# zbMATH — the first resource for mathematics

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
##### Keywords:
epidemic model; homoclinic bifurcations; limit cycles
AUTO
Full Text:
##### References:
 [1] Bogdanov, R.I., Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. anal. appl., 9, 144-145, (1975) · Zbl 0447.58009 [2] Chow, S.-N.; Li, C.; Wang, D., Normal forms and bifurcation of planar vector fields, (1994), Cambridge University Press Cambridge [3] Doedel, E.J., Auto: a program for the automatic bifurcation analysis of autonomous systems, Congr. numer., 30, 265-284, (1981) [4] Dumortier, F.; Roussarie, R.; Sotomayor, J.; Zoladek, H., Bifurcations of planar vector fields, Lecture notes in mathematics, 1480, (1991), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0755.58002 [5] Feng, Z.; Thieme, H., Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. biosci., 128, 93-130, (1995) · Zbl 0833.92017 [6] Guckenheimer, J.; Holmes, P.J., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied mathematical sciences, 42, (1983), Springer-Verlag New York [7] Perko, L., Differential equations and dynamical systems, Texts in applied mathematics, 7, (1996), Springer-Verlag New York [8] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, Texts in applied mathematics, 2, (1990), Springer-Verlag New York · Zbl 0701.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.