Periodic epidemic outbursts explained by local saturation of clusters

Adding the notion of spatial locality to the susceptible-infected-recovered (or SIR) model, allows to capture local saturation of an epidemic. The resulting minimum model of an epidemic, consisting of five ordinary differential equations with constant model coefficients, reproduces slowly decaying periodic outbursts, as observed in the COVID-19 or Spanish flu epidemic. It is shown that if immunity decays, even slowly, the model yields a fully periodic dynamics.

Figure 1

Representation of the SBIGR model. The susceptible part of the population is subdivided into individuals $B$ inside the blob, and $S$, outside the blob. The recovered are also subdivided into individuals $G$ inside and individuals $R$ outside the blob. The blob population is constituted by $N^{*}=B+I+G$. The dashed arrow from the $R$ to the $S$ compartment reflects the temporal decrease of immunity.

Figure 2

Graphical representation of the evolution of an (a) accelerating or (b) decelerating epidemic. The “blob” around the infected $I$ represents all people that are in close contact with contagious individuals. The blob expands when the local concentration of $I$ is large. The blob will shrink if little infected are present. $S$ and $B$ are susceptibles, $G,R$ are recovered. Change in surface, $\delta A$ during a time-interval $\delta t$ is indicated as a shaded area.

Figure 3

(a) Numerical integration of the SBIGR model [Eqs. (12)]. The figure shows a stackplot of the quantities $B,I,G$. Slowly decaying oscillations are observed for this set of parameters. In the inset it is illustrated that the time-interval $T$ between two peaks of the number of infected individuals (indicated by dots in the main plot) is set by the parameter $\kappa /\zeta$.

Figure 4

(a) Comparison of the results of the SBIGR model [Eqs. (12)] with data of newly reported cases for South Africa. Results for different values of $\kappa$ are shown for a fixed value of $\kappa /\zeta$.

Figure 5

Phase-space plot of the SBIGR model [Eqs. (12)] with decrease of immunity $\chi >0$. For the current parameters, for values of $\chi >{\chi }_{\text{crit}}=7.15\times {10}^{-4}$, corresponding to a typical immunity decrease time $t_{1/2}\approx 1000days$ leads to a limit cycle, representing a nonvanishing periodic epidemic. For $\chi <7.15\times {10}^{-4}$, the system reaches a stable point.