Past exam of the topics in applied mathematics course of the University of Oxford / 2009 / A / 2

The law of mass action assigns infection rate $rIS$, death rate $aI$, and vaccination rate $pIS$. Vaccination consumes a susceptible animal and produces a vaccinated one while leaving the triggering infected animal present. In dimensional time $T$,All compartment counts remain nonnegative. Adding the equations gives $(I+S+V+R{)}_T=0$; deaths are retained in the bookkeeping compartment rather than removed from that conserved count. There is no recovery class of live immune survivors in the infection reaction: in this particular model every animal that becomes infected ultimately dies.

Use $t=aT$ and scale each count by the susceptible infection threshold $a/r$:For example $I_T=(a^2/r)q_t$, while $rIS=(a^2/r)qs$. Dividing all equations by $a^2/r$ therefore givesThus $\eta$ measures the vaccination response relative to transmission in this mass-action model. The total dimensionless count $q+s+v+m$ is conserved, and the instantaneous infection growth threshold is $s=1$.

While $q>0$ and $s>0$, division of the equations gives $dq/ds=(1-s)/[(1+\eta )s]$ and $dv/ds=-\eta /(1+\eta )$. Integrating from the initial data yieldsLet $s_{\infty }$ be the final susceptible level. Using $q_{\infty }=0$ givesThis is the smaller positive root selected by decreasing susceptibility; the function $s-\log s$ is decreasing on $(0,1)$, so that selected root is unique. Its upper bound below $s_0$ also follows directly because the added positive term moves the decreasing-branch root left.

Conservation of the total count, or integration of $m_t=q$, then gives the number killed, including those initially infected:The remaining susceptible loss is split between infection and vaccination in fixed ratio $1:\eta$; this explains the factor multiplying $s_0-s_{\infty }$.

Rearrange the final-size equation as $s_{\infty }=s_0\exp [s_{\infty }-s_0-(1+\eta )q_0]$. With fixed $q_0>0,s_0>0$ and $\eta \to \infty$, the selected root is exponentially small:Substitution into the death count yieldsThus strong vaccination suppresses most secondary fatal infections but cannot save the $q_0$ animals already infected, as required by the stated reaction scheme. The limit holds for the fixed positive initial infection assumed here; it need not be uniform if $q_0$ is allowed to shrink with increasing $\eta$.

For the vaccinated epidemic-wave final size, write $s_{-}=s(-\infty )$. From the travelling-wave equations, $dv/ds=-\eta /(1+\eta )$ and $dm/ds=-1/[(1+\eta )s]$. Using the ahead-of-wave boundary data givesAdding the four wave equations yields $q^{\prime \prime }+c(q+s+v+m{)}^{\prime }=0$. Integrate once and use $q^{\prime }(\infty )=0$ to obtain $q^{\prime }+c(q+s+v+m)=c{s}^{\ast }$. At minus infinity the assumptions $q=q^{\prime }=0$ therefore give $s_{-}+v_{-}+m_{-}=s^{\ast }$.

Combining these identities shows that $s_{-}$ satisfiesThe nontrivial positive epidemic-wave root has $0<s_{-}<1<s^{\ast }$; the other root $s_{-}=s^{\ast }$ is the disease-free solution. The assumed positive wave consequently requires $s^{\ast }>1$; for $s^{\ast }\le 1$ the displayed relation has no distinct smaller root.

At the back of the pulse there are no infectives, so the live biological population consists of susceptible and vaccinated animals:The final-size root is independent of vaccination strength in this travelling-wave problem; vaccination instead changes the split between deaths and survivors for the total biological population lost from susceptibility.

From the preceding conservation identity,The nontrivial root $s_{-}$ depends only on $s^{\ast }$, through $\log (s^{\ast }/s_{-})=s^{\ast }-s_{-}$. Therefore the coefficient $s^{\ast }-s_{-}$ is independent of $\eta$, proving the exact requested proportionality to $(1+\eta {)}^{-1}$, conditional on existence of the positive wave. This is stronger than an asymptotic estimate and differs from the spatially homogeneous problem with a fixed positive initially infected count.