Backward bifurcation and control in transmission dynamics of arboviral diseases

• A compartmental model for the control of arboviral diseases is presented and analyzed.
• Main control implemented: imperfect vaccine combined with individual protection and some vector control strategies (adulticide and larvicide).
• Principal causes of the occurrence of backward bifurcation phenomenon: Incidence rates and disease–induced death.
• The spread of the disease can be reduced if the level of each control is maintained at a high level, and this, over a long period.

In this paper, we derive and analyze a compartmental model for the control of arboviral diseases which takes into account an imperfect vaccine combined with individual protection and some vector control strategies already studied in the literature. After the formulation of the model, a qualitative study based on stability analysis and bifurcation theory reveals that the phenomenon of backward bifurcation may occur. The stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the reproduction number, R0,R0, is less than unity. Using Lyapunov function theory, we prove that the trivial equilibrium is globally asymptotically stable. When the disease–induced death is not considered, or/and, when the standard incidence is replaced by the mass action incidence, the backward bifurcation does not occur. Under a certain condition, we establish the global asymptotic stability of the disease–free equilibrium of the principal model. Through sensitivity analysis, we determine the relative importance of model parameters for disease transmission. Numerical simulations show that the combination of several control mechanisms would significantly reduce the spread of the disease, if we maintain the level of each control high, and this, over a long period.