Past exam of the short options course of the University of Oxford / 2017 / 9 / a

The susceptible compartment retains $S_t$ individuals, gains $B$ recruits per generation and loses $\alpha S_t{I}_t$ through infection. The mass-action interaction term represents new infections generated by encounters between susceptible and infectious individuals. Those new cases form the next infectious compartment, $I_{t+1}$; the current infectious generation leaves that compartment after one time step.

The initial conditions $S_0,I_0$ specify the starting susceptible and infectious counts. The parameter $\alpha$ is the transmission coefficient in this deterministic approximation, while $B$ supplies new susceptible individuals. This is the discrete susceptible-infective model with recruitment.

At a fixed point the first equation gives $\alpha S_{\ast }{I}_{\ast }=B$. The second gives $I_{\ast }=\alpha S_{\ast }{I}_{\ast }$, so $I_{\ast }=B>0$ andThere is no disease-free fixed point: with $I=0$, susceptibles still increase by $B$ each step.

Set $r=\alpha B>0$. The Jacobian matrix of the update at the fixed point isIts characteristic equation and multipliers areFor $0<r<4$, these are distinct complex conjugates with product one, so both have modulus one. The linearized iterates are bounded and oscillatory: neutral linear stability holds for $0<\alpha B<4$, with no linear asymptotic attraction.

At $r=4$, the repeated multiplier is $-1$, but $M\ne -I$. Writing $M=-I+N$ gives $N\ne 0$, $N^2=0$ and $M^n=(-1{)}^n(I-nN)$, so generic linearized perturbations grow linearly. For $r>4$, a real multiplier has modulus greater than one. Thus the equilibrium is linearly unstable for $\alpha B\ge 4$.

For the unit-modulus range, the nonlinear behavior is not determined by this calculation: unit-modulus multipliers do not determine nonlinear stability.

Yes, susceptible replenishment allows repeated outbreaks. While $I_t>0$, the infection ratio isInfection grows above the susceptible threshold $S_t=1/\alpha$ and declines below it. A wave depletes susceptibles; subsequent recruitment can restore the threshold and permit renewed growth. In the neutral linear regime, perturbations around the positive fixed point oscillate, so repeated small waves occur for sufficiently small perturbations over any fixed finite observation period.

For example, in scaled variables $x=\alpha S$, $y=\alpha I$, the choice $r=\alpha B=0.2$, $x_0=1.03$, $y_0=0.19$ produces the repeated positive waves illustrated below. This finite trajectory illustrates recurrence of outbreaks without claiming a nonlinearly attracting periodic orbit from the linearization alone. If $I_0=0$ exactly, the deterministic model has no infectious seed and $I_t$ stays zero.