Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks

Abstract

We investigate the effects of delaying the time to recovery (delayed recovery) and of nonuniform transmission on the propagation of diseases on structured populations. Through a mean-field approximation and large-scale numerical simulations, we find that postponing the transition from the infectious to the recovered states can largely reduce the epidemic threshold, therefore promoting the outbreak of epidemics. On the other hand, if we consider nonuniform transmission among individuals, the epidemic threshold increases, thus inhibiting the spreading process. When both mechanisms are at work, the latter might prevail, hence resulting in an increase of the epidemic threshold with respect to the standard case, in which both ingredients are absent. Our findings are of interest for a better understanding of how diseases propagate on structured populations and to a further design of efficient immunization strategies.

Keywords: Complex networks; Disease spreading; Heterogeneous mean-field approach; SIS model.

Fig. 1

The figure shows the cycle of infection of a susceptible individual. We assume that after the initial infection, the newly infected node will remain infectious during a time window of $T+1$ time steps, after which the node recovers and gets back to the susceptible state.

Fig. 2

The effect of degree-dependent spreading rates on the propagation of diseases in complex networks. In this case, the delayed recovery mechanism is not present. (a) BA scale-free model. (b) Random WS model ($p=1$). All data points are obtained after averaging 100 independent runs and the dashed lines are a guide to the eyes. The arrows denote the epidemic thresholds as given by the analytical results. The parameter $A$ in these figures plays the role of a scaling factor for the critical point. When $A=1$, the contact process is recovered for the case α=0 and so the critical point is ${\lambda }_0^c=1$.

Fig. 3

Effect of the delayed recovery in the absence of the degree-dependent spreading rates scenario (i.e. α=1.0). (a) BA scale-free model. (b) Random WS model ($p=1$). All data points are obtained after averaging 100 independent runs and the dashed lines are a guide to the eyes. The arrows denote the epidemic thresholds as given by the analytical results.

Fig. 4

Same results as in Fig. 3 but for α=0 and $A=3$. (a) BA scale-free model. (b) Random WS model ($p=1$). All data points are obtained after averaging 100 independent runs and the dashed lines are a guide to the eyes. The arrows denote the epidemic thresholds as given by the analytical results.

Fig. 5

Numerical results when both mechanisms studied are concurrently active. (a) BA scale-free model. (b) Random WS model ($p=1$). The value of α has been set to 0.5. Arrows identify the epidemic thresholds as given by the mean-field approach.

Fig. 6

Comparison of critical threshold between the numerical simulations and analytical results in Eq. (20) in which α is fixed to be 1.0 and the network parameters are set to be $m=m0=4$ in BA model.