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Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. The modelling can help decide which intervention/s to avoid and which to trial, or can predict future growth patterns, etc.

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. The modelling can help decide which intervention/s to avoid and which to trial, or can predict future growth patterns, etc.

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We define $\lambda=\beta/\gamma$ as the effective infection rate of this model. From the above equation, we can find that when $\lambda>1$, the steady-state value of I is $i=(\beta-\gamma)/\beta=1-1/\lambda$, an endemic disease state is obtained in this case (indicates that there will be a number of infected individuals in the population at the final state). However, when $\lambda<1$, we have$i\to0(t\to\infty)$, which means that there will be no I state individuals in the population finally and is called a healthy state. Therefore, $\lambda=1$ is a threshold value of the SIS model, which is also known as the basic reproduction number <ref>Heffernan J M, Smith R J, Wahl L M. Perspectives on the basic reproductive ratio[J]. Journal of the Royal Society Interface, 2005, 2(4): 281-293.{{url = https://www.ncbi.nlm.nih.gov/pmc/articles/pmc1578275/ }}</ref>

We define $\lambda=\beta/\gamma$ as the effective infection rate of this model. From the above equation, we can find that when $\lambda>1$, the steady-state value of I is $i=(\beta-\gamma)/\beta=1-1/\lambda$, an endemic disease state is obtained in this case (indicates that there will be a number of infected individuals in the population at the final state). However, when $\lambda<1$, we have$i\to0(t\to\infty)$, which means that there will be no I state individuals in the population finally and is called a healthy state. Therefore, $\lambda=1$ is a threshold value of the SIS model, which is also known as the basic reproduction number <ref>Heffernan J M, Smith R J, Wahl L M. Perspectives on the basic reproductive ratio[J]. Journal of the Royal Society Interface, 2005, 2(4): 281-293.</ref>

在这里，我们将$\lambda=\beta/\gamma$定义为该模型的有效感染率。 从上式可知，当$\lambda>1$时，I的稳态值为$i=(\beta-\gamma)/\beta=1-1/\lambda$，表明终态时人群中存在一定比例的受感染者，也就是说当$\lambda>1$时，整体人群处于地方性疾病状态。 但是，当$\lambda<1$时，我们有$i\to0(t\to\infty)$，这意味着最终人群中将没有I状态个体，整体人群被称为健康状态。 综上，$\lambda=1$是SIS模型的传播阈值，也称为基本再生数。

在这里，我们将$\lambda=\beta/\gamma$定义为该模型的有效感染率。 从上式可知，当$\lambda>1$时，I的稳态值为$i=(\beta-\gamma)/\beta=1-1/\lambda$，表明终态时人群中存在一定比例的受感染者，也就是说当$\lambda>1$时，整体人群处于地方性疾病状态。 但是，当$\lambda<1$时，我们有$i\to0(t\to\infty)$，这意味着最终人群中将没有I状态个体，整体人群被称为健康状态。 综上，$\lambda=1$是SIS模型的传播阈值，也称为基本再生数。

=== SIR模型 ===  

=== SIR模型 ===  

The SIR model is introduced to explain the epidemics with permanent immunity in the population. Different from the SI and SIS model, a recovered state (R) is given in this model. Individuals in S state would be infected by the I state individuals with probability $\beta$, whereas the I state individuals would recover to R state with a recovery probability $\gamma$. Thus, the ordinary differential equations of SIR model are

The SIR model is introduced to explain the epidemics with permanent immunity in the population. Different from the SI and SIS model, a recovered state (R) is given in this model. Individuals in S state would be infected by the I state individuals with probability $\beta$, whereas the I state individuals would recover to R state with a recovery probability $\gamma$. Thus, the ordinary differential equations of SIR model are

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