# A Mathematical Model for the Population Dynamics of Malaria with a Temperature Dependent Control

• A. Nwankwo
• D. Okuonghae
Original Research

## Abstract

In this work, we develop a mathematical model for malaria transmission that investigates the impact of temperature on the efficacy of ITNs. The disease free state of the autonomous version of the formulated model is seen to be globally asymptotically stable in the absence of disease induced mortality when the associated reproduction number is less than unity. Also, when the associated reproduction number of the non-autonomous model is greater than unity, the disease persist in the population. Sensitivity and uncertainty analysis of the autonomous model, using the associated reproduction number, number of infected humans and vectors as response functions, show that the probability of transmission from human to vectors ($$\beta _{hv}$$), maximum biting rate of vectors ($$b_{max}$$), vector carrying capacity ($$k_v$$) as well as ITN efficacy and coverage ($$\varepsilon _b$$ and $$b_o$$) have the most impact on all three response functions. In the non-autonomous model, for temperature values in the range $$[18\pm 4-32\pm 4]\,^\circ$$C and using the number of infected humans and vectors as response functions, it can be seen that the bed-net coverage lost its significance for temperature values in the range $$[22\pm 4-30\pm 4]\,^\circ$$C and $$[22\pm 4-26\pm 4] \,^\circ$$C respectively. Numerical simulations of the time-averaged reproduction number for the case where the efficacy of the bed-net is temperature-dependent showed that temperature values in the range [22–27] $$\,^\circ$$C effects the efficacy of the bed-nets most. Numerical simulations of the non-autonomous model for the case where ITN efficacy is temperature-dependent, we determine temperature ranges that make for increase in the effectiveness of the bed-nets in the fight against malaria.

## Keywords

Malaria Non-autonomous model Efficacy (ITNs) Global stability Sensitivity analysis

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© Foundation for Scientific Research and Technological Innovation 2019

## Authors and Affiliations

1. 1.Department of MathematicsUniversity of BeninBenin CityNigeria