A Hybrid Lagrangian-Eulerian Model For Vector-Borne Diseases

We propose a multi-patch and multi-group vector–borne disease model to study the effects of human commuting and/or mosquito migration on disease spread where the move-ments of humans and mosquitoes follow Lagrangian and Eulerian approaches,respectively.We first define the basic reproduction number of the model,R₀,which completely deter-mines the global dynamics of the model system.Namely,ifR₀?1,then the disease–free equilibrium is globally asymptotically stable,and ifR₀>1,then there exists a unique endemic equilibrium which is globally asymptotically stable.Then,we show that the ba-sic reproduction number has upper and lower bounds which are independent of the host residence times matrix and the vector migration matrix.In particular,nonhomogeneous mixing of hosts and vectors in a homogeneous environment increases disease persistence and the basic reproduction number of the model attains its minimum when the distribu-tions of humans and mosquitoes are proportional.Moreover,R₀can also be estimated by the basic reproduction numbers of isolated patches in the case where the environment is homogeneous.In two-group and two-patch case,we numerically analyze the dependence of the basic reproduction number and the total number of infected people with respect to the residence times matrix and investigate the optimal mosquito control strategy with limited vector control resources.