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| Classical disease transmission models typically have a unique locally asymptotically stable equilibrium whenever a certain threshold, known as the basic reproduction number ($\mathcal{R}_0$) is less than unity. However the situation when the model undergoes a backward bifurcation at $\mathcal{R}_0=1$ is different, since in this case for values of $\mathcal{R}_0$ less than one the stable disease free fixed point coexists with one stable positive and one unstable positive equilibrium.
We consider an SIVS (susceptible -- infected -- vaccinated -- susceptible) model to describe the spread of an infectious disease in a population of individuals in two regions. Assuming the disease dynamics exhibits the phenomenon of backward bifurcation in both sub-populations, we incorporate the possibility of travel between the regions and give condition for the existence of backward bifurcation in the full system. The mathematical analysis reveals an unusually rich dynamical behavior: from triple transcritical and double saddle-node bifurcation points eight possible endemic equilibria bifurcate besides the disease free steady state. We investigate the global dynamics and the stability of fixed points with analytical tools and rigorous numerical computations. |
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