[1] [2] [3] INTRODUCTION MODEL We propose some simple models of _hybrid_ type for the spreading of an epidemic disease (Zika virus) in a coupled system of two-populations (humans and mosquitoes). The virus is spread among humans through sexual contact, and between humans and mosquitoes through mosquitos bites. The mosquitoes do not transmit the virus amongst themselves, but they transmit it to their offspring. The human population is viewed as heterogeneous, and is represented by a sexual network across which the infection spreads following a SEIR dynamics model. The mosquitos are modeled as a homogeneous populations, across which the disease spreads following a SEI dynamics model. The passing of the virus from humans to mosquitoes and vice-versa is captured by a two-way coupling between the two systems. Members of either population who have not previously got the virus has the status of S–susceptible. They change the status to E–exposed, in the period immediately after they have got the virus; at this stage they cannot transmit the virus further. At the next stage, they change the status to infected, when they are capable to transmit the virus. The humans change the status to R–recovered, at the next stage. There is no R–status for the mosquitoes, due to their short life-span. The diagram of the virus transmission is in Fig. []. In our model, the human population is modeled by a network whose nodes represent individuals, and edges represent (unprotected) sexual contact. Sexual network For the purpose of modeling the transmission of the virus infection, we devise a model of a multi-partite sexual network. There are many existing models for sexual networks , but here we restrict to a simple one, as the focus is the study the competition of two mechanisms for the infection epidemics. The nodes of the network are of four distinct categories, and the we attach edges between nodes according to rules specific to these different categories, as follows: - W-nodes, represented steady, monogamous relationships. Each such node is connected with one other node from the same category. One example of people in this category would be married couples. The M-nodes can become infected when bitten by an infected mosquito, and can only transmit the infection between partners or to mosquitoes. - X-nodes, unsteady, monogamous relationships. Each such node is connected with one other node from the same category, and, additionally, it connects to one additional node chosen at random either from the X-nodes, or from the Z-nodes, described below. - Y-nodes, representing sexually inactive members of the network. They can only become infected by mosquitoes but they cannot transmit the infection to other humans. - Z-nodes, representing non-monogamous relationships. Pairs of nodes within this category are connected at random, with some fixed probability. Additionally, some of the Z-nodes are connected with X nodes, as described above. The members of this population can get infected either through mosquito bites or through sexual contact. Dynamics of epidemics The equations of the spreading of the infection among the human population are: {dt}&=&-\beta_hA_{i,j}S^h_i I^h_j-\beta_t \zeta (S^h_i/N^h) I^m P^m,\\ {dt}&=&\beta_hA_{i,j}S^h_i I^h_j+\beta_t \zeta (S^h_i/N^h) I^m P^m-\sigma_h I^h_i,\\ {dt}&=&\sigma_h I^h_i-\gamma_h I^h_i,\\ {dt}&=&\gamma_h I^h_i. The equations of the spreading of the infection among the mosquito population are: {dt}&=&-\beta_t\zeta S^m I^h-\rho_m I^m,\\ {dt}&=&\beta_t\zeta S^m I^h-\sigma_m E^m,\\ {dt}&=&\sigma_m E^m +\rho_m I^m. The matrix A = (Ai, j)i, j = 1, …, Nh represents the adjacency matrix of a human sexual network. The variables at time t have the following meaning: - Sih= probability that node-i of the human network is susceptible, - Eih= probability that node-i is exposed, - Iih= probability that node-i is infected, - Rih= probability that node-i is recovered, - Nh= total number of nodes in the human network, - $I^h=(^{N^h}I^h_i)/N^h$ average probability of infection over the human population, which approximately equals the ratio of infected humans, - Sm= proportion of susceptible mosquitoes, - Em= proportion of exposed mosquitoes, - Im= proportion of infected mosquitoes, - Pm= total number of female mosquito population. The parameters of the model are as follows: - βh= sexual transmission rate of the virus between humans (i.e., for a given pair of individuals who are sexually connected), - βt= transmission rate of the virus between humans and mosquitoes (same value for both directions of transmission), - σh= rate of infectiousness (from non-infectious exposed state) for humans, - σm= rate of infectiousness (from non-infectious exposed state) for mosquitoes, - γh human recovery rate, - ρm= net growth rate of infected mosquito by transovarial transmission (from adult female to eggs) - ζ= mosquito bite rate (i.e., number of bites given by a female mosquito per unit of time). For each i = 1, …, Nh, we have Sih + Eih + Iih + Rih = 1. The ratio of susceptible (resp. exposed, infected, recovered) out of the total human population is the expected value of the probability distribution $S^h=(^{N^h} S^h_i)/N^h$ (resp. $E^h=(^{N^h} E^h_i)/N^h$, $I^h=(^{N^h} I^h_i)/N^h$, $R^h=(^{N^h} R^h_i)/N^h$). Note that Sh + Eh + Ih + Nh = 1. The population of mosquitoes varies in time following seasonal cycles, depending on geographic area and meteorological conditions. For realistic modeling, statistical projections on the size of the mosquito population can be obtained . There are also mathematical models . We consider a very simple model for the time-evolution of the size of the female mosquito population: P^m(t)=A-B\cos\left ({p}\right ). The rate at which mosquitoes bite humans is proportional to the number of mosquitoes but independent of the number of people – as mosquitoes only need a certain number of blood meals, as long as there are sufficiently many humans around . We denote by ζ the number of bites per mosquito per unit of time. Thus, the total number of bites that all infected mosquito can give per unit of time is ζImPm, which amounts to ζ(1/Nh)ImPm bites of infected mosquitoes per human (node). This gives ζ(Sih/Nh)ImPm the probability of a bite per unit of time per susceptible human (node). This is the explanation for the second term in . When the number of humans (nodes in the network), for a given number of mosquitoes, the probability of a susceptible human to be bitten decreases as Nh increases. The parameter ρm will be assumed very small, since the role of transmission of the virus from infected female mosquitoes to their offspring in sustaining transmission of the virus to humans has not yet been defined. . ACKNOWLEDGEMENT The authors are grateful to Yael Eiferman and Anita Levy for their contribution to literature review. [1] ‡ [2] ♭ [3] † Research of M.G. was partially supported by NSF grant DMS-0635607.