Analysis of a vector-borne disease model with human and vectors immigration
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Analysis of a vector-borne disease model with human and vectors immigration Ali Traoré1 Received: 14 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract We study the stability analysis of a vector-borne disease model. A wind-borne longdistance immigration of vectors and human immigration are considered. We assume a nonlinear incidence function including mass action and saturating incidence as special cases. There is no disease-free equilibrium and therefore no basic reproduction number. The only equilibrium is an endemic equilibrium. By the Lyapunov method, we show that this endemic equilibrium is globally asymptotically stable. We established that when the fraction of infective human and vectors immigrants approaches a small value, there is a threshold for which the disease can be reduced in the community. Keywords Vector-borne diseases · Immigration · Global stability · Lyapunov function Mathematics Subject Classification 34D20 · 34D23 · 34D45 · 37C75 · 92B05
1 Introduction Vector-borne diseases still remain a public health problem despite control efforts. Mortality and morbidity rates from these diseases are highest in the subtropical regions. Vector-borne diseases are transmitted by vectors, among which the best known are mosquitoes. Mosquitoes are responsible for a number of infectious diseases such as malaria, dengue fever and West Nile virus. The upsurgence of vector-borne diseases is due to several factors including immigration of humans, insecticide and drug-resistance of vectors, rapid development of urbanization (see [6,9,15]). Human migration remains one of the reasons for the rapid spread of vector-borne diseases, particularly in Africa. These migrations are often linked to economic problems. Peo-
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Ali Traoré [email protected] LAboratoire de Mathématiques et Informatique (LAMI), Département de Mathématiques, Université Joseph KI-ZERBO, 03 BP 7021 Ouagadougou 03, Burkina Faso
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ple move around in order to find a job. People also migrates due to civil unrest and military conflicts. This creates large refugee populations living in the endemic area for long periods of time under poor conditions. Migrations and travels of populations sometimes take place from endemic regions to healthy regions (see [13,19,23]). This situation favors the introduction of pathogens into areas where they were previously absent [24]. However, beyond human migration, the migration of vectors should not be ignored in the disease control. The immigration of certain species of Anopheles mosquitoes over a long distance remains one of the factors for the persistence of malaria in sub-Saharan region of Africa (see [3,10]). In [10], the authors have shown that these migrations are carried out overnight by using the wind and are one of the consequences of the presence of the disease in regions where surface water is absent for 3 to 8 months of year. These Anopheles after having taken their blood meal travel a distance of up to 300 km and this for 9 hours. It has
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