Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 3

The SI model has two compartments and one transition,where $S$ and $I$ are the numbers susceptible and infectious and $\beta >0$ is the per-infective transmission-rate parameter under frequency-dependent homogeneous mixing.

The ordinary differential equations areThe conservation of population identity $S+I=N$ follows by adding the equations.

Using $S=N-I$ reduces the system to the logistic differential equationSeparation of variables and the initial values giveDifferentiation yields the incidence

Writing $x=\theta e^{-\beta t}$ gives $\Lambda =N\beta x/(1+x{)}^2$. Its maximum occurs at $x=1$, orIf time is restricted to $t\ge 0$ and $\theta <1$, the unconstrained maximizer precedes the initial time, so the maximum on the observed interval is instead at $t=0$.

Set $\tau =t-t_{\ast }$. The Logistic solution of the SI model becomesIt is a symmetric, unimodal bell-shaped curve about the incidence peak $\tau =0$.

The removal rate is $\dot{R}=\gamma I$. Because $\gamma$ is constant, its interior peak occurs when $I$ is maximal. At such a point,and $I>0$, so

Dividing the $I$ equation by the $S$ equation givesThereforeis a first integral. Define the basic reproduction number by $R_0=\beta S(0)/\gamma$. At the removal peak, $S=S(0)/R_0$, and henceand, using $S+I+R=N$,

The same first integral, evaluated initially and after the epidemic when $I(\infty )=0$, givesWhen $I(0)$ is negligible and $R(0)=0$, conservation of population gives $S(\infty )\simeq S(0)-R(\infty )$. Since $\beta /\gamma =R_0/S(0)$, this is equivalent to the final size relation for an epidemic