Modeling epidemic spreading in star-like networks

Our source of inspiration is the spreading of vector-borne pathogens with a particular focus on non-systemic transmission. Such transmission, known as cofeeding, occurs between vectors feeding on the same host simultaneously without the host becoming viremic. For instance, the Tick-Borne Encephalitis virus (TBEv) is mainly maintained by this transmission route in a natural cycle involving as vectors ticks of Ixodes species, and as hosts different animal species, in particular rodents. However, TBEv is also of interest for the human health since it causes the most important arboviral infection of the human central nervous system in Europe and Russia, which can result in long-term sequelae and, in some cases, to death. We model this transmission process using a Susceptible-Infectious- Susceptible model (SIS) on a dynamical contact network. Specifically, to describe and analyze cofeeding dynamics we consider a bipartite network composed by a collection of disconnected star-like structures. In such network, nodes are divided in two sets, A and B, that, in our biological source of inspiration represent rodents and ticks, respectively. Nodes of set B, B-nodes, are tied only with an A-node that represents the center of star-like structure. Moreover, in these bipartite structures, we specify the degree, i.e. the number of neighbors, of an A-node by p, a probability density function. A-nodes are not susceptible to the pathogen, while -nodes are divided in susceptible and infectious according to their status. Thus, the pathogen spreading may occur only between B-nodes and only if connected through a common A-node. The transmission between an infectious individual and a susceptible one occurs with probability b, while the recovery takes place with probability m. At each iteration t we take into account the fraction of infected nodes of type B - i.e. the prevalence among set B - which is function of the prevalence at time t-1, of the transmission probability b, of the recovery probability m and of the degree probability function p. To model the dynamical nature of this network model, at every time step B-nodes are reshuffled and are randomly connected to A-nodes keeping the star-like structure (in our source of inspiration the reshuffling of ticks over rodents). By studying the system of differential equations describing both the dynamical star-like contact network and the epidemiological dynamics over the B-nodes, we analytically depict a necessary condition for which the pathogen remains endemic among B-nodes. The necessary transmission probability for the pathogen to reach endemicity is proportionally inverse to the second moment of the degree probability function. In other words, the larger the heterogeneity of the degree distribution the smaller transmission probability is needed by the pathogen to be endemic. Furthermore, we confirm our results by stochastically simulating the epidemic spreading on a number of synthetic networks generated using several degree probability distributions. It is worth to stress that this is the first time that such result is found out on the peculiar star-like networks. For the non-systemic transmission of vector-borne diseases it means that the larger the heterogeneity in the number of vectors feeding on a host (i.e. the greater is the aggregative behavior of ticks on hosts) is, the higher the probability that the disease becomes endemic through the vector population. However, such result could be extended to other transmission processes. For instance, the spreading occurring among people using transport means could be modeled by the approach just described. In such scenario passengers become B-nodes and means of transportation become A-nodes. Once again, a larger the heterogeneity among the number of passengers results in a lower the probability transmission needed for the pathogen to be maintained in the population.