Rich Bifurcation Structure in a Two-Patch Vaccination Model

Abstract

We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptibleinfected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.

Dynamical analysis and control strategies of an SIVS epidemic model with imperfect vaccination on scale-free networks

“…Remark 11 Theorem 6 focuses on the necessary conditions of the optimal control problem. In the objective Table 1 Values of parameters and initial conditions of system (1) γ = 0.0125 φ = 0.03 ϕ = 0.04 σ = 0.001 T = 2000 β = 0.015 β = 0.0007 (14), it is difficult to evaluate the weights D k and W k for k = 1, . .…”

“…In 2000, Kribs-Zaleta and Velasco-Hernández [11] put forward an SIVS model with multiple endemic states which exhibited a backward bifurcation for some parameter values and presented a complete analysis of its behavior. Thereafter, many researchers considered various cases based on this model, such as nonlinear incidence rate [12], age of vaccination [13], spatial dispersal of individuals [14], and state-dependent pulse vaccination [15]. However, the basic assumption for the above works is that individuals in each compartment are homogeneously mixed.…”

A study of rigorous ODE integrators for multi-scale set-oriented computations

“…Some additional experiments show that implementing both improvements may result in considerably better performance of the method. Indeed, it was possible to deal with some ODEs in an effective way; for example, when the hyperbolicity was strong (like in the Vanderpol equations) and the dynamics was simple (like in [13], where all the chain recurrent dynamics consisted of a small number of equilibria). However, dealing with more subtle dynamical systems seems to be still beyond the computing possibilities, and thus prompts further improvements and development of more effective methods.…”

“…The approach described above provides an automatic method for the analysis of global dynamics in multi-parameter discrete-time dissipative systems induced by continuous maps; see [1,3,4,13,14,27] for the introduction of the method and applications to specific models in population dynamics, theoretical physics and epidemiology. We emphasize the wide applicability of this method, which is due to the mild assumptions on the system, as well as admitting the situations where the generator of the dynamical system is not known precisely; indeed, the the only essential input to the set-oriented construction and topological computations is the multivalued map on the grid.…”

Persistent homology as a new method of the assessment of heart rate variability

“…We believe that novel methods of HRV assessment might provide additional insight into poststroke heart rate changes not only related to the change of cardiovascular system, but also from different origins, including respiratory or humoral ones. Indeed, application of topological methods to the analysis of biological, medical, or physical models may provide new insights and valuable results that might compete with the results of analysis conducted by classical methods; see [29][30][31] for some examples. As a consequence, in order to claim our research successful, we aim at providing experimental evidence that the results obtained using the topological parameters are at least as good as the results obtained using the standard hrv parameters.…”