Age-Dependent Survival Rates in SIR-SI Dengue Transmission Model and Its Application Considering Human Vaccination and Wolbachia Infection in Mosquitoes

In this paper, an SIR-SI mathematical model in the form of a system of integral equations describing the transmission of dengue disease between human and mosquitoes is proposed and analyzed. Age-dependent functions are used to describe the survival of individuals in human and mosquito populations. The basic reproduction number is derived and its relationship to the equilibria is also explored. The results show that the existence of the positive endemic equilibrium is determined by a threshold number. This threshold number is also the same one that determines the global stability of the equilibrium. The threshold acts like the known basic reproduction number in the counterpart differential equations model and also follows the same rule for the critical level of intervention. Furthermore, as an application, the effect of wolbachia infection is explored, such as how this infection changes the resulting threshold and what the consequence of its presence is in the dynamics of the disease. In this case, the decrease of the mosquitoes’ life expectancy and biting rate are used to reflect the effect of wolbachia bacterial infection on the mosquitoes. In other words, a mosquito which is infected by wolbachia has a lower life expectancy than a normal mosquito. The results, both from mathematical analysis and numerical examples, show that the presence of wolbachia has the potential as a biological control agent to eliminate the dengue in the human population. A comparison of the wolbachia introduction into the mosquito population with the existing strategy, such as vaccination, is also presented.