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 (susceptible--infected--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.


  1. backward bifurcation
  2. patch model
  3. equilibrium
  4. numerical simulations
  5. rigorous numerics
  6. computational dynamics
  7. set-oriented computations
  8. ODE
  9. time-1 map
  10. Conley index
  11. Morse decomposition

MSC codes

  1. Primary
  2. 92D30; Secondary
  3. 92-08

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 980 - 1017
ISSN (online): 1536-0040


Submitted: 3 November 2014
Accepted: 24 March 2015
Published online: 11 June 2015


  1. backward bifurcation
  2. patch model
  3. equilibrium
  4. numerical simulations
  5. rigorous numerics
  6. computational dynamics
  7. set-oriented computations
  8. ODE
  9. time-1 map
  10. Conley index
  11. Morse decomposition

MSC codes

  1. Primary
  2. 92D30; Secondary
  3. 92-08



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