Figure 3

Type of the equilibria for constant transmission and recovery rates. For the ancestral COVID-19 parameters \(r=1/14\) and \(g=0.2\), the possible stability outcomes depend only on the value of T. (a) Line indicating which values of T result in the three outcomes in terms of the intersections with the \(\Gamma\) and \(\Delta\) curves. (b–d) Shows the changes in proportions infected (red) and cooperating (blue) over five years time for different values of T. (b) Proportion infected and cooperating curves for \(T=0.2\). For any value of \(T < 0.61\) we expect the proportions infected and cooperating to exhibit damped oscillations while converging to equilibrium 4 \((T,1+r/g(T-1))\), and with the parameters used here that would be (0.2, 0.55) which is reflected in the figure. (c) Proportion infected and cooperating curves for \(T=0.62\). Because T satisfies \(0.61< T < 0.64\) we expect the proportions infected and cooperating to approach equilibrium 4 \((T,1+r/g(T-1))\), here (0.62, 0.06), which is reflected in the figure. We note that initially the infection grows rapidly towards the standard SIS model equilibrium (\(1-r/g=0.64\)) and only when it passes \(T=0.62\) does cooperation kick in and eventually drags the infection down to 0.62 where it stabilizes. (d) Proportion infected and cooperating curves for \(T=0.7\). For any value of \(T > 0.64\) equilibrium 3 \((1-r/g,0)\) is stable, i.e. the proportion infected I will approach the standard SI model equilibrium \(1-r/g=0.64\) and cooperation C will approach 0. We see this reflected in the figure. This is a consequence of the fact that the threshold T is larger than the standard SIS model equilibrium \(1-r/g\), so I will never exceed T, and therefore cooperation will decrease towards 0.