Global dynamics of SVIR epidemic model with distributed delay and imperfect vaccine

Abstract

In this research, we investigate an SVIR system with distributed delay. We first provide some preliminary results on the proprieties of the Volterra function and the well-posedness of the solution, also the existence of a global compact attractor denoted D. Then we determine the global behavior of the solution in detail, where it is characterized in two different cases in terms of the basic reproduction number R0. For R01, we prove the uniform persistence and D is restricted to the set {E0}∪ℭ∪D1, with ℭ is the set of points that connects orbits from E to D1, and D1attracts all points with initial infection force. For proving that D just consists of a positive equilibrium we used a Lyapunov approach, where we provided the relationship between the Lyapunov function for the distributed delayed system and the Lyapunov function for the system with the differential system. The results are confirmed using some graphical representations.

Keywords

• Lyapunov functional
• Uniform persistence
• Global attractor