## Sensitivity and Mathematical Analysis of Malaria and Cholera Co-Infection

Published: 2022-07-07

Page: 425-452

Department of Physical Sciences, Chrisland University, P.M.B. 2131, Abeokuta, Ogun State, Nigeria.

Department of Mathematics and Statistics, Federal University, Wukari, P.M.B. 1020, Taraba State, Nigeria.

S. O. Ajao

Department of Computer and Mathematics, Elizade University, P.M.B. 002, Ilara-Mokin, Ondo State, Nigeria.

Department of Mathematics/Statistics, Osun State College of Technology, Esa-Oke, Osun State, Nigeria.

Department of Mathematics and Statistics, LASUSTECH, Ikorodu, Lagos State, Nigeria.

Department of Pure and Applied Mathematics, LAUTECH, P.M.B. 2000, Ogbomoso, Oyo State, Nigeria.

*Author to whom correspondence should be addressed.

### Abstract

In order to better understand the factors affecting the dynamical spread of cholera and malaria in a community, we defined a new fourteen (14) compartmental mathematical model in this study. The model is examined for each of the factors influencing the spread of the disease in an effort to identify those that are most sensitive. First, full models of malaria-cholera co-infection were studied, then sub models of malaria and cholera alone. For the purpose of determining whether disease-free and endemic equilibrium points exist, the stability of the malaria model alone, the cholera model alone, and the entire model of malaria-cholera co-infection were all examined. The disease-free equilibrium point has been demonstrated to be locally asymptotically stable whenever the threshold is below unity and unstable whenever the threshold exceeds unity using the Next Generation Matrix Method (NGM). Calculations are made to determine the model's relative sensitivity solution for each parameter. To demonstrate the impact of each parameter on the dynamical spread of malaria-cholera co-infection, numerical solutions utilizing the differential transformation approach were performed by Maple software. Sensitivity analyses showed that the transmission rate was the most sensitive of all the model's parameters. Sensitivity analyses revealed that out of all the parameters involved in the model, the transmission rate ($$\beta$$ M), low immunity rate ($$\varepsilon$$1 ) and mosquito biting rate ($$\alpha$$) for Malaria model, transmission rate  ($$\beta$$ c) and low immunity rate  ($$\varepsilon$$2 ) for Cholera model, transmission rate ($$\beta$$ Mc) and low immunity rate  ($$\varepsilon$$3) for the co-infection model  were of high influence on their basic reproduction numbers. While ($$\tau$$4) is the rate of treatment for co-infection with malaria and cholera, which has a negative impact and lowers the basic reproduction rate. The consequence is that it will be challenging to eradicate or stop the spread of the two illnesses in the population if the reproduction rate for malaria/cholera co-infection keeps rising. The rate of treatment  ($$\tau$$4)  reduction in fundamental reproduction of Malaria/Cholera Co-infection, which makes it convenient to stop the spread of the two infections. Therefore, efforts must be made to decrease the rate at which each person comes into contact with the two diseases, and ongoing care for those who are infected in the community is necessary to maintain a disease-free environment.

Keywords: Malaria, cholera, reproduction number, critical points, sensitivity analysis, stability

#### How to Cite

Adeniran, G. A., Olopade, I. A., Ajao, S. O., Akinrinmade, V. A., Aderele, O. R., & Adewale, S. O. (2022). Sensitivity and Mathematical Analysis of Malaria and Cholera Co-Infection. Asian Journal of Pure and Applied Mathematics, 4(1), 425–452. Retrieved from https://globalpresshub.com/index.php/AJPAM/article/view/1625

### References

World Health Organization. 10 facts on malaria, WHO online; 2011.

World Health Organization, World Malaria report, WHO press Switzerland; 2012.

Magambedze G, Chiyaka C, Mukandavive. Optimal control of malaria chemotherapy. Nonlinear analysis: Modeling and Control. 2011;16(4):415-434.

Adewale SO, Ajao SO, Olopade IA, Adeniran GA, Oyewumi AA. Effect of chemoprophylaxis treatment on the dynamical spread of malaria. International Journal of Scientific and Engineering Research. 2016;7(1).

ISSN. 2229-5518

Adewale SO, Adeniran GA, Olopade IA, Mohammed IT. Mathematical analysis of the effect of growth rate of vibrio-cholera in the dynamical spread of cholera. Research Journal of Mathematics 2015;2(3).

ISSN 2349-5375

Chitnis N, Cushing JM, Hyman M. Bifurcation analysis of mathematical model for malaria transmission. Society for Industrial and Applied Mathematics. (SIAM) Journal of Applied Mathematics. 2006;67(1):24–45.

Hartley DM, Morris Jr. JG, Smith DL. Hyper infectivity: A critical element in the ability of V.cholera to cause epidemic. PLos Med. 2006;3(1):63-69.

Ochoche JM. A mathematical model for the transmission dynamics of cholera with control strategy. International Journal of Science and Technology. 2013;2(11).

Nelson EJ, Chowdhuy A, Flynn J, Schild S, Bourassa L. Transmission of vibrio Cholera antagonized by lytic phage and Entry into the Aquatic Environment. PLos Pathogens. 2008;4(10):1-15.

Colwell RR, Huq A. Environmental reservoir of vibro cholera, the causative agent of cholera. Ann. N.Y Acad. Sci. 1994;740:44-53.

Zuckerman JN, Rombo L, Fish A. The true burden risk of cholera: Implications for prevention and control. Lancet Infect. Dis. 2007;7(8):521-530.

Politzer R. “Cholera studies” Bulletin of the World Health Organization. 1995;13(1):1-25.

Sarkar BL. “Cholera studies,” Bulletin of the world Health Organization. 2002;13(1):1-25.

Saymov RM. Treatment and Prophylaxis of cholera with bacteriophage. Bulletin of the world Health Organization. 1963;28:361-367.

Summers WC. Bacteriophage therapy. Annual Review of Microbiology. 2001;55:437-451.

Mahalanabis A, Lopez AL, Sur D, Deen Jr, Manna B. A randomized, placebo controlled trial of the bivalent killed, whole-cell, oral cholera vaccine in adults and children in a cholera endemic area in Kolkata, Indian, PLos One. 2008;3(6):17.

Longini IM, Nizam A, Ali M, Yunus M, Shenvi N, Clements JD. Controlling endemic cholera with oral vaccines. PLos Med. 2007;4(11):1776-1783.

Seidlein LV. Vaccines for Cholera Control: Does herd immunity play a role? PLos Med. 2007;4(11): 1719-1721.

Sur D. Efficacy and Safety of a modified killed-whole-cell oral cholera vaccine in India; an interim analysis of a cluster-randomized, double-blind, Placebo-controlled trial. Lancet. 2009;349:1694-1702.

Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibrium for compartmental moels of disease transmission, Mathematical Biosciences. 2002;180:29-48.

Lakshmkanthan V, Leela S, Martynyuk AA. Stability analysis of non-linear system. Marcel Dekker, Inc., New York and Basel; 1989.

Castillo-Chavez C, Feng Z, Huang W. On the computation of R0 and its role on global stability; 2002.

Available:math.laasu.edu/chavez/2002/JB276.pdf