除条件:
推导出以下的根除条件:
===脉冲疫苗接种策略===
===脉冲疫苗接种策略===
这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:
这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:
\end{align}
\end{align}
</math>
</math>
很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：
很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：
:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>
:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>
而根除条件是:
而根除条件是:
==年龄的影响：年龄结构模型==
==年龄的影响：年龄结构模型==
年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:
年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:
:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>
:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>
(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:
(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:
:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>
:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>
其中:
其中:
:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>
:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>
是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。
是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。
新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:
新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:
:<math>i(t,0)=r(t,0)=0</math>
:<math>i(t,0)=r(t,0)=0</math>
但对于易感新生儿的密度来说，这是非局部的:
但对于易感新生儿的密度来说，这是非局部的:
:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>
:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>
其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。
其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。
此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:
此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:
在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:
在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:
:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>
:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>
人口均衡是
人口均衡是
:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>
:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>
能够自动保证无疾病时解的存在:
能够自动保证无疾病时解的存在:
基本再生数可以通过求解适当的函数算子的谱半径来计算。
== 分区传染病模型中的其他考虑事项==
=== 垂直传染 ===
=== 垂直传染 ===
就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>
就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>
===水平传染===
===水平传染===
人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。<ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>
人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。<ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>
=== 其他 ===
=== 其他 ===
在建立流行病模型时可能需要考虑的其他事件包括以下事项:
在建立流行病模型时可能需要考虑的其他事件包括以下事项:
*Variable infectivity
*不同类型的疾病混合
*Distributions that are spatially non-uniform
*动态变化的传染率
*Diseases caused by macroparasites
*空间上的不均匀分布
*由大型寄生虫引起的疾病
==确定性和随机流行病模型==
必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。
必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>
更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。仓室模型的可靠性仅限于区分人群类别的应用。
更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。仓室模型的可靠性仅限于区分人群类别的应用。
平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>
平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>