On the dynamical behaviour of the epidemiological processes on bipartite networks

The main aim of this work is to improve the understanding of the effective complexity of the transmission of a vector-borne disease. Vector-borne diseases involve two families of actors: vectors and hosts. Vectors are intended as the pathogen transmitter from a host to another. Most reported vectors are invertebrates such as mosquitoes or ticks. On the other hand, hosts are those animals, usually vertebrate, which could be infected by vectors, thus becoming source of infection for vectors successively feeding on them. For the heterogeneous nature of the transmission dynamics between vectors and hosts the use of homogeneous mixing models is not appropriate, while a network approach is highly motivated. The dynamic network approach is a non-reductionist approach enabling analysis of systems as a whole, which makes it an ideal tool for epidemiological systems. The transmission network of a number of important vector-borne diseases could be naturally described by a bipartite network, structures where nodes are divided in two different groups and edges are allowed only between nodes belonging to different groups. In order for this model to faithfully describe those epidemiological systems, pathogen transmission may occur only between hosts and vectors or, in other words, the transmission between individuals of the same family does not happen or at least it is negligible. Examples that satisfy such constraints are pathogens such as those responsible for Tick Borne Encephalitis or Lyme disease, two of the most reported diseases transmitted by ticks [1]. The transmission of a pathogen over a bipartite network was first studied by Gomez-Gardenes and colleagues [2], where the dynamics of a sexually transmitted disease over a heterosexually contact network was taken into account. For a Susceptible-Infected-Susceptible (SIS) dynamical model, the expression of the epidemic threshold and its dependence with the system size was derived. These results were generalized by Bisanzio and coauthors [1] and applied to vector borne diseases. In this article a very important results was proven: if until then the aggregative behavior of vectors on hosts were usually described by negative-binomial distributions, they showed that on two different datasets, best fitting was obtained by power law distributions, with exponents around 2.5. This implies an important consequence: the epidemic threshold vanishes in the limit of large network sizes. In other terms, the condition for an endemic behavior, observed in several countries around the world, seems to be well supported by field data. Moreover, in a recent manuscript [3] we showed that a pathogen agent spreading on a bipartite scale-free network can have some evolutionary benefits with respect to its diffusion on classical scale-free unipartite networks. In fact, on a bipartite network in which the two families of nodes have degrees distributed as power laws, the condition for the endemic situation is satisfied for transmission probabilities smaller than those needed on the same networks where the bipartite structure constraint does not hold. This means that pathogens of epidemiological frameworks such as sexually transmitted diseases in heterosexual populations or vector-borne diseases such as Tick Borne Encephalitis or Lyme borreliosis could take advantage of the peculiar transmission route naturally represented by bipartite networks. This situation could result in an evolutionary advantage for the pathogen as its transmission probability is less subject to selection pressure. In the perspective of epidemiological inference, we were interested in examining a more complex situation: the case of a disease spreading over a direct multipartite graph or in other terms where the individual are divided into $n$ families and connections are allowed only between nodes of different families. In particular, we considered a class of multipartite networks in which meta-structure connecting the different families of nodes is cyclic. Under this assumption and adopting a mean-field approach, we calculated the epidemic threshold for a SIS dynamics, demonstrating that the critical condition for the occurrence of the endemic situation is: \[\frac{\lambda_1}{\mu_1}\dots\frac{\lambda_n}{\mu_n}>\frac{\ave{k_1}}{\ave{k_1^2}}\dots\frac{\ave{k_n}}{\ave{k_n^2}}\] Hence to have a vanishing epidemic threshold is sufficient that at least one of families has an heterogeneous behavior. The epidemic threshold reckon on cyclic multipartite behavior is not only an interesting extension of results gained by previous studies on bipartite networks dynamics [1,2]. Indeed it can be used to describe a number of eco-epidemiological systems in which different species of vectors are competent for different species of hosts. In conclusion, we demonstrated that infections can have some advantage if spreading on bipartite scale-free networks instead of spreading on unipartite scale-free networks. Moreover, we obtained the epidemic threshold expression for a cyclic multipartite network which can be used for a number of important eco-epidemiological situations. The dynamical behaviour of these networks is relevant for all those epidemiological processes that model the spreading of pathogenic agents that transmit between different animal species, as it is the case of vector-borne diseases.