Control of epidemics via social partnership adjustment

Epidemic control is of great importance for human society. Adjusting interacting partners is an effective individualized control strategy. Intuitively, it is done either by shortening the interaction time between susceptible and infected individuals or by increasing the opportunities for contact between susceptible individuals. Here, we provide a comparative study on these two control strategies by establishing an epidemic model with nonuniform stochastic interactions. It seems that the two strategies should be similar, since shortening the interaction time between susceptible and infected individuals somehow increases the chances for contact between susceptible individuals. However, analytical results indicate that the effectiveness of the former strategy sensitively depends on the infectious intensity and the combinations of different interaction rates, whereas the latter one is quite robust and efficient. Simulations are shown to verify our analytical predictions. Our work may shed light on the strategic choice of disease control.

Figure 1

SI control of epidemics, i.e., increasing the breaking probability between susceptible and infected individuals. For the uniform interaction case ($w_{SS}=w_{SI}=w_{II}$), the model degenerates to the conventional SIS model. There is only one internal equilibrium and it is stable provided ${\lambda }_e>1$. Here we solely adjust $w_{SI}$ such that the duration time of $SI$ links is shorter than the other two types of links, i.e., $w_{SI}>w_{II}=w_{SS}$. The three panels show the phase diagrams in the $(w_{SI},{\lambda }_e)$-plane. The quality of the SI control is sensitively dependent on the reference uniform breaking probabilities. (a) When they are small ($w_{SS}=w_{II}=0.05$), decreasing the interaction between susceptible and infected individuals makes the phase diagram change from endemic state (red) to bistable state (yellow) and then to final extinct state (blue). (b) When $w_{SS}=w_{II}=0.5$, there is no bistable state (yellow) any more. It becomes harder to eradicate disease when the population is even more mobile. (c) The right panel shows that the SI control is incapable of eradicating the disease provided the population is intrinsically highly mobile ($w_{SS}=w_{II}=0.95$).

Figure 2

Equilibrium position of infected fraction as a function of $w_{SI}$. Here $w_{SS}=w_{II}=0.5$, and ${\lambda }_e=7$. The disease cannot be eradicated by the SI control, and the level of infection in the equilibrium state declines very slowly (from 0.857 to 0.854) by increasing $w_{SI}$ (from 0.5 to 1).

Figure 3

SS control, i.e., decreasing the breaking probability between susceptibles. We start from the uniform case $w_{SS}=w_{SI}=w_{II}$. Here the disease is solely controlled by increasing the duration time of the social ties between susceptibles, i.e., $w_{SS}<w_{II}=w_{SI}$. These phase diagrams are similar for all the reference uniform interaction rates: i) for small ${\lambda }_e$, the SS control makes the disease change from endemic state (red) directly to extinction state (blue); ii) for large ${\lambda }_e$, the SS control can still eradicate the disease, but the phase diagram has to pass from endemic (red) to bistable state (yellow) and finally to extinction (blue).

Figure 4

Equilibrium position of infected fraction as a function of $w_{SS}$. Here $w_{SI}=w_{II}=0.5$, and ${\lambda }_e=7$. Increasing the interaction time between susceptibles (i.e. decreasing $w_{SS}$) effectively eradicates the disease. In particular, for $0.033<w_{SS}<0.08$ (bistability), the disease dies out provided the initial infection is few in number. Even when the initial number of infection is large, the final level of infection is still lower than the case with $0.08<w_{SS}<0.5$. For $w_{SS}<0.033$, the disease is eradicated no matter what the initial state is.

Figure 5

Extinction. The discrete points are obtained from simulations, and the red lines represent the analytical results based on time scale separation. All of the four panels show that the disease dies out no matter how many infected individuals are present in the beginning. The parameters in all the four panels are from the blue regions in Figs. 1 and 3. Thus they are consistent with the analytical predictions. (Common parameters: ${\lambda }_e=1.5, N=100, \alpha =0.01$.)

Figure 6

Endemic. The simulation results (discrete points) show that no matter what the initial fraction of infective individuals is (except for the zero case which is an absorbing state of the agent-based model), the population would end up with a constant positive fraction of infected individuals. However, the analytical predictions would somewhat overestimate the simulation results. This is mainly because the running time for the simulation is beyond the quasistationary time scale of our theoretical analysis. Parameters are from the red regions in Figs. 1 and 3. (Common parameters: $N=100, \alpha =0.01$.)

Figure 7

Bistability. Discrete points represent simulation results. Dashed lines are via the analytical approximations: Red horizontal lines represent the stable fixed points, whereas green vertical lines represent the unstable fixed point (critical value). By comparison, the simulation results show qualitative agreement with the analytical predictions. In particular, the critical initial fraction of infected individuals, ensuring a dramatic outbreak of epidemics, is consistent with the unstable fixed point predicted by the analytical result. However, disagreements are present, where analytical results underestimate the final fraction of infective individuals when the infective individuals are rare initially. It is noteworthy that, despite this quantitative inconsistency, the salient feature of the bistable dynamics is still captured by the analytical predictions. Parameters are from the yellow regions in Figs. 1 and 3. (Common parameters: ${\lambda }_e=7, N=100, \alpha =0.01$.)

Figure 8

The effect of the population size on the accuracy of the analytical approximation. Population size with $N=100$ is enough to capture the salient feature of the bistability, compared with $N=500$. (Common parameters: ${\lambda }_e=7, w_{SS}=0.04, w_{SI}=w_{II}=0.5$ and $\alpha =0.01$.)