Abstract
There are several types of deterministic compartmental models for disease epidemiology such as SIR, SIS, SEIS, SEIR, etc. The exposed population group in, for example SEIS or SEIR, usually represents individuals in the incubation class. Time delays (of which there are several types) when incorporated into a SIR or SIS model, also fulfil the role of the incubation period without necessarily adding another compartment to the model. This paper incorporates time delays into a SIS model that describes the transmission dynamics of cutaneous leishmaniasis. The time lags account for the incubation periods within the sandflies vector, the human hosts and the different animal groups that serve as reservoir hosts. A threshold value, R 0 , of the model is computed and used to study the disease-free equilibrium and endemic equilibrium of the system. Analysis demonstrating local and global stability of the disease-free equilibrium when R 0 < 1 is provided for all n + 1 population groups involved is provided. The existence of an endemic equilibrium is only guaranteed when R 0 > 1 and numerical analysis of the endemic equilibrium for a human host, a vector host and a single animal reservoir host that is globally stable is also provided.
Highlights
Leishmaniasis is primarily a zoonotic disease caused by protozoan parasites that are transmitted to humans by female sandflies
Cutaneous leishmaniasis (CL) is the most dominant form of the disease leading to skin lesions that can lead to lifelong scars and disabilities
We continue our analysis of the equilibriums by considering the case n = 2 of the system (3), where the x1 variable represents a proportion of infectious human population, x2 variable represents a proportion of infectious animal population and y is a proportion of infectious sandflies
Summary
Leishmaniasis is primarily a zoonotic disease caused by protozoan parasites that are transmitted to humans by female sandflies. Recognized as one of the most important and widespread parasitic disease in the world, CL prevention and control remains a challenge for health authorities in some countries [4] Their model does not take into consideration time delays in humans (the incidental) host or sandflies (vector) host. Because within a single locality many different animals can serve as reservoir for CL, Agyingi et al [19] developed a susceptible-infectious model that describes the transmission dynamics of cutaneous leishmaniasis. Because the leishmaniasis parasite undergoes an incubation period within the animal reservoirs, sandflies vector and human host, this paper builds on the work in [19] by incorporating time delays in all populations involved, distinguishing it from previous models.
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