Mathematical modeling of mosquito borne diseases with vertical transmissions as applied to Dengue

Aedes aegypti mosquitoes transmit important mosquito borne diseases that include dengue, yellow fever, zika, chikungunya, rift valley, and west nile among others. The dynamics of these diseases are influenced by various factors such as population dynamics of humans and mosquitoes, mosquito behaviour, and transmission modes. This study focuses on multiple transmissions, where both vertical and horizontal modes are considered with application to dengue virus. We therefore present a model that incorporates vertical transmission within the mosquito population. Threshold quantities for the model are computed, with the mosquito extinction equilibrium being globally asymptotically stable when the basic offspring number (N0) is less than one, also, the disease free equilibrium is shown to be locally asymptotically stable when the basic reproduction number (R0) is less than one. The model is shown to undergo backward bifurcation, and conditions under which the disease free equilibrium would be globally asymptotically stable is presented. Type reproduction numbers are also computed. Some results of numerical simulations, and sensitivity (both local and global) analysis of the model parameters are shown and computed respectively.