On Behaviour of a Host-vector Epidemic Model with Non-linear Incidence

1Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA

2Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA

3Avdeling for N?rings-og sosialfag, H?gskolen i Finnmark, Follums vei 31, N-9509 ALTA, NORWAY?Corresponding author.

Address: Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA Received 5 October, 2011; accepted 18 January, 2012

In this paper wefind the possible phase portraits and bifurcationsfor a generalclass of host-vectorepidemic models with non-linear incidence function generalizing the Ross model.

Epidemics; Non-linear incidence; Global analysis; Bifurcations MSC2010: 34C05, 34C23, 34D23, 92D30 Bismark Akoto, Emmanuel Kwame Essel, Gunnar S¨oderbacka (2012). On Behaviour of A Host-vector Epidemic Model with Nonlinear Incidence. Studies in Mathematical Sciences, 4(1), 6-17. Available from: URL: /index.php/sms/ article/view/j.sms.1923845220120401.1369DOI: /10.3968/j.sms.1923845220120401.1369

INTRODUCTION

1.MAIN THEOREM

Equations in (1) are considered for 0≤x,y≤1 because x and y are supposed to correspond to the relative

infectious population of host and vector. We assume that the functions gi,hiandμisatisfy the following

conditions.

CONCLUSION

We have examined a generalized Ross model for a large class of non-linear incidence functions and found possible phase portraits and bifurcations. Many known incidence functions are inside this class. There are three types of structurally stable types of phase portraits. One type has the disease-free origin as a global attractor. A second one has the endemic equilibriumas a global attractor. In the third type both disease-free origin and endemic equilibrium are attractors and there is a saddle equilibrium with stable set forming the boundary between the basins of attractions of the both attractors. The possible bifurcations are the usual saddle-node and transcritical bifurcations.

ACKNOWLEDGEMENTS

We wish to thank Professors V A Osipov and O Staffans for checking some details in the proofs for misprints. We also thank I Hauge and T Utsi, the dean and the head of Department of Natural Science of University College of Finnmark for support in cooperation. Many sincere thanks also go to the Institute of Mathematical Sciences, Ghana, and the Department of Mathematics and Statistics of University of Cape Coast, Ghana, for ensuming a fruitful collaboration with the University College of Finnmark.

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