Threshold Dynamics Of A Reaction-diffusion Malaria Model With Distributed Latencies

It is well-known that the transmission of malaria is caused by the bites of mosquitoes.Since the life habit of mosquitoes is influenced by seasonal factors such as temperature,humidity and rainfall,the transmission of malaria presents clear seasonable changes.In this paper,in order to take into account the incubation periods in humans and mosquitoes,we study the threshold dynamics of two periodic reaction-diffusion malaria models with distributed delay in terms of the basic reproduction number.We first propose a time-periodic vector-bias model which incorporates extrinsic in-cubation period?EIP?in mosquitoes.Taking the diffusion,latency and seasonality into consideration,we derive a periodic reaction-diffusion malaria model with distributed delay and study the well-posedness of this model.Secondly,we introduce the basic reproduction number₀via the next generation operator method and characterize₀by the spectral radius of the Poincar?e mapping of a related linear system.By using the comparison principle and the uniform persistence theory,we further obtain the threshold dynamics of the malaria model.It is shown that the disease-free equilibrium is globally attractive if₀<1,while the disease will be persistent if₀>1.To investigate how intrinsic incubation period?IIP?within humans affect the transmission of this disease,we finally incorporate the IIP into the vector-bias model.Furthermore,the basic reproduction num-ber₀is introduced by virtue of the next generation operator method and the Poincar?e mapping of a linear system.Moreover,the threshold dynamics is established in terms of₀.It is proved that if₀<1,then the disease-free periodic solution of the model is globally asymptotically stable;and if₀>1,then the disease is persistent.