| Multi-group models have been proposed to describe the transmission dynamical ofinfectious disease in heterogeneous populations. It divides the host population into ndistinct subgroup. Every infectious individual in every subgroup can infect the everysusceptible individual by a property. Every subgroup is homogeneity that is theindividual in the same subgroup has the same contact and recover rate.In this paper, we investigate a multi-group epidemic model with distributed delayand a general incidence functions. First, we investigate the dynamical behavior of thedelay functional equations. First, an explicit estimation of0based on the basicreproduction of sub-group0kcan be showed, which indicates that ignoring theheterogeneity of multi-group type may lead to an underestimation of the basicreproduction ratio. Then by the methods of constructing the Lyapunov functional, thegraph-theoretic approach for large-scale systems and uniform persistence theory, theglobal dynamics of the system based on the basic reproduction number0can beobtained. More specifically, if R0≤1, then the disease-free equilibrium is globallyattracting, and it means that the disease will die out; if R0﹥1then the uniqueinfection equilibrium exists and globally attractively, and it means that the disease willpersistent exist in every sub-group. Due to every suspected has a low positive bounded,then if there exist a sub-group, such that the disease free, then the disease willextinguish in all subgroup.At last, we consider the model in two-group case with limited vaccine resourceswith the purpose of constructing the optimal distribution strategy between the twogroups. By the implication of the basic reproduction rate in biology, the standard wechoose to choose the strategy is to make the basic reproduction ratio get the minimalwith the variety of distribution ratio. There are some theoretical results. It suggeststhat the optimal distribution strategies may change dramatically due to the variance ofheterogeneity. |
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