Epidemic spreading in localized environments with recurrent mobility patterns

The spreading of epidemics is very much determined by the structure of the contact network, which may be impacted by the mobility dynamics of the individuals themselves. In confined scenarios where a small, closed population spends most of its time in localized environments and has easily identifiable mobility patterns—such as workplaces, university campuses, or schools—it is of critical importance to identify the factors controlling the rate of disease spread. Here, we present a discrete-time, metapopulation-based model to describe the transmission of susceptible-infected-susceptible-like diseases that take place in confined scenarios where the mobilities of the individuals are not random but, rather, follow clear recurrent travel patterns. This model allows analytical determination of the onset of epidemics, as well as the ability to discern which contact structures are most suited to prevent the infection to spread. It thereby determines whether common prevention mechanisms, as isolation, are worth implementing in such a scenario and their expected impact.

Figure 1

Total fraction of infected individuals $\rho$ in the steady state as a function of the infectivity probability $\beta$, for four values of the mobility probability. Solid lines are the results of our model, while symbols are the Monte Carlo simulations. The dashed vertical lines indicate the epidemic threshold as calculated by Eq. (13). For this plot, the number of subpopulations of type residential is $D=25$ and there are $C=5$ common sites, with equal-sized residential sites of 100 individuals each. The isolation mechanism is inactive ($\gamma =1$, $p=p^{\prime }$), the recovery probability $\mu =0.1$, and the connectivity matrix is an unweighted, fully connected bipartite network.

Figure 2

Epidemic threshold ${\beta }_c$ as a function of the mobility probability $p$, for different configurations of the number of residential and common sites. We observe in all of them a non-monotonic behavior of ${\beta }_c$ which increases up to an optimum value of the mobility parameter ($p*$) that makes the epidemic threshold maximum (see dashed lines). Here the recovery probability $\mu =0.2$, all residential subpopulations are of the same size $n=25$, the isolation mechanism is disabled ($\gamma =1$) and the underlying topology is an unweighted fully connected bipartite network.

Figure 3

Epidemic threshold curve as a function of mobility probability $p$, for different degrees of heterogeneity of subpopulation sizes, controlled by their variance. (In print, lighter hues of gray correspond to lower values of the variance.) Here $D=10$, $C=5$ and the total number of agents is $N=100$. We vary how individuals are distributed among residential sites, ranging from the homogeneous case (all residential sites are of equal size) to the most heterogeneous setup (all individuals reside in the same node). We see that as the heterogeneity increases, the epidemic threshold gets smaller, and the effect of the optimum mobility $p*$ is diluted.

Figure 4

Epidemic threshold ${\beta }_c$ as a function of the mobility parameter $p$, for different values of the isolation factor $\gamma$. (In print, lighter hues of gray correspond to lower values of the isolation factor.) For this plot we have used a fully connected unweighted bipartite network consisting of ten residential sites and ten common locations. All residential patches are of size 100 and the recovery probability is $\mu =0.1$. We observe that the curves cross at exactly the expected value $p*=0.5$ given that there are exactly the same number of residences and common sites (see main text for a broader explanation).