In this paper, we consider a discrete-time Susceptible-Infected-Vaccinated-Susceptible (SIVS) epidemic model with vaccinations and a variable population size. The model includes the recruitment of susceptible and vaccinated individuals, vaccination of susceptible individuals, loss of vaccine-induced immunity, disease-induced mortality, and two possible outcomes after infection. We prove that the solutions remain nonnegative and that the total population is ultimately bounded. The disease-free and endemic equilibria are obtained, together with the basic reproduction number $ R_0 $. By applying the Jury criterion, we show that the disease-free equilibrium is locally asymptotically stable when $ R_0 < 1 $. The endemic equilibrium exists uniquely when $ R_0 > 1 $, and is locally asymptotically stable whenever the corresponding Jury conditions are satisfied. A Lyapunov function is used to obtain a sufficient condition for the global asymptotic stability of the disease-free equilibrium. Taking the transmission rate as the bifurcation parameter, we verify the transversality and quadratic nondegeneracy conditions and establish a nondegenerate forward transcritical bifurcation at $ R_0 = 1 $. A numerical example within the biologically admissible parameter region confirms disease extinction for $ R_0 < 1 $, convergence to a positive endemic equilibrium for $ R_0 > 1 $, and the exchange of stability at the threshold. Additional simulations illustrate oscillatory and irregular dynamics after the admissibility restrictions are relaxed; these behaviors are interpreted as mathematical extensions of the discrete map.
Citation: Jia Xu, Ke Wang, Xiaoling Han. Dynamic analysis of a discrete SIVS epidemic model with vaccination[J]. AIMS Mathematics, 2026, 11(8): 25897-25923. doi: 10.3934/math.20261038
In this paper, we consider a discrete-time Susceptible-Infected-Vaccinated-Susceptible (SIVS) epidemic model with vaccinations and a variable population size. The model includes the recruitment of susceptible and vaccinated individuals, vaccination of susceptible individuals, loss of vaccine-induced immunity, disease-induced mortality, and two possible outcomes after infection. We prove that the solutions remain nonnegative and that the total population is ultimately bounded. The disease-free and endemic equilibria are obtained, together with the basic reproduction number $ R_0 $. By applying the Jury criterion, we show that the disease-free equilibrium is locally asymptotically stable when $ R_0 < 1 $. The endemic equilibrium exists uniquely when $ R_0 > 1 $, and is locally asymptotically stable whenever the corresponding Jury conditions are satisfied. A Lyapunov function is used to obtain a sufficient condition for the global asymptotic stability of the disease-free equilibrium. Taking the transmission rate as the bifurcation parameter, we verify the transversality and quadratic nondegeneracy conditions and establish a nondegenerate forward transcritical bifurcation at $ R_0 = 1 $. A numerical example within the biologically admissible parameter region confirms disease extinction for $ R_0 < 1 $, convergence to a positive endemic equilibrium for $ R_0 > 1 $, and the exchange of stability at the threshold. Additional simulations illustrate oscillatory and irregular dynamics after the admissibility restrictions are relaxed; these behaviors are interpreted as mathematical extensions of the discrete map.
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Jia Xu, Ke Wang, Xiaoling Han. Dynamic analysis of a discrete SIVS epidemic model with vaccination[J]. AIMS Mathematics, 2026, 11(8): 25897-25923. doi: 10.3934/math.20261038
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