Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation

Dengue fever is a virus-caused disease in the world. Since the highinfection rate of dengue fever and high death rate of its severeform dengue hemorrhagic fever, the control of the spread of thedisease is an important issue in the public health. In aneffort to understand the dynamics of the spread of the disease,Esteva and Vargas [2] proposed a SIR v.s. SIepidemiological model without crowding effect and spatialheterogeneity. They found a threshold parameter $R_0,$ if $R_01,$ then the disease willalways exist. To investigate how the spatial heterogeneity and crowding effectinfluence the dynamics of the spread of the disease, we modify theautonomous system provided in [2] to obtain areaction-diffusion system. We first define the basic reproductionnumber in an abstract way and then employ the comparison theorem andthe theory of uniform persistence to study the global dynamics ofthe modified system. Basically, we show that the basic reproductionnumber is a threshold parameter that predicts whether the diseasewill die out or persist. Further, we demonstrate the basicreproduction number in an explicit way and construct suitableLyapunov functionals to determine the global stability for thespecial case where coefficients are all constant.