Spreading of persistent infections in heterogeneous populations

Up to now, the effects of having heterogeneous networks of contacts have been studied mostly for diseases which are not persistent in time, i.e., for diseases where the infectious period can be considered very small compared to the lifetime of an individual. Moreover, all these previous results have been obtained for closed populations, where the number of individuals does not change during the whole duration of the epidemics. Here, we go one step further and analyze, both analytically and numerically, a radically different kind of diseases: those that are persistent and can last for an individual’s lifetime. To be more specific, we particularize to the case of tuberculosis’ (TB) infection dynamics, where the infection remains latent for a period of time before showing up and spreading to other individuals. We introduce an epidemiological model for TB-like persistent infections taking into account the heterogeneity inherent to the population structure. This sort of dynamics introduces new analytical and numerical challenges that we are able to sort out. Our results show that also for persistent diseases the epidemic threshold depends on the ratio of the first two moments of the degree distribution so that it goes to zero in a class of scale-free networks when the system approaches the thermodynamic limit.

Figure 1

(Color online) Set of allowed transitions in the epidemic model within each connectivity class.

Figure 2

Stationary proportion of sick individuals as a function of $\lambda$ $\left(∊\left[0.25,0.5\right]\right)$ for $p=0.07$, $r=0.002$, ${\mu }_{tb}=0.2$, $\mu =0.009$. and $b=0.01$. The arrow marks the position of the epidemic threshold $\left({\lambda }_c=0.305\right)$ as given by analytical calculations using the previous values for the parameters and the degree distribution. Error bars are smaller than symbol size. The initial size of the network, characterized by a degree distribution $P\left(k,t_o\right)\sim k^{-3}$, is $N_0={10}^6$. Each point is an average over at least 1000 values of ${⟨t⟩}^{\ast }$.