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A mathematical model of malaria transmission with media-awareness and treatment interventions

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A Correction to this article was published on 30 July 2024

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Abstract

Malaria, a lethal protozoan disease transmitted through the bites of female Anopheles mosquitoes infected with Plasmodium parasites, remains a significant global health concern. This study introduces a compartmental mathematical model to explore the impact of insecticide use and malaria treatment based on awareness initiatives. The model incorporates the influence of media-based awareness on the effectiveness of insecticide utilization for malaria control. Key mathematical properties, such as positivity, boundedness of solutions, feasibility, and stability of equilibria, are systematically investigated. Our analysis demonstrates that all solutions to the system are positive and bounded within a specified set of initial conditions, establishing the mathematical soundness and epidemiological relevance of the model. The basic reproduction number \(R_0\) is determined through the next-generation matrix method. Stability analysis reveals that the disease-free equilibrium is globally asymptotically stable when \(R_0\) is less than one, while it becomes unstable if \(R_0\) exceeds one. Global stability of the endemic equilibrium is established using an appropriate quadratic Lyapunov function in cases where \(R_0\) surpasses one. We identify the most sensitive parameters of the model through normalized forward sensitivity indices. In addition, numerical simulations employing the Runge–Kutta method in Python software further validate our findings. Both analytical and numerical results collectively suggest that the integration of awareness-based insecticide usage with malaria treatment holds the potential for malaria elimination. This comprehensive approach not only contributes to the mathematical rigor of the model but also underscores its practical implications for effective malaria control strategies.

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Acknowledgements

The authors would like to thank Adama Science and Technology University (ASTU) for offering resources and steadfast support during this research article.

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  1. Legesse Lemecha Obsu and Feyissa Kebede Bushu have contributed equally to this work.

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  1. Department of Applied Mathematics, Adama Science and Technology University, 1888, Adama, Oromia, Ethiopia

    Andualem Tekle Haringo, Legesse Lemecha Obsu & Feyissa Kebede Bushu

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Appendix A: Routh–Hurwitz stability criteria for the characteristic polynomial (26)

Appendix A: Routh–Hurwitz stability criteria for the characteristic polynomial (26)

Based on the relationship between the coefficients and the trace/determinant, we can write the expression for the characteristic polynomial coefficients as follows:

$$\begin{aligned} C_1 =&tr(J)=\vartheta -(J_{11}+J_{22}+J_{33}+J_{44}+J_{55}+J_{66}+J_{77})>0,\\ C_2 =&-(tr(J)C_1 - tr(J^2))/2,\\ C_3 =&(tr(J)C_2 - tr(J^2)C_1 + tr(J^3))/3,\\ C_4 =&-(tr(J)C_3 - tr(J^2)C_2 + tr(J^3)C_1-tr(J^4))/4,\\ C_5 =&(tr(J)C_4 - tr(J^2)C_3 + tr(J^3)C_2 - tr(J^4)C_1 + tr(J^5))/5,\\ C_6 =&-(tr(J)C_3 - tr(J^2)C_4 + tr(J^3)C_3 - tr(J^4)C_2 + tr(J^5)C_1 -tr(J^6))/6,\\ C_7 =&(tr(J)C_6 - tr(J^2)C_5 + tr(J^3)C_4 - tr(J^4)C_3 + tr(J^5)C_2 - tr(J^6)C_1 + tr(J^7))/7,\\ C_8 =&-\text {det}(J) \end{aligned}$$

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Haringo, A.T., Obsu, L.L. & Bushu, F.K. A mathematical model of malaria transmission with media-awareness and treatment interventions. J. Appl. Math. Comput. 70, 4715–4753 (2024). https://doi.org/10.1007/s12190-024-02154-9

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