A Model of the Transmission Dynamics of Leishmaniasis

Abstract

In this paper we present a susceptible–infectious–susceptible (SIS) model that describes the transmission dynamics of cutaneous Leishmaniasis. The model treats a vector population and several populations of different mammals. Members of the human population serve as the incidental hosts, and members of the various animals populations serve as reservoir hosts. We establish the basic reproduction number and the equilibrium conditions of the system. We use a generalization of the Lyapunov function approach to show that when the basic reproduction number is less than or equal to one, the diseases-free equilibrium is a global attractor, and that when it is greater than one the endemic equilibrium is a global attractor. We present numerical simulations that demonstrate the dynamics of the model for a system containing a human population and a single animal population.

Data collection to characterise the impact of canine leishmaniosis and modelling of the role of animals in spreading Leishmania infantum within the European Union

“…Article ID 10. (Agyingi et al, 2011) i. Description: This is a deterministic SIS model; therefore infected individuals can become susceptible after infection.…”

Analysis of a Model of Leishmaniasis with Multiple Time Lags in All Populations

“…In a more recent paper by Roy et al [18], a model for CL that focuses on the human host and sandflies vector is provided with a single delay incorporated only in the human 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. The model incorporated a single vector population, multiple animal populations that serve as reservoirs [20] and a human host population.…”

“…The model incorporated a single vector population, multiple animal populations that serve as reservoirs [20] and a human host population. 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. The delays represent the time duration between inoculation of susceptible individuals and them becoming infectious.…”

Understanding Visceral Leishmaniasis Disease Transmission and its Control—A Study Based on Mathematical Modeling

“…From the 19th century onwards, mathematical and statistical models of infection transmission have provided a greater understanding of infection persistence in a population and have played an imperative role in providing solutions to curb the spread of infection [13]. A number of mathematical and statistical modeling studies have been published [5,[14][15][16][17][18][19][20][21][22][23][24][25][26] to explore the dynamics of leishmaniasis disease transmission and its incidence in populations. Most of these models have focused on transmission of cutaneous leishmaniasis through different populations [14,[21][22][23][24][25].…”

“…A number of mathematical and statistical modeling studies have been published [5,[14][15][16][17][18][19][20][21][22][23][24][25][26] to explore the dynamics of leishmaniasis disease transmission and its incidence in populations. Most of these models have focused on transmission of cutaneous leishmaniasis through different populations [14,[21][22][23][24][25]. Only a few studies that explicitly model visceral leishmaniasis transmission exist [15,16,19,20,27,28].…”