Impact of the infectious period on epidemics

The duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread. We consider an epidemic model that is network based and non-Markovian, containing classic Kermack-McKendrick, pairwise, message passing, and spatial models as special cases. For this model, we prove a monotonic relationship between the variability of the infectious period (with fixed mean) and the probability that the infection will reach any given subset of the population by any given time. For certain families of distributions, this result implies that epidemic severity is decreasing with respect to the variance of the infectious period. The striking importance of this relationship is demonstrated numerically. We then prove, with a fixed basic reproductive ratio ($R_0$), a monotonic relationship between the variability of the posterior transmission probability (which is a function of the infectious period) and the probability that the infection will reach any given subset of the population by any given time. Thus again, even when $R_0$ is fixed, variability of the infectious period tends to dampen the epidemic. Numerical results illustrate this but indicate the relationship is weaker. We then show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics. For Poissonian contact processes, and arbitrarily distributed infectious periods, we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds, respectively, for the probability that any given individual has been infected by any given time.

Figure 1

We consider a special case of the stochastic model where the graph is a square lattice of 900 individuals and $\mathcal{X}$ is mutually independent; ${\omega }_{ji}\sim \text{Exp}\left(1\right)$ for all $i\in \mathcal{V},j\in {\mathcal{N}}_i$; ${\nu }_i=0$ for all $i\in \mathcal{V}$; ${\mu }_i\sim \mathrm{\Gamma }(k,3/4k)$ for all $i\in \mathcal{V}$ $\left[\mathrm{\Gamma }\right(k,3/4k)$ is the gamma distribution with shape parameter $k$ and scale parameter $3/4k]$; every individual is independently initially infectious with probability 0.01 and initially susceptible otherwise. In (a) we have approximated the expected number susceptible against time for $k=1,2,4,\text{and}4000$, corresponding to variances of the infectious period of approximately 0.56, 0.28, 0.14, and 0.00014, while in (b) we have approximated the expected number infectious against time for $k=1,2,4,\text{and}4000$. Each approximation was computed as the average of 1000 stochastic simulations. Here, the mean infectious period is the same for all individuals and kept constant at 3/4. In (c) we have plotted the probability density function for the infectious period for each value of $k$.

Figure 2

We consider the same scenario as for Fig. 1 except with ${\mu }_i\sim \mathrm{\Gamma }(k,e^{3/4k}-1)$ for all $i\in \mathcal{V}$, and plot the expected number susceptible (a) and the expected number infectious (b) against time. Here, the transmission probability is the same for all ordered pairs of neighbor and kept constant at $1-e^{-3/4}\approx 0.53$, giving $R_0\approx 3\times 0.53=1.59$. For $k=1,2,4,\text{and}4000$, the mean of the infectious period is approximately 1.1, 0.91, 0.82, and 0.75, with variance 1.2, 0.41, 0.17, and 0.00014, respectively. In (c) we have plotted the probability density function for the infectious period for each value of $k$. It is straightforward that the transmissibility variable (which is a function of the infectious period) here decreases in convex order as $k$ increases.