Uniform persistence and backward bifurcation of vertically transmitted vector-borne diseases

Vertical transmissions of vector-borne diseases such as dengue virus, malaria, West Nile virus and Rift Valley fever are among challenging factors in disease control. In this paper, we used a system of differential equations to study the impacts of vertical transmission (in the vector) and horizontal (vector-to-host) transmissions cycle on the uniform persistence of a vertically transmitted disease. This is given analytically when the secondary reproduction number is greater than unity. Additional numerical results indicate that vertical transmission rates raise the epidemic level. However, we provide analytical results to reveal that vertical transmission of the disease in vectors alone without horizontal transmission, could not make the disease endemic. Furthermore, by transforming the system of differential equations to a nonlinear eigenvalue equation, the existence of a backward bifurcation is established. The directions of bifurcation is forward when the disease-induced death rate of hosts is reduced below a threshold. It is shown that the bifurcation is backward when the disease-induced death rate in the host is above a critical value. Additionally, increased disease transmission parameters also contribute to backward bifurcation. Numerical results are also provided to highlight that, depending on the initial conditions, the disease could be persistent even when the secondary reproduction number is less than unity.