A CLASS OF LYAPUNOV FUNCTIONS AND THE GLOBAL STABILITY OF SOME EPIDEMIC MODELS WITH NONLINEAR INCIDENCE

Jianquan Li; Yali Yang; Yanni Xiao; Shuo Liu

doi:10.11948/2016004

2016 Volume 6 Issue 1

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Jianquan Li, Yali Yang, Yanni Xiao, Shuo Liu. A CLASS OF LYAPUNOV FUNCTIONS AND THE GLOBAL STABILITY OF SOME EPIDEMIC MODELS WITH NONLINEAR INCIDENCE[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 38-46. doi: 10.11948/2016004

Citation:

Jianquan Li, Yali Yang, Yanni Xiao, Shuo Liu. A CLASS OF LYAPUNOV FUNCTIONS AND THE GLOBAL STABILITY OF SOME EPIDEMIC MODELS WITH NONLINEAR INCIDENCE[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 38-46. doi: 10.11948/2016004

A CLASS OF LYAPUNOV FUNCTIONS AND THE GLOBAL STABILITY OF SOME EPIDEMIC MODELS WITH NONLINEAR INCIDENCE

1 Science College, Air Force Engineering University, 710051 Xi'an, P. R. China;
2 College of Mathematics and Information Science, Shaanxi Normal University, 710062, Xi'an, P. R. China;
3 Centre for Disease Modelling, York University, M3J 1P3, Toronto, Canada;
4 Department of Applied Mathematics, Xi'an Jiaotong University, 710049 Xi'an, P. R. China;
5 School of Biomedical Engineering, the Fourth Military Medical University, 710032 Xian, P. R. China

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Abstract

Abstract

In this paper, by investigating an SIR epidemic model with nonlinear incidence, we present a new technique for proving the global stability of the endemic equilibrium, which consists of introducing a variable transformation and constructing a more general Lyapunov function. For the model we obtain the following results. The disease-free equilibrium is globally stable in the feasible region as the basic reproduction number is less than or equal to unity, and the endemic equilibrium is globally stable in the feasible region as the basic reproduction number is greater than unity. The generality of the technique is illustrated by considering certain nonlinear incidences and SIS and SIRS epidemic models.
- Epidemic model /
- nonlinear incidence /
- global stability /
- Lyapunov function /
- equilibrium
MSC: 34K20;92D30

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References

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