Journal
JOURNAL OF BIOLOGICAL DYNAMICS
Volume 8, Issue 1, Pages 99-116Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/17513758.2014.912682
Keywords
Lyapunov functional; 34D30; global stability; multi-group epidemic model; relapse distribution; 92D30
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Funding
- National Natural Science Foundation of China [11201128, 11271303]
- Science and Technology Research Project of the Department of Education of Heilongjiang Province [12531495]
- Natural Science Foundation of Heilongjiang Province [A201211]
- Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province
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In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.
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