A novel epidemic network model incorporating infections induced by exposed individuals and environments

This paper is concerned with the modeling and control problems of an epidemic network system, in which the disease spreads through multiple routes. A novel heterogeneous annealed network model is developed to describe the disease transmission from the complex networks point of view. Different from the prior multi-path epidemic transmission annealed network models, the occurrence of the considered disease spreading is also induced via the exposed individuals which are infectious within the incubation period. By using the next-generation matrix method, the basic reproduction number is provided. The existence and stability of equilibria are analyzed accordingly. In particular, the sufficient conditions for the monotonic variation of basic reproduction number with incubation period and the exposure transmission intensity are theoretically obtained, which is related to the heterogeneity of human contact. Furthermore, detailed explanations about this condition are provided for regular network, random network and scale-free network. The dynamic of vaccination network is considered, and the effective vaccination rate threshold to ensure the disease disappears is proposed. Moreover, an optimal control strategy is designed to trade off the minimum infection and control cost by Pontryagin’s Maximum Principle. Finally, some numerical analysis is performed to verify the theoretical results. Compared with the perfect immunization measures, optimized control minimizes the number of infections and achieves better control effect.