The generalization of an n-patch model for Leishmaniasis

Abstract

Leishmaniasis is a vector borne disease, which is caused by a protozoan parasite. Infected female sandflies \textit{(Phlebotomine sp.)} are responsible for such disease transmission. People can carry some species of Leishmania for long periods and the incubation period for Leishmaniasis may be few weeks or several months. Migration depends on host immunological status also. Leishmaniasis is endemic globally in ninety eight countries. Spread of the disease is intensely dependent on migration of the disease among human and vector between various regions or countries. Here, we formulate an $n$-patch mathematical model (such patches can be some states, regions or countries) considering each patch have susceptible, infected human as well as susceptible and infected vector population. We have derived basic reproduction ratio for each patch as well as the general basic reproduction ratio for the system and shown that there exists a disease-free equilibrium that is locally asymptotically stable. Further, we study the system analytically and numerically when the migration between each patch of individual class of humans and vector effects the spreading of the disease. We also established the results taking into account for different number of patches for $n=2$. Our result reveals that the movement of human and vector from each patch to another patch plays an important role for spreading of the disease.

https://doi.org/10.28919/cmbn/3361