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Homoclinic bifurcation in an SIQR model for childhood diseases. (English) Zbl 0969.34042
The authors study an SIQR model for childhood disease, where \(S, I, Q, R\) denote the number of susceptible, infected, isolated (quarantined) and recovered individuals, respectively. A center manifold reduction at a bifurcation point has the normal form for this model: \[ \dot x=y, \quad \dot y=axy+bx^2y+0(4), \] which indicate that there is a bifurcation of codimension greater than two for some critical parameter values. In particular, the authors find an unfolding which gives rise to a codimension two bifurcation and prove analytically the existence of limit cycles and homoclinic orbits. They also confirm that there exist Hopf and homoclinic bifurcations of the model by several numerical computations.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
Software:
AUTO
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References:
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