Bifurcation and Sensitivity Analysis of a Malaria Model with Isolated Drug Resistant Population

A malaria model with isolated drug resistant population after the first line of treatment is presented using six systems of first order nonlinear differential equations. The disease free equilibrium point and the basic reproduction number are determined. Local stability of the disease free equilibrium is determined and the conditions for the existence of endemic equilibrium. Bifurcation analysis reveals the existence of backward bifurcation. Sensitivity analysis is used to determine the impact of the model parameter on the basic reproduction number. Early detection and using correct dosage will go a long way to prevent drug resistance. DOI: https://dx.doi.org/10.4314/jasem.v24i11.15 Copyright: Copyright © 2020 Sangotola. This is an open access article distributed under the Creative Commons Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dates: Received: 25 January 2020; Revised: 05 November 2020; Accepted: 14 November 2020

Malaria is caused by the protozoan parasites of genus Plasmodium. In humans it is caused by Plasmodium falciparum, Plasmodium malariae, Plasmodium ovale, and Plasmodium vivax. Of these, P.falciparum is the most common cause of infection in Africa and South East Asia (Mandal et al., 2011). The symptoms of malaria includes chills, fever, vomiting and headache. The first mathematical model of malaria was developed by Sir Ronald Ross, while serving at the Indian Medical Service in 1890's. He developed a simple mathematical model now known as the classical "Ross model" which explained the relationship between the number of mosquitoes and incidence of malaria in humans. Over the years, various mathematical models have developed to effectively understand the dynamics of malaria (Olaniyi et al., 2014 andNgwa et al., 2000). Drug resistance occurs due to genetic mutation that allows the organism to survive treatment. As a result, the drug becomes less effective and infections persist in the body, increasing the risk of spread to others. Improper use of drugs and taking smaller than recommended doses are among the major causes of drug resistance. According to the world health organization, coordinated action is needed to reduce the emergence and spread of antimicrobial resistance.
Most research work on this subject involves non isolated drug resistant population (Okosun et al., 2011, Cai et al., 2013and Ronoh et al., 2016. Here, we propose a model where those with initial drug resistance are isolated and cannot transmit the disease during this period until they are effectively treated.

MATERIALS AND METHOD
The model divides the total human population into susceptible humans S , infected humans I , isolated drug resistant humans R , and recovered humans R . The vector population is divided into susceptible mosquitoes S , and infected mosquitoes R . The exposed stage is omitted in both human and vector population because we assumed that they will progress to the infectious stage. The dynamics of the model is such that susceptible individual are recruited into the human population at input rate Λ . Every class of human population is decreased by natural death except for the infectious class and isolated drug resistant class which has a per capita disease induced death rate and respectively . A susceptible human becomes infected after being bitten by an infectious mosquito with contact rate and transmission rate . After the first line of treatment, those that respond to treatment move to the recovered class while the ones who do not respond move to the isolated drug resistant class for further treatment. However, the recovered humans develop a temporary acquired immunity against the disease and later loses this immunity to become susceptible again at per capita rate c. Mosquitoes are recruited into the population at rate Λ but decreased through interaction with infectious humans with transmission rate . Both the susceptible and infectious mosquitoes are decreased by natural death while the infectious mosquitoes are further reduced as a result of the parasite at rate . The following systems of first order differential equations describe the model.
Proof: The total human population size is given by Basic reproduction number: The next generation matrix approach by Driessche and Watmough (2002) is applied to obtain the basic reproduction number.
The linearized matrices F and V, computed at the disease-free equilibrium from (8) and (9)  The basic reproduction is given by ( ) where is the spectral radius. Thus, Local stability of disease-free equilibrium: One of the most important concerns in the analysis of epidemiological models is the determination of the asymptomatic behaviour of their solutions which is usually based on the stability of the associated equilibrium.
Theorem 2: (Local stability of disease-free equilibrium). The disease-free equilibrium for the system (1) − (6) is locally asymptomatically stable if < 1 and unstable otherwise. Proof: The Jacobian matrix evaluated at the diseasefree is given by  Bifurcation Analysis: To demonstrate the possibility of the co-existence of the equilibria of the model (1) − (6) at < 1 but near = 1, the centre manifold theory described in (Chavez et al., 2004) is applied. This theory can be used to establish the local stability of the endemic equilibrium near the threshold parameter Let be a bifurcation parameter such that = * when = 1.Then * can be obtained from (10)  Using the definition of and from (Chavez et al., 2004) and solving gives < 0 and > 0.
Sensitivity analysis: Sensitivity analysis is used to determine dependencies between input parameters and results of the model. The normalised forward sensitivity index of a variable, , that depends on a parameter, q, is defined as:

Γ = ×
Sensitivity index of the basic reproduction number of the basic reproduction number with respect to the model parameters are computed below: 2( + + + ) − 2( + ) − 2( + ) Γ = 1 revealed that a 10 percent in transmission rate corresponds to 10 percent increase in the basic reproduction number. The negative sign of the sensitivity analysis shows inverse proportionality between the parameter and the basic reproduction number.

Conclusion:
We present a mathematical model for malaria dynamics with isolated drug resistant population after the first line of treatment. The basic reproduction number of the model and the local stability of the disease free equilibrium are determined. The model is found to exhibit a backward bifurcation which implies that keeping the basic reproduction number below unity is not sufficient for the eradication of the disease. The contribution of the model parameters on the basic reproduction number were also highlighted through sensitivity analysis.