Adv Syst Sci Appl 2023; 03:91–107 Published online at https://ijassa.ipu.ru. Global Stability Analysis of Malaria Model with Prophylactic Treatment Saheed Ajao1∗, Isaac Olopade2, Sunday Adewale3, Adelani Adesanya1 1Department of Mathematics and Computer Science, Elizade University, Ondo State, Nigeria 2Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria 3Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Oyo State, Nigeria Abstract: In this paper, we present a mathematical model of the interaction between the human population and the vector (mosquito) population to study the stability of a malaria model in the presence of prophylactic treatment. This study is aimed at investigating the effect of the prophylactic treatment and the long-term dynamics of the solutions of the model. The graph-theoretic method was used to obtain the basic reproduction number (R0). We obtained the disease-free equilibrium for the model which is locally and globally asymptotically stable when the basic reproduction number is less than unity. Moreover, we showed that there exists a unique endemic equilibrium whenever R0 > 1, and the Lyapunov function was used to establish that the endemic equilibrium is globally asymptotically stable whenever R0 > 1. The simulations show the impact of prophylactic treatment on the population of infectious individuals and the findings of this study suggest that prophylactic treatment is an effective approach to reduce the transmission of malaria as prophylactic treatment reduces the population of infectious individuals. Further numerical simulations carried out conformed with the analytic results. Keywords: malaria, prophylactic treatment, basic reproduction number, graph-theoretic method, global stability, Lyapunov function 1. INTRODUCTION Malaria is one of the most serious public health problems in the world. In many developing countries, it is one of the main causes of death and disease [6]. Globally, 247 million cases of malaria were recorded in 2021 and malaria claimed the lives of 619,000 people [46]. In 2020, malaria cases stood at 241 million cases globally, and an estimate of 627,000 deaths was also recorded with the WHO Africa region having the larger share of the global malaria burden [44]. Malaria is a fatal disease and one of the main causes of death for children under five years of age. It is caused by a parasite called Plasmodium which is transmitted through the bite of an infected female Anopheles mosquito. The parasite requires two hosts for the completion of its life cycle: namely, a female Anopheles mosquito and a human [29]. The presence of malaria depends on the climatic factors of an area, such as rainfall, humidity, and temperature, and this can be linked to the prevalence of malaria in tropical and subtropical areas where the conditions allow mosquitoes to survive, multiply, and complete their life cycle [7]. When parasites enter the human body through the bites of infected mosquitoes, the symptoms manifest themselves after about 10 to 15 days, ∗Corresponding author: saheed.ajao@elizadeuniversity.edu.ng 92 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA and children under the age of five, pregnant women, non-immune travellers, HIV/AIDS patients and people with low immune systems are more likely to develop the disease [45]. If one is exposed to the disease, it can cause morbidity and fatality if not treated in time. In recent times, the modelling of the transmission of infectious diseases is now influencing the theory and practice of disease management and control. Mathematical modelling now plays an important role in policy decision-making regarding the epidemiology of disease in many countries [21]. Mathematical models have become useful tools for analysing and controlling the transmission of infectious diseases, and they are used to compare, plan, implement, evaluate and optimize different detection, prevention, and control programs [16]. Several models have been formulated to explain and show the interaction between the host and vector, its transmission and control (see [1, 3, 4, 8–10,17,19,26,37,38,40,42]). Massad et al. [30] studied the malaria infection risk for travellers visiting the Amazonian region of Brazil using a mathematical modelling approach. The study reveals that the risk is high, heterogeneous and depends on the length of stay, arrival time to a locality, entomological potential, and force of infection for the population of the local vector. Mohammed-Awel and Gumel [31] worked on the mathematics of an epidemiology-genetic model to assess the effect of insecticide resistance on the dynamics of malaria transmission. The epidemiology model of the disease is coupled with the genetics of the vector population and incorporates various fitness costs associated with the insecticide resistance for the malaria model. The results of their simulations show that the Insecticides Treated Nets and Indoors Residual Spraying (ITNs & IRS) control strategies reduce the disease effectively in both moderate and high malaria transmission communities if the ITNs coverage level is high enough, but fail to manage insecticide resistance effectively in the community. Woldegerima et al. [43] studied the mathematical analysis of transmission-blocking drugs’ impact on the population dynamics of malaria. Rehman et al. [35] proposed and analysed a mathematical model of malaria with multiple compartments with an associated learning mechanism between the vector-to-host and vice-versa incorporating memory, reinfection, and relapse. Traoré et al. [39] presented a mathematical model of malaria transmission with four different metamorphic stages of mosquitoes and the model also features seasonality. Collins and Duffy [11] studied the mathematical model for the dynamics of malaria and its control in Nigeria. Epidemiological features such as drug resistance, mosquito nets use and malaria treatment were introduced into the model. The results show that the disease will remain in the country if good control measures are not geared toward dominant resistant strains, mosquito net use, and treatment. Kuddus and Rahman [22] developed a human-mosquito model for the dynamical transmission of malaria. The sensitivity analysis was carried out and the contact rates for the human and the mosquito were found to have the greatest influence on the prevalence of malaria. Moreover, the impacts of other features of the model like progression rate, disease-related death rate, recovery rate, and the rate of losing immunity were also examined. Olaniyi et al. [34] worked on the mathematical analysis of a social hierarchy-structured model for the transmission dynamics of malaria. The human population of the model was classified based on two social classes namely: low and high, and the optimal control analysis was also carried out. Beretta et al. [5] proposed a mathematical model for the transmission of malaria which features two-age and asymptomatic classes, and it was found that decreasing the biting rate and increasing the mortality rate of the mosquito can cause the endemic situation to fade off. One of the important aspects of modelling in epidemiology is to examine the global property of the equilibrium, and the Lyapunov function, which serves as a basic and potent tool, has been widely used to study and establish the global stability of the equilibrium. Korobeinikov and Wake [20] presented the Lyapunov functions for the SIR, Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 93 SIRS, and SIS models to establish the global stability of the endemic states of the model. Vargas-De-Leon [41] studied the global stability of infectious disease models with relapse by constructing Lyapunov functions in providing the conditions for the stability of the models with relapse. Li et al. [25] presented an approach that is algebraic in nature to prove the global stability of a class of epidemic models and showed how the coefficients ai can be chosen for the classical Lyapunov function ∑n i=1 ai(xi − x∗ i − xi ln xi/x∗ i ) so that the derivative of the Lyapunov function is negative definite or semidefinite. Khan et al. [18] examined the global stability of vector-host disease with variable population sizes, and numerical simulations were carried out to justify the theoretical results. Gebremestal [15] studied the global stability of the transmission of a malaria model with logistic growth, and the simulations examined how changes in the sustainable level of the vectors affect the population of humans. The study of endemic global stability is not only of importance to Mathematics but also helps in forecasting the long-term dynamics of the disease to design effective prevention and intervention strategies to combat the disease [36]. Most of the studies in the literature failed to consider the prophylactic treatment in malaria models. Hence, the novelty of this work is the focus on the exploration, using the Lyapunov function, of the global stability properties of a malaria model incorporating prophylactic treatment. Furthermore, it is assumed that a fraction of newly infected individuals with low immunity progress to the exposed class while the remaining fraction moves to the infectious class. The paper is organized as follows: section 2 has the method of model design/formulation and model analysis, sections 3 and 4 contain the results and discussion of results, respectively, while the conclusion follows in section 5. 2. METHODS 2.1. Model design In this subsection, we describe how the model is designed and formulated. The malaria model has two populations, namely the human population and the vector (mosquito) population. The overall human population (denoted by Nh) is subdivided into susceptible individuals (Sh), exposed individuals (EM), infectious individuals (IM), and those that recovered (RM). Also, the total mosquito population (denoted by Nv) consists of three subpopulations, namely susceptible mosquitoes (Sv), exposed mosquitoes (Ev) and infectious mosquitoes (Iv). The susceptible humans are assumed to be recruited into the population at the rate π and get infected with malaria after being bitten by infected mosquitoes at the rate λM , which is given by λM = βMaIv, where βM is the probability of transmission from mosquitos to humans provided that there is significant contact between the mosquito and the human, and a is the number of mosquito bites that one person has per unit time. The susceptible population has those who lost their immunity at the rate ϕ in the recovered class, and the population decreases due to natural death which does occur in all human subpopulations at the rate µh. It is assumed that a fraction ϵ of newly infected individuals with low immunity moves to the exposed class of humans EM , while the remaining fraction moves to the infectious compartment IM , the exposed humans progress into the infectious class at the rate κ, and the prophylactic treatment is given at the rate τ1 to the exposed humans making them move to the recovered class. Those that are treated in the infectious class at the rate τ2 progress to the recovered class and death due to the disease causes a reduction in the population of the infectious compartment. The recovered population diminishes as a result of loss of immunity at the rate ϕ and natural death at the rate µh which occurs in all human subpopulations. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 94 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA The susceptible mosquitoes Sv come into the population at a constant rate (recruitment rate πv), and they become infected after effective contact with people infected with malaria at the rate λv, where λv = βvb(EM + ηIM) and η ≥ 1 is the amplification parameter which accounts for the relative infectiousness of infectious individuals when compared to the exposed individuals, b is the number of humans bitten by mosquito per unit time and βv is the transmission probability from humans to mosquitoes. The newly infected mosquitoes move to the exposed class of mosquitoes, and there is a progression from the exposed class to the infectious class of mosquitoes at the rate σ. The death of mosquitoes does occur in all compartments of mosquitoes at the rate µv. The schematic diagram of the model showing the interaction among the classes is depicted in figure 2.1 and table 2.1 shows the definitions of the parameters. Then, the system of equations of the model is as follows: Fig. 2.1. Flow chart of the malaria model Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 95 dSh dt = πh − λMSh − µhSh + ϕRM , dEM dt = ϵλMSh − (κ + τ1 + µh)EM , dIM dt = (1 − ϵ)λMSh + κEM − (τ2 + δ + µh)IM , dRM dt = τ1EM + τ2IM − (ϕ + µh)RM , dSv dt = πv − λvSv − µvSv, dEv dt = λvSv − (σ + µv)Ev, dIv dt = σEv − µvIv, (2.1) where λM = βMaIv, and λv = βvb(EM + ηIM). For convenience, we can rewrite the equations above as below: dSh dt = πh − λMSh − µhSh + ϕRM , dEM dt = ϵλMSh − T1EM , dIM dt = (1 − ϵ)λMSh + κEM − T2IM , dRM dt = τ1EM + τ2IM − T3RM , dSv dt = πv − λvSv − µvSv, dEv dt = λvSv − T4Ev, dIv dt = σEv − µvIv, (2.2) where T1 = κ + τ1 + µh, T2 = τ2 + δ + µh, T3 = ϕ + µh, and T4 = σ + µv. 2.2. Model Analysis The analysis of the model is carried out here, and the threshold for the eradication and persistence of malaria is determined and explored. 2.2.1. The invariant region Theorem 2.1: The closed set D = { (Sh, EM , IM , RM , Sv, Ev, Iv) ∈ R7 + : Nh ≤ πh µh , Nv ≤ πv µv } , is positively invariant with non-negative initial values in R7 +. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 96 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA Table 2.1. Description of parameters used in the model (2.1) Parameter Description Value Source πh Recruitment rate of humans 0.3913 [3] βM Probability of transmission from mosquitoes to humans 0.001 [24] µh Natural death rate of humans 0.00003913 [3] a Number of mosquito bites per unit time 0.29 [17] ϕ Rate of loss of immunity 0.25 [3] ϵ Fraction of newly infected people with low immunity 0.6 Assumed τ1 Prophylactic treatment for the exposed class 0.002 Assumed κ Progression rate of humans from exposed to infectious class 0.0714 [32] b Number of humans bitten by mosquitos per unit time 0.2980 [4] η Amplification factor 2.5 Assumed δ Death due to the disease 0.00009 [8] τ2 Treatment rate for the infectious 0.0092 [33] µv Natural death rate of mosquitoes 0.0476 [3] πv Recruitment rate of mosquitoes 0.7 Assumed σ Progression rate of exposed mosquitoes 0.0555 [24] βv Probability of transmission from humans to mosquitos 0.0001 [24] Proof Consider the feasible region D = Dh × Dv ⊂ R7 + with Dh = { (Sh, EM , IM , RM) ∈ R4 + : Nh ≤ πh µh } , and Dv = { (Sv, Ev, Iv) ∈ R3 + : Nv ≤ πv µv } , and by adding the human and mosquito compartments of (2.1) separately with δ = 0 we have dNh dt = πh − µhNh and dNv dt = πv − µvNv. It follows that dNh dt ≤ πh − µhNh and dNv dt ≤ πv − µvNv. Then Nh(t) ≤ Nh(0)e−µht + πh µh (1 − e−µht) and Nv(t) ≤ Nv(0)e−µvt + πv µv (1 − e−µvt). Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 97 If Nh(0) ≤ πh µh , then Nh(t) ≤ πh µh . Also, Nv(0) ≤ πv µv , then Nv(t) ≤ πv µv . Hence, all solutions of the model with initial values in D stay there for t > 0. This means that D is positively invariant and in this region, and the model is considered to be epidemiologically meaningful and mathematically well-posed. 2.2.2. Disease-free equilibrium The disease-free equilibrium of equation (2.2) is obtained by setting all the rates of change to zero. Hence, the disease-free equilibrium is given by E1 = (Sh, EM , IM , RM , Sv, Ev, Iv) = ( πh µh , 0, 0, 0, πv µv , 0, 0 ) . (2.3) 2.2.3. Basic reproduction number We use the graph-theoretic method as given by de-Camino-Beck et al. [13] to obtain the basic reproduction number of the model. The digraph reduction procedure of the digraph Fλ−1 − V for the model at disease-free is shown in figure 2.2. Fig. 2.2. Digraph reduction procedure of the model From Figure 2d, − 1 + βvbπvσβMaπh(η − ϵη)λ−2 µ2 vµhT2T4 + ϵβMaπhβvbπvσ(κη + T2)λ−2 µ2 vµhT1T2T4 = 0, ∴ R0 = λ = √√√√βvbπvσβMaπh(T1η − T1ϵη + ϵκη + ϵT2) µ2 vµhT1T2T4 . (2.4) The threshold R0 is the basic reproduction number, which is the average number of new infection cases emanating from a single infection source when introduced in a population consisting of only susceptibles [14]. This threshold determines whether malaria will die out or persist. In the next subsection, we try to show that a small influx of infected Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 98 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA mosquitoes/infectious individuals into the community/environment will not generate a substantial malaria outbreak when R0 < 1. 2.2.4. Local stability of the disease-free equilibrium Theorem 2.2: The disease-free equilibrium of the system (2.2) given by (2.3) is locally asymptotically stable if R0 < 1 and unstable when R0 > 1. Proof The Jacobian matrix J(E1) of the system (2.2) evaluated at disease-free equilibrium (2.3) is as given below: J(E1) =  −µh 0 0 ϕ 0 0 −βMaπh µh 0 −T1 0 0 0 0 ϵβMaπh µh 0 κ −T2 0 0 0 (1 − ϵ)βMaπh µh 0 τ1 τ2 −T3 0 0 0 0 −βvbπv µv −βvbηπv µv 0 −µv 0 0 0 βvbπv µv βvbηπv µv 0 0 −T4 0 0 0 0 0 0 ϕ −µv  . (2.5) The first three eigenvalues of (2.5) are −µh, −T3, and −µv. The remaining submatrix is given by J(E1) =  −T1 0 0 ϵβMaπh µh κ −T2 0 (1 − ϵ)βMaπh µh βvbπv µv βvbηπv µv −T4 0 0 0 σ −µv  . (2.6) The characteristic polynomial of the matrix (2.6) is given by X4λ 4 + X3λ 3 + X2λ 2 + X1λ + X0 = 0. (2.7) Where X4 = 1, X3 = µv + T4 + T2 + T1, X2 = T1T2 + T1T4 + T1µv + T2T4 + T2µv + T4µv, X1 = T2T4µv + T1T4µv + T1T2µv + T1T2T4 [ 1 − R2 0µv (η(1 − ϵ) + ϵ) [ϵ(ηκ + T2) + T1η(1 − ϵ)] ] , and X0 = T1T2T4µv(1 − R2 0). The characteristic equation (2.7) has negative roots if it satisfies Routh-Hurwitz criteria, such that Xi > 0 for i = 0, 1, 2, 3, 4 and X1(X2X3 − X1) > X0X 2 3 . Clearly, X4, X3 and X2 > 0, X1 > 0 if ϵ(ηκ + T2) + T1η(1 − ϵ) > µv(η(1 − ϵ) + η) and X0 > 0 for R0 < 1. Hence, the disease-free equilibrium is locally asymptotically stable whenever R0 < 1. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 99 This theorem 2.2 implies that there is possibility of malaria eradication (when R0 < 1) if the initial subpopulation sizes are in the basin of attraction of the disease-free equilibrium E1. To ensure that malaria eradication is not dependent on the initial subpopulations sizes, it is vital to show that the disease-free equilibrium of malaria is globally asymptotically stable. 2.2.5. Global stability of disease-free equilibrium Theorem 2.3: The disease-free equilibrium given by (2.3) is globally asymptotically stable when R0 < 1. Proof We construct the Lyapunov function to prove the global stability of the disease-free as thus: L = (κη + T2)EM + T1ηIM + T2T1R0 βvbSv Ev + T4T2T1R0 σβvbSv Iv. (2.8) Differentiating equation (2.8) gives L̇ = (κη + T2) ˙EM + T1η ˙IM + T2T1R0 βvbSv Ėv + T4T2T1R0 σβvbSv İv. (2.9) L̇ = (κη + T2)[ϵβMaIvSv − T1EM ] + T1η[(1 − ϵ)βMaIvSh + κEM − T2IM ] + T2T1R0 βvbSv [βvb(EM + ηIM)Sv − T4Ev] + T4T2T1R0 σβvbSv [σEv − µvIv]. L̇ = κηϵβMaIvπh µh + T2ϵβMaIvπh µh − T1T2EM + T1η(1 − ϵ)βMaIvπh µh − T1ηT2IM + T2T1R0EM + T2T1ηR0IM − T4T2T1R0Ivµ2 v σβvbπv . L̇ = [√ T1T2T4βMaπhµ2 v σβvbπvµh Iv + T1T2EM + T1T2ηIM ] R0 − 1. (2.10) Therefore, L̇ < 0 if R0 < 1 with L̇ = 0 if and only if EM = IM = Iv = 0. Also, the largest invariant set in {(Sh, EM , IM , RM , Sv, Ev, Iv) ∈ D : L̇ = 0} is the singleton E1. By the LaSalle Invariance Principle [23], every solution that starts in D approaches E1 as t → ∞. Therefore, the disease-free equilibrium is globally asymptotically stable whenever R0 < 1. The implication of theorem 2.3 epidemiologically is that malaria can be eradicated regardless of the initial sizes of the subpopulations of the model whenever the basic reproduction number is less than unity. 2.2.6. Existence of endemic equilibrium The condition for which the endemic equilibrium exists is investigated here. Let E∗ 2 = (S∗ h, E∗ M , I∗ M , R∗ M , S∗ v , E∗ v , I∗ v ), represents an arbitrary endemic equilibrium of the system of equations (2.2). Also, let λM and λv at endemic steady state be denoted by λ∗ M and λ∗ v and given by λ∗ M = βMaI∗ v , (2.11) λ∗ v = βvb(E∗ M + ηI∗ M). (2.12) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 100 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA When we solve the equations of model (2.2) at steady state in terms of the forces of infection λ∗ M and λ∗ v , we will have S∗ h = πh µh + λ∗ M(1 − ϕP3) , E∗ M = P1λ ∗ MS∗ h, I∗ M = P2λ ∗ MS∗ h, R∗ M = P3λ ∗ MS∗ h, (2.13) S∗ v = πv λ∗ v + µv , E∗ v = P4λ ∗ vS∗ v , I∗ v = P5λ ∗ vS∗ v . (2.14) where P1 = ϵ T1 , P2 = (1 − ϵ)T1 + κϵ T1T2 , P3 = T2τ1ϵ + τ2((1 − ϵ)T1 + κϵ) T1T2T3 , P4 = 1 T4 , P5 = σ µvT4 . and T1, T2, T3, T4, and T5 are defined in (2.2) above. By substitution of (2.13) and (2.14) into (2.11) and (2.12), we have λ∗ M = βMaP5λ ∗ vS∗ v , (2.15) λ∗ v = βvb(P1λ ∗ MS∗ h + ηP2λ ∗ MS∗ h). (2.16) Further substitution leads to cλ∗ M − d = 0, where d = µvµh(R0 2 − 1) and c = βvbπh(P1 + ηP2) + µv(1 − ϕP3). Then λ∗ M = d c . (2.17) From (2.17), it shows clearly that c > 0 and d > 0 if R0 > 1. Then, λM is positive (greater than zero) whenever R0 > 1. Hence, the following result is established: Lemma 2.1: There exists a unique endemic equilibrium for the model if the basic reproduction number R0 > 1. 2.2.7. Global stability of endemic equilibrium The global stability of the endemic equilibrium of the model is considered here for the case when ϕ = 0, and we take ϵ = 1. Theorem 2.4: The unique endemic equilibrium of the model with ϕ = 0 and ϵ = 1 is globally asymptotically stable whenever R0|ϕ=0,ϵ=1 > 1. Proof The Volterra type of Lyapunov function stated in 2.18 is used to prove the global stability of the endemic equilibrium. L = ( Sh − S∗ hln Sh S∗ h ) + ( EM − E∗ M ln EM E∗ M ) + T1 κ ( IM − I∗ M ln IM I∗ M ) + ( Sv − S∗ v ln Sv S∗ v ) + ( Ev − E∗ v ln Ev E∗ v ) + T4 σ ( Iv − I∗ v ln Iv I∗ v ) . (2.18) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 101 Taking the derivative of (2.18), we have L̇ = Ṡh − S∗ h Sh Ṡh + ˙EM − E∗ M EM ˙EM + T1 κ ( ˙IM − I∗ M IM ˙IM ) + Ṡv − S∗ v Sv Ṡv + Ėv − E∗ v Ev Ėv + T4 σ ( ˙IM − I∗ M IM ˙IM ) . (2.19) L̇ = 2λMS∗ h + 2µhS∗ h − µhSh − λMS∗2 h Sh − µhS∗2 h Sh − λMShE∗ M EM + T1E ∗ M − T1E ∗ MIM IM − T1EMI∗ M IM + T1EM + 2λvS∗ v + 2µvS∗ v − µvSv − λvS∗2 v Sv − µvS∗2 v Sv − λvSvE∗ M Ev + T4E ∗ v − T4E ∗ MIv I∗ v − T4EvI∗ v Iv + T4E ∗ v . L̇ = T1E ∗ M ( 2 − S∗ h Sh ) + µhS∗ h ( 2 − S∗ h Sh ) − µhSh − T1E ∗2 M Sh S∗ hEM + T1E ∗ M − T1E ∗ MIM I∗ M − T1EMI∗ M IM + T1E ∗ M + T4S ∗ v ( 2 − S∗ v Sv ) + µvS∗ v ( 2 − S∗ v Sv ) − µvSv − T4E ∗2 v Sv SvEv + T4E ∗ v − T4E ∗ vIv I∗ v − T4EvI∗ v Iv + T4E ∗ v . (2.20) By simplification, (2.20) becomes L̇ = µhS∗ h ( 2 − S∗ h Sh − Sh S∗ h ) + T1E ∗ M ( 4 − S∗ h Sh − E∗ MS∗ h EMS∗ h − EMI∗ M E∗ MIM − IM I∗ M ) + µvS∗ v ( 2 − S∗ v Sv − Sv S∗ v ) + T4E ∗ v ( 4 − S∗ v Sv − E∗ vSv EvS∗ v − Iv I∗ v − EvI∗ v E∗ vIv ) . (2.21) Since the arithmetic mean is greater than the geometric mean, then 2 − S∗ h Sh − Sh S∗ h ≤ 0, 4 − S∗ h Sh − E∗ MS∗ h EMS∗ h − EMI∗ M E∗ MIM − IM I∗ M ≤ 0, 2 − S∗ v Sv − Sv S∗ v ≤ 0, 4 − S∗ v Sv − E∗ vSv EvS∗ v − Iv I∗ v − EvI∗ v E∗ vIv ≤ 0. With all the parameters of the model being positive, therefore, L̇ ≤ 0 for R0|ϕ=0,ϵ=1 > 1. Hence, by the LaSalle Invariance Principle [23], every solution to the model equations approaches the endemic equilibrium E2 as t → ∞ whenever R0|ϕ=0,ϵ=1 > 1. The epidemiological implication of theorem 2.4 is that malaria will persist in the community irrespective of the initial sizes of the populations of the model whenever R0 > 1. 3. RESULTS The results of the numerical simulations of the model are obtained using the ODE45 solver of MatLab and the values of the parameters used for the simulations are given in Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 102 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA table 2.1. We investigate the influence of prophylaxis on the dynamics of malaria and the long-term dynamics of the model. The results of the simulations are depicted in figures 3.3 - 3.6. 0 5 10 15 20 25 30 5 10 15 20 25 30 35 Time (Days) In fe ct io u s In d iv id u al s no prophylaxis with prophylaxis Fig. 3.3. Graph of the effect of the prophylactic treatment on the infectious population 0 5 10 15 20 25 30 14 16 18 20 22 24 26 28 30 32 Time (Days) In fe ct io u s In d iv id u al s τ 1 =0.02 τ 1 =0.06 τ 1 =0.10 τ 1 =0.14 Fig. 3.4. Graph of the infectious population at different values of prophylactic treatment rate (τ1) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY ANALYSIS OF MALARIA MODEL 103 0 500 1000 1500 2000 0 20 40 60 80 100 120 140 160 180 Time (Days) In fe ct io u s In d iv id u a ls Fig. 3.5. Plot of the infectious individuals at different initial conditions when R0 = 0.65 and the parameters used are given in Table 2.1 0 500 1000 1500 2000 0 50 100 150 200 250 Time (Days) In fe ct io u s In d iv id u al s Fig. 3.6. Plot of the infectious individuals at different initial conditions when R0 = 12.4, with ϵ = 1, ϕ = δ = 0, βM = 0.004, βv = 0.0004, τ1 = 0.001, τ2 = 0.004, µv = 0.0175 and other parameters used are given in Table 2.1 4. DISCUSSION OF THE RESULTS A mathematical model of malaria with prophylactic treatment is presented, and its global asymptotic stability properties are studied. The threshold which determines whether malaria will fade out or persist is established. The qualitative analysis of the stability carried out showed that the disease-free equilibrium is locally asymptotically Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 104 S. AJAO, I. OLOPADE, S. ADEWALE, A. ADESANYA stable whenever R0 < 1. Also, there exists an endemic equilibrium whenever the basic reproduction number goes beyond unity. The Lyapunov functions were used to establish that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable whenever R0 < 1 and R0 > 1, respectively. The graph of figure 3.3 illustrates the population of infectious individuals when prophylactic treatment is present and when there is no prophylactic treatment. The population of infectious individuals is at minimum when there is prophylaxis as compared to when there is none. Hence, the presence of prophylactic treatment lowers the number of infectious individuals as it prevents exposed individuals to progress to the infectious class/state. The increasing effect of prophylactic treatment rate on the infectious population is depicted in figure 3.4. The population of infectious individuals diminishes as the prophylactic treatment rate increases and thereby reducing the burden of the disease. Therefore, prophylactic treatment plays an important role in reducing the population of the infectious individuals. The practical situation of the study is the recent implementation of seasonal malaria chemoprophylaxis in Nigeria by the Malaria Consortium [27,28] to reduce the disease burden. Some states in Nigeria were selected for the pilot phase and every child in the community under five years old is given prophylactic treatment every month. A study by de Cola et al. [12] reported that the treatment programme reduces the prevalence of malaria in the country, which is in line with the result of this current study. The work of Ambe et al. [2] also supports the result we obtained. We carried out numerical simulations on global stability to validate the analytic results. The plot in Fig. 3.5 shows the long-term dynamics of the population of infectious individuals at different initial conditions when R0 < 1. The infectious population converges to disease-free equilibrium and vanishes, which means that the disease dies out of the community irrespective of the different initial sizes of the infectious class. Also, Fig. 3.6 shows a scenario of the population of the infectious class with different initial conditions when the basic reproduction number R0 > 1. In this case, the disease does clear out from the population, but persists, and this is in line with the analytic result. 5. CONCLUSION We formulated a mathematical model for the dynamics of malaria with prophylaxis to study the global stability properties of the model. The basic reproduction number was obtained using the graph-theoretic method. 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World Health Organization (WHO). (2023, February 19). Malaria: Key facts. [Online]. Available http://www.who.int/news-room/fact-sheets/detail/malaria. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) https://doi.org/10.1016/j.nonrwa.2019.103081 https://doi.org/10.1016/j.nonrwa.2019.103081 https://doi.org/10.1016/j.amc.2021.126005 https://doi.org/10.1016/j.amc.2021.126005 http://www.who.int/news-room/fact-sheets/detail/malaria http://www.who.int/news-room/fact-sheets/detail/malaria http://www.who.int/news-room/fact-sheets/detail/malaria Introduction Methods Model design Model Analysis The invariant region Disease-free equilibrium Basic reproduction number Local stability of the disease-free equilibrium Global stability of disease-free equilibrium Existence of endemic equilibrium Global stability of endemic equilibrium Results Discussion of the Results Conclusion