Abstract-In this paper, we propose a model of transmission of arboviruses, which takes into account a future vaccination strategy in human population. A qualitative analysis based on stability 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 associated reproduction number, R 0 , is less than unity. We show that the backward bifurcation phenomenon is caused by the arbovirus induced mortality. Using the direct Lyapunov method, we prove the global stability of the trivial equilibrium. Through a global sensitivity analysis, we determine the relative importance of model parameters for disease transmission. Simulation results using a nonstandard qualitatively stable numerical scheme are provided to illustrate the impact of vaccination strategy in human communities.

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, R 0 , is less than unity. Using Lyapunov function theory, we prove that the trivial equilibrium is globally asymptotically stable; When the diseaseinduced 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 full 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.

In this paper, we derive and analyse a model for the control of arboviral diseases which takes into account an imperfect vaccine combined with some other control measures already studied in the literature. We begin by analysing the basic model without control. We prove the existence of two disease-free equilibrium points and the possible existence of up to two endemic equilibrium points (where the disease persists in the population). We show the existence of a transcritical bifurcation and a possible saddle-node bifurcation and explicitly derive threshold conditions for both, including defining the basic reproduction number, [Formula: see text], which provides whether the disease can persist in the population or not. The epidemiological consequence of saddle-node bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for disease elimination from the population. It is further shown that in the absence of disease-induced death, the model does not exhibit this phenomenon. The model is extended by reformulating the model as an optimal control problem, with the use of five time dependent controls, to assess the impact of vaccination combined with treatment, individual protection and two vector control strategies (killing adult vectors and reduction of eggs and larvae). By using optimal control theory, we establish conditions under which the spread of disease can be stopped, and we examine the impact of combined control tools on the transmission dynamic of disease. The Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations and efficiency analysis show that, vaccination combined with other control mechanisms, would reduce the spread of the disease appreciably.

We prove that any group in the class of one-relator groups given by the presentation 〈a,b;[am,bn]=1〉, where m and n are integers greater than 1, is cyclic subgroup separable (or πc). We establish some suitable properties of these groups which enable us to prove that every finitely generated abelian subgroup of any of such groups is finitely separable