Vaccination strategies of an SIR pair approximation model with demographics on complex networks

Pair approximation model is an effective tool to study epidemic spread on complex networks. It can more accurately capture the effects of network structure on the spreading process. That then helps us grasp the spreading laws of epidemics on networks and further make effective prevention and control measures. Vaccination, an important measure for prevention and control of infectious disease, has made great achievements in public health. In this paper we study vaccination strategies with the help of pair approximation epidemic model with demographics. We firstly introduce constant vaccination into SIR pair approximation model. The reproduction number and endemic prevalence of disease are investigated, the critical vaccination rate which can help to control disease transmission is also given. Considering the restriction of financial resources, it is necessary to control disease transmission simultaneously to reduce vaccination cost. To this end, we further investigate optimal vaccination of SIR pair approximation model by use of optimal control theory. The existence of optimal solution is established and optimality system is derived. Finally, a series of stochastic simulations on different initial networks are performed to demonstrate our theoretical models and some numerical simulations are provided to observe and analyze different vaccination strategies.