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'''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 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.
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]].<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>
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 origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.
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。
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>
==The SIR model==
==SIR模型==
The '''SIR model'''<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> 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:
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:
'''<font color="#ff8000">SIR 模型</font>'''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''': 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 '''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.
This model is reasonably predictive<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> 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]].
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.
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。]]
===Transition rates===
===传染率===
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]]<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>).
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).
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).
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>''')。
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>)。
===The SIR model without vital dynamics===
===缺少生命动力学的SIR模型===
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="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><ref name="Beckley"/>
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:
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:
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:
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>
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.<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> 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>
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>
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> ，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>
(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)。
This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely
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 transcendental equation has a solution in terms of the Lambert function, namely
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
====The force of infection====
====感染力====
Note that in the above model the function:
Note that in the above model the function:
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> 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.
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.
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。
====Exact analytical solutions to the SIR model====
====SIR 模型的精确解析解====
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.<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
An equivalent analytical solution found by Miller<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> yields
An equivalent analytical solution found by Miller yields
An equivalent analytical solution found by Miller yields
An equivalent analytical solution found by Miller yields
等价的'''<font color="#ff8000">解析解analytical solution</font>'''由米勒发现
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
===The SIR model with vital dynamics and constant population===
===具有生命动力学和稳定人口的SIR模型===
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<ref name="Beckley"/>. The model with mass-action transmission is:
==Variations on the basic SIR model==
==基础SIR模型的变化==
===The SIS model===
===SIS模型===
[[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]]
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),<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> 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
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
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>''') ，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:
===The SIRD model===
===SIRD模型===
[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]
[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]
Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>
Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>
[图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</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>).]]
[[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>).]]
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>).
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>)。]
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>
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.
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.
这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。
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>
===The MSIR model===
===MSIR模型===
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.
MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:
MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:
[图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 仓室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:]
===Carrier state===
===病原携带状态===
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 [[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.
A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.
A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.
[图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。]
===SEIR模型===
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).
SEIR compartmental model
SEIR compartmental model
[图10：SEIR compartmental modelSEIR传染病模型]
=== The SEIS model ===
=== SEIS模型 ===
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.
=== The MSEIR model ===
=== MSEIR模型 ===
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.
=== The MSEIRS model ===
=== MSEIRS模型 ===
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.
===Variable contact rates===
===可变接触率===
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.
==Modelling vaccination==
==疫苗接种模型==
The SIR model can be modified to model vaccination
The SIR model can be modified to model vaccination
The SIR model can be modified to model vaccination
SIR 模型可以经过修改，对疫苗接种进行建模
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>
Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
===Vaccinating newborns===
===新生儿疫苗接种===
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>:
===Vaccination and information===
===疫苗接种与信息===
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:
===Vaccination of non-newborns===
===非新生儿疫苗接种===
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:
===Pulse vaccination strategy===
===脉冲疫苗接种策略===
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:
==The influence of age: age-structured models==
==年龄的影响：年龄结构模型==
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:
分区传染病模型中的其他考虑事项
分区传染病模型中的其他考虑事项
=== Vertical transmission ===
=== 垂直传染 ===
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.<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>
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.
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.
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。
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===
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" /><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>
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.
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.
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。
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 ===
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:<ref name=":2" />
One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<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>
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.
One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.
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。
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>
==参见==
*[[Mathematical modelling in epidemiology]]
*[[Mathematical modelling in epidemiology]]
==References==
==参考文献==
{{reflist}}
{{reflist}}
==Further reading==
==深入阅读==
*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}
*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}
==External links==
==外部链接==
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]
==编辑推荐==
==编辑推荐==
===集智文章推荐===
[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]
[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]
[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型 用Links建模网络动力学]
[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型 用Links建模网络动力学]
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[[分类: 流行病学]] [[分类:科学建模]]