Past exam of the topics in applied mathematics course of the University of Oxford / 2008 / A / 2

The term $rIS$ is a mass-action incidence rate: each infective individual makes potentially transmitting contacts with susceptible individuals, and the total rate is proportional to the product of the two compartment sizes. An infection transfers one animal from $S$ to $I$, accounting for opposite signs in their equations. The per-capita mortality rate of an infective individual is $a$, giving transfer $aI$ from $I$ to $R$.

There are no births, recoveries or additional deaths in this model. Adding the three equations gives $\dot{S}+\dot{I}+\dot{R}=0$, so the total initial biological population is conserved when the dead compartment is included. Before quarantine, $S$ decreases whenever $I,S>0$; after effective isolation no further transmission is permitted.

Write $S_0=S_{initial}$ and $I_0=I_{initial}$. Until quarantine and while $I,S>0$, division of the equations givesIntegration from the initial state yields the first integralLet $S_{\ast }$ be the susceptible count at the first time $I=I_{\ast }$. It is the root of an equation reached on the decreasing-$S$ outbreak trajectory ofFor a strictly rising threshold crossing this is the larger admissible root of an equation, with $a/r<S_{\ast }<S_0$; if the threshold equals the peak, $S_{\ast }=a/r$. The assumption that quarantine is reached selects the physically relevant branch and excludes a merely hypothetical declining-branch crossing after an earlier one.

Since there is no incubation period here, all removals from $S$ have already become infection cases by quarantine. No further susceptible individuals are infected after isolation. The requested number among the initially susceptible individuals is therefore

The delayed influx $rI(t-\tau )S(t-\tau )$ represents animals exposed one incubation period earlier who enter the infective individual compartment now. They die at the instantaneous rate $aI(t)$ once in that compartment. The displayed bookkeeping removes an animal from $S$ at the same delayed time that it enters $I$, preserving $S+I+R$.

For this bookkeeping, $S$ must include the exposed animals which have not yet completed incubation period. If $S$ were instead defined as strictly unexposed susceptible individuals, its loss term would be the current exposure rate and a separate latent compartment would be needed. The calculations here follow the printed delay differential equation. A full initial-value problem for a delay differential equation also needs a history on $[-\tau ,0]$; the following small-delay expansion of an epidemic transfer rate concerns a smooth compatible history and the regular regime assumed by the question.

In the first-order Taylor expansion correction it suffices to use the zeroth-order derivatives. They giveThe delayed product is consequentlyMultiplying by $r$ and substituting into the two transfer equations producesThe opposite signs of $f$ and $g$ are also required by conservation law of $S+I+R$: this correction changes the timing of the transfer between $S$ and $I$, while leaving the instantaneous death term unchanged.

At the retained order define $K=1+\tau r(I-S)$. Neglecting terms of order $a\tau$ in the incidence correction givesDividing and expanding consistently,Thus the susceptible count $S_{\ast }$ at the first threshold crossing obeys the same retained-order equation as before:It is important to distinguish completed case transfers at quarantine from all infections already initiated. Let the quarantine time be $T$. Isolation prevents exposures after $T$, but animals exposed during $(T-\tau ,T)$ still finish their incubation period during $(T,T+\tau )$. Their number, in the printed bookkeeping, isThe infection cases transferred by time $T$ among the initial $S$ class number $S_0-S_{\ast }$. Adding those pending infection cases gives the eventual number who catch the infectious disease:This adds only infections caused by exposures before isolation; it does not allow secondary transmission after quarantine. If one asks instead only for the transfers completed by the instant of quarantine, the answer is $S_0-S_{\ast }$. The distinction is necessary because the model keeps incubating animals in its $S$ compartment.