### Abstract

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co-infection is proposed and analyzed. Results show that both diseases can be eliminated if R_{0}, the basic reproduction number of the co-infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co-exists with the disease-free equilibrium when one of R_{m} or R_{l}, the basic reproduction numbers of malaria-only and visceral leishmaniasis-only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease-free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease-free equilibrium is globally asymptotically stable if VL and post-kala-azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_{m} and R_{l} are greater than unity, then we have co-existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state.

Original language | English |
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Pages (from-to) | 4334-4353 |

Number of pages | 20 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 39 |

Issue number | 15 |

DOIs | |

Publication status | Published - Oct 1 2016 |

Externally published | Yes |

### Keywords

- backward bifurcation
- basic reproduction number
- co-infection
- malaria
- PKDL
- visceral leishmaniasis

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Mathematical Methods in the Applied Sciences*,

*39*(15), 4334-4353. https://doi.org/10.1002/mma.3864