Adv Syst Sci Appl 2023; 04:90–103 Published online at https://ijassa.ipu.ru. Global Stability and Sensitivity Analysis of Basic Reproduction Number of a Malaria Model Adesoye I. Abioye1, Olumuyiwa J. Peter1,2*, Festus A. Oguntolu3, Tawakalt A. Ayoola4, Asimiyu O. Oladapo4 1Department of Mathematical and Computer Sciences, University of Medical Sciences, Ondo City, Ondo State, Nigeria 2Department of Epidemiology and Biostatistics, School of Public Health, University of Medical Sciences, Ondo City, Ondo State, Nigeria 3Department of Mathematics, Federal University of Technology, Minna, Niger State, Nigeria 4Department of Mathematical Sciences, Osun State University, Osogbo, Osun State, Nigeria Abstract: This paper explores a mathematical model of malaria, focusing on the basic reproduction number (R0) and employing Lyapunov functions to assess the global stability of disease-free and endemic equilibria. Sensitivity analysis of key parameters is conducted to evaluate their impact on disease control. The results indicate an active malaria outbreak with decreasing human classes signifying disease progression and increasing mosquito classes suggesting heightened transmission risk. Effective control measures, including mosquito control and treatment of infected individuals, are essential to mitigate the outbreak. Keywords: malaria, sensitivity, global stability analysis, numerical simulations, Lyaponov function 1. INTRODUCTION Malaria is a life threatening tropical disease. It is both preventable and treatable. A case of simple malaria, however, can escalate to a severe form of the disease, which is generally fatal if not treated promptly. This risk of infection is higher in some locations than others, depending on a variety of circumstances, including the species of local mosquitoes. It may also vary according to season, with the danger being greatest in tropical nations during the rainy season [6, 17]. The risk of disease can be lowered by preventing mosquito bites with mosquito nets and insect repellents, or by using mosquito-control techniques such as spraying insecticides. Malaria is still frequent in tropical and subtropical nations, despite the fact that it is uncommon in temperate climes. Globally, the World Health Organization estimates that 241 million clinical cases of malaria will occur in 2020, with 627,000 deaths, the majority of whom would be children in Africa. The first signs of malaria usually appear after 10-15 days of being bitten by an infected mosquito. Fever, headache, and chills are common symptoms, however they could be minor and difficult to distinguish from malaria. In malaria-endemic settings, people who have developed partial immunity may become infected yet exhibit no symptoms [12, 13]. Over the years, precise frameworks for comprehending the dynamics of malaria transmission in the human population have been developed using mathematical ∗Corresponding author: peterjames4real@gmail.com GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 91 models [3–5, 11]. The model developed by [2] ia considered in this study in order to determine the effect of the sensitive parameters to the disease control. Further studies on malaria models can be found in [2].The motivation for this study stems from the persistent threat of malaria as a life-threatening disease transmitted by mosquitoes, posing a significant public health challenge for many years. Recognizing the gravity of the issue, this research endeavors to explore a sophisticated mathematical model of malaria. By incorporating the fundamental epidemiological indicator, the basic reproduction number (R0), and employing Lyapunov functions, we aimed to gain insights into the global stability of disease-free and endemic equilibria. Furthermore, the study’s motivation extended to conducting a sensitivity analysis of key parameters, shedding light on their influence in the context of disease control. The findings revealing an active malaria outbreak, characterized by diminishing human populations and burgeoning mosquito populations, underscore the urgency for effective control measures. Ultimately, the study’s motivation lies in the imperative need to address this ongoing health crisis through measures such as mosquito control and timely treatment of infected individuals, with the ultimate goal of mitigating the outbreak and safeguarding public health. The paper is structured as follows: section two deals with the models formulation, section three deals with the basic analysis of the model, section four deals with the sensitivity analysis of the model and discussion of result. Finally, we concluded the study in section five. 2. MODEL FORMULATION This model developed is an extension of [2] by considering the global stabilities and sensitivity analysis of the model. According to [2], the total population of humans, Nh(t) which is known as the host comprises of Susceptible class of human (Sh(t)), Exposed class of human (Eh(t)), Infectious class of human (Ih(t)) and Recovered class of human (Rh(t)). This can be defined as Nh(t) = Sh(t) + Eh(t) + Ih(t) +Rh(t) Similarly, the total population of mosquitoes, Nv(t) which is also known as vector comprises of Susceptible class of mosquito (Sv(t)), Exposed class of mosquito (Ev(t)) and Infectious class of mosquito (Iv(t)). This can be defined as Nv(t) = Sv(t) + Ev(t) + Iv(t) Moreover, the parameters of human and mosquito populations show the movement of individual at different rates from one class to another. This can be defined below as follows; Λh and Λv represent the recruitment rates of humans and mosquitoes respectively, σh and σv represent the the Probability of transmission of malaria per mosquito bite to susceptible humans and from infected humans to susceptible mosquitoes respectively. µ and η represent natural mortality rates of humans and mosquitoes respectively. Also, υ is the developing rate of exposed humans and ψ is the rate of the newborn birth with malaria from the mother’s womb. εh and εv represent the proportion of an antibody produced by human in response to the incidence of infection caused by mosquito and produced by mosquito in response to the incidence of infection caused by human respectively. δh and δv represent the disease-induced mortality rates of humans and mosquitoes respectively. ξ is the biting rate of mosquitoes and α is the developing rate of exposed mosquitoes. γ1 and γ2 represent the rate of loss of immunity in humans and relapse rate of humans respectively. ω is the recovery rate of of humans. 2.1. Flow Chart of the Model The diagram below is the flow chart of the model. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 92 A.I. ABIOYE, ET. AL Fig. 2.1. Flow chart diagram 2.2. Differential Equations of the Model The first order differential equations can be derived from the flow chart model as follows: dSh dt =Λh − ξσhIvSh 1 + ϵhIv − µSh + γ1Rh dEh dt = ξσhIvSh 1 + ϵhIv − (υ + µ)Eh dIh dt =υEh − (ω + µ+ δh)Ih + ψIh + γ2Rh dRh dt =ωIh − (γ1 + γ2 + µ)Rh dSv dt =Λv − ξσvIhSv 1 + ϵvIh − ηSv dEv dt = ξσvIhSv 1 + ϵvIh − (α + η)Ev dIv dt =αEv − (η + δv)Iv (2.1) 3. EXISTENCE OF EQUILIBRIUM POINTS It is very important to study the equilibrium points of the model (2.1) which comprises of Disease Free Equilibrium (DFE) point and Endemic Equilibrium Point (EEP). Therefore, to show that the DFE exists by setting the right hand sides of each of the equations in model (2.1) to be zero and substituting Eh = Ih = Rh = Ev = Iv = 0 to obtain (Sh, Eh, Ih, Rh, Sv, Ev, Iv) = ( Λh µ , 0, 0, 0, Λv η , 0, 0 ) (3.2) Also, for the endemic, setting the right hand sides of the equations in model (2.1) to be zero and substituting Sh = S∗ h, Eh = E∗ h, Ih = I∗h, Rh = R∗ h, Sv = S∗ v , Ev = E∗ v and Iv = I∗v to Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 93 obtain S∗ h = (Λh + γ1R ∗ h)(1 + ϵhI ∗ v ) µ(1 + ϵhI∗v ) + ξσhI∗v E∗ h = ξσhI ∗ vS ∗ h (1 + ϵhI∗v )(υ + µ) I∗h = υE∗ h + γ2R ∗ h ω + µ+ δh − ψ R∗ h = ωI∗h γ1 + µ+ γ2 S∗ v = Λv(1 + ϵvI ∗ h) η(1 + ϵvI∗h) + ξσvI∗h E∗ v = ξσvI ∗ hS ∗ v (1 + ϵvI∗h)(α + η) I∗v = αE∗ v η + δv (3.3) Therefore, eq. (3.2) has shown that we can have two equilibriums in the non-negative of R7 + thus, the existence of Disease Free Equilibrium (DFE) point E0 = ( Λh µ , 0, 0, 0, Λv η , 0, 0 ) and eq. (3.3) has shown the existence of Endemic Equilibrium point (EEP), E1 = (S∗ h, E ∗ h, I ∗ h, R ∗ h, S ∗ v , E ∗ v , I ∗ v ) 3.1. Basic Reproduction Number (R0) This section describes how to derive basic reproduction number R0 of the disease free equilibrium (DFE) using next generation matrix method. According to [18], if F denotes the rate of appearance of new infections and V denotes the rate of transfer of the DFE, then FV−1 is called ”Next Generation Matrix”. Therefore F(X) =  ξσhIvSh 1 + ϵhIv 0 ξσvIhSv 1 + ϵvIh 0 0 0 0  and V(X) =  (υ + µ)Eh (ω + µ+ δh − ψ)Ih − υEh − γ2Rh (α + η)Ev (η + δv)Iv − αEv µSh − Λh − γ1Rh (γ1 + µ+ γ2)Rh − ψIh ηSv − Λv  Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 94 A.I. ABIOYE, ET. AL Solving the above matrices using Jacobian Matrix at the disease free equilibrium, E0 =( Λh µ , 0, 0, 0, Λv µ , 0, 0 ) , we obtain F =  0 0 0 ξσhΛh µ 0 0 0 0 0 ξσvΛv η 0 0 0 0 0 0  and V =  υ + µ 0 0 0 −υ ω + µ+ δh − ψ 0 0 0 0 α + η 0 0 0 −α η + δv  respectively. Thus, [18] basic reproduction number R0 means the predominant eigenvalue equivalent to the Spectral radius of matrix FV−1. Mathematically, R0 = ρ(FV−1) where ρ is the Spectral radius. Therefore; R0 = √ υξσhΛh µ(υ + µ)(ω + µ+ δh − ψ) αξσvΛv η(α + η)(η + δv) (3.4) 3.2. Global Stability Analysis of Disease-Free Equilibrium Theorem 3.1: If R0 ≤ 1 then the disease free equilibrium E0 given by equation (3.4) is globally asymptotically stable. Otherwise, it is unstable. Proof Let us formed Lyapunov function of the type L = υEh (υ + µ)(ω + µ+ δh − ψ) + Ih (ω + µ+ δh − ψ) + ηR0Ev ξσvΛv + η(α + η)R0Iv ξσvαΛv (3.5) It is easy to identify that the coefficient of each state variables of the above eq. (3.5) are non-negative. Thus, differentiating (3.5) as well as substitution model (2.1) to obtain L̇ = υ (υ + µ)(ω + µ+ δh − ψ) ( ξσhIvSh 1 + ϵhIv − (υ + µ)Eh ) + 1 (ω + µ+ δh − ψ) (υEh − (ω + µ+ δh − ψ)Ih + γ2Rh)+ ηR0 ξσvΛv ( ξσvIhSv 1 + ϵvIh − (α + η)Ev ) + η(α + η)R0 ξσvαΛv (αEv − (η + δv)Iv) (3.6) It is assume that if ϵh = 0 then the saturated incidence ( ξσhIvSh 1 + ϵhIv ) becomes bilinear incidence (ξσhIvSh). Similarly, if ϵv = 0 then the saturated incidence ( ξσvIhSv 1 + ϵvIh ) becomes bilinear incidence (ξσvIhSv) [4, 7, 10, 14]. Epidemiologically, it is assumed that, in humans, Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 95 the rate of antibodies produced against the antigens of human is zero (i.e. ϵh = 0) and in mosquitoes, the rate of antibodies produced against the antigens of mosquitoes is zero (i.e. ϵv = 0). Thus, eq. (3.6) becomes L̇ = υ (υ + µ)(ω + µ+ δvh− ψ) (ξσhIvSh − (υ + µ)Eh) + 1 (ω + µ+ δh − ψ) (υEh − (ω + µ+ δh − ψ)Ih + γ2Rh) + ηR0 ξσvΛv (ξσvIhSv − (α + η)Ev) + η(α + η)R0 ξσvαΛv (αEv − (η + δv)Iv) L̇ = ( υξσhSh (υ + µ+ δh)(ω + µ+ δh − ψ) − η(α + η)(η + δv)R0 ξσvΛvα ) Iv + ( ηR0Sv Λv − 1 ) Ih + ( γ2 (ω + µ+ δh − ψ) ) Rh ≤ [ υξσhΛh µ(υ + µ+ δh)(ω + µ+ δh − ψ) − η(α + η)(η + δv)R0 ξσvΛvα ] Iv + (R0 − 1) Ih = [(√ υσhΛhη(α + η)(η + δv) ασvΛvµ(υ + µ+ δh)(ω + µ+ δh − ψ) ) Iv + Ih ] (R0 − 1) If L̇ ≤ 0 on condition that R0 ≤ 1 in addition to L̇ = 0 as long as R0 = 1 or Ih = 0 and Iv = 0 from the above result. It shows that the supreme invariance set in {(Sh, Eh, Ih, Rh, Sv, Ev, Iv) ∈ R7 + : L̇ = 0} is the singleton disease free equilibrium (E0). According to [9], Disease free equilibrium (E0) is globally asymptotically stable in R7 +. 3.3. Global Stability Analysis of Endemic Equilibrium Theorem 3.2: If R0 > 1, then equation (2.1) has a unique endemic equilibrium, then E1 = (S∗ h, E∗ h, I∗h, R∗ h, S∗ v , E∗ v , I∗v ) every time R0 > 1 and otherwise there is no endemic equilibrium. Proof According to [4, 7, 10, 14], we have defined in eq. (3.3), the existence of EEP as E1 = (S∗ h, E ∗ h, I ∗ h, R ∗ h, S ∗ v , E ∗ v , I ∗ v ). It is assume that if ϵh = 0 then the saturated incidence( ξσhIvSh 1 + ϵhIv ) becomes bilinear incidence (ξσhIvSh). Similarly, if ϵv = 0 then the saturated incidence ( ξσvIhSv 1 + ϵvIh ) becomes bilinear incidence (ξσvIhSv) [4,7,10,14]. Epidemiologically, it is assumed that, in humans, the rate of antibodies produced against the antigens of human is zero (i.e. ϵh = 0) and in mosquitoes, the rate of antibodies produced against the antigens of mosquitoes is zero (i.e. ϵv = 0). Therefore, eq. (3.3) becomes; S∗ h = Λh + γ1R ∗ h µ+ ξσhI∗v ; E∗ h = ξσhI ∗ vS ∗ h (υ + µ) ; I∗h = υE∗ h + γ2R ∗ h ω + µ+ δh − ψ ; R∗ h = ωI∗h γ1 + µ+ γ2 ; S∗ v = Λv η + β∗ v ; E∗ v = β∗ vS ∗ v (α + η) ; I∗v = αE∗ v η + δv (3.7) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 96 A.I. ABIOYE, ET. AL Let β∗ h = ξσhI ∗ v and β∗ v = ξσvI ∗ h (3.8) From eq. (3.7) and eq. (3.8), make I∗h subject of formula by substitution method to obtain I∗h = (υΛha6 + υγ1ωI ∗ h + a2ωγ2I ∗ h) β ∗ h + a2µωγ2I ∗ h a2a3a6(µ+ β∗ h) (3.9) where a1 = ξ2υΛhσhΛvσvα, a2 = (υ + µ), a3 = (ω + µ+ δh − ψ), a4 = (α + η), a5 = (η + δv) and a6 = (γ1 + γ2 + µ). Similarly, From eq. (3.7) and eq. (3.8), β∗ h can also be written as β∗ h = ξ2ΛvσhσvαI ∗ h (η + δv)(α + η)(η + ξσvI∗h) (3.10) Therefore, eq. (3.9) and eq. (3.10) can now be written as P1I ∗ h + P2 = 0 (3.11) where P1 = (a3η (υγ1ω + a2γ2ω − a2a3a6)R2 0 + ξυΛhσv(ωγ2 − a3a6)) and P2 = υΛhη (a3a6(R2 0 − 1) + ωγ2) From the above result, eq. (3.11) can be rewritten as I∗h = −P2 P1 ≤ 0 if P2 ≥ 0 at R0 ≤ 1, and there is no endemic equilibrium. Furthermore, I∗h = P2 P1 > 0 if P2 < 0 at R0 > 1. Thus, ∃ an endemic equilibrium only at R0 > 1. This means that eq. (2.1) has a unique endemic which is non-negative equilibrium anytime R0 > 1. Theorem 3.3: If R0 > 1, then E1 which is the endemic equilibrium defined as E1 = (S∗ h, E ∗ h, I ∗ h, R ∗ h, S ∗ v , E ∗ v , I ∗ v ) is globally asymptotically stable in the neighborhood of the region R7 +. Proof Let us construct Goh-Volterra type Lyapunov function [4, 7, 10, 14]: G =Sh − S∗ h − S∗ h ln Sh S∗ h + Eh − E∗ h − E∗ h ln Eh E∗ h + c1 ( Ih − I∗h − I∗h ln Ih I∗h ) + c2 ( Sv − S∗ v − S∗ v ln Sv S∗ v ) + c3 ( Ev − E∗ v − E∗ v ln Ev E∗ v ) + c4 ( Iv − I∗v − I∗v ln Iv I∗v ) where c1 = ξσhS ∗ hI ∗ v υE∗ h c2 = c3 = σhS ∗ hI ∗ v σvS∗ vI ∗ h and c4 = ξσhS ∗ hI ∗ v αE∗ v . The derivation of G with respect to time is; Ġ = ( 1− S∗ h Sh ) Ṡh + ( 1− E∗ h Eh ) Ėh + ξσhS ∗ hI ∗ v υE∗ h ( 1− I∗h Ih ) İh + σhS ∗ hI ∗ v σvS∗ vI ∗ h( 1− S∗ v Sv ) Ṡv + σhS ∗ hI ∗ v σvS∗ vI ∗ h ( 1− E∗ v Ev ) Ėv + ξσhS ∗ hI ∗ v αE∗ v ( 1− I∗v Iv ) İv (3.12) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 97 To simplify the eq. (3.3) and consider Rh → R∗ h as time, t→ ∞, then we have; Λh =ξσhS ∗ hI ∗ v + µS∗ h (υ+µ) = ξσhS ∗ hI ∗ v E∗ h (ω+µ+ δh − ψ) = υE∗ h I∗h Λv =ξσvS ∗ vI ∗ h + ηS∗ v (α+η) = ξσvS ∗ vI ∗ h E∗ v (η+δv) = αE∗ v I∗v (3.13) Substituting eq. (2.1) and eq. (3.13) into eq. (3.12) when (Rh → R∗ h) as time, t→ ∞ to obtain Ġ =µS∗ h ( 2− S∗ h Sh − Sh S∗ h ) + ξσhS ∗ hI ∗ v ( 1− S∗ h Sh ) + ξσhS ∗ hI ∗ v ( 1− ShIvE ∗ h S∗ hI ∗ vEh ) + ξσhS ∗ hI ∗ v ( 1− I∗hEh IhE∗ h ) + ησhS ∗ hI ∗ v σvI∗h ( 2− S∗ v Sv − Sv S∗ v ) + ξσhS ∗ hI ∗ v ( 1− S∗ v Sv ) + ξσhS ∗ hI ∗ v ( 1− SvIhE ∗ v S∗ vI ∗ hEv ) + ξσhS ∗ hI ∗ v ( 1− I∗vEv IvE∗ v ) (3.14) Thus, this can be reduced to Ġ =µS∗ h ( 2− S∗ h Sh − Sh S∗ h ) + ησhS ∗ hI ∗ v σvI∗h ( 2− S∗ v Sv − Sv S∗ v ) + ξσhS ∗ hI ∗ v( 6− S∗ h Sh − I∗hEh IhE∗ h − ShIvE ∗ h S∗ hI ∗ vEh − S∗ v Sv − I∗vEv IvE∗ v − SvIhE ∗ v S∗ vI ∗ hEv ) (3.15) Finally, if Arithmetic mean (AM) ≥ Geometric mean (GM), then the following inequalities satisfy; 2−S ∗ h Sh − Sh S∗ h ≤ 0, 2− S∗ v Sv − Sv S∗ v ≤ 0,( 6− S∗ h Sh − I∗hEh IhE∗ h − ShIvE ∗ h S∗ hI ∗ vEh − S∗ v Sv − I∗vEv IvE∗ v − SvIhE ∗ v S∗ vI ∗ hEv ) ≤ 0 (3.16) As a result of this, Ġ ≤ 0 for R0 > 1. Thus, all the parameters are positive with Ġ = 0 as long as Sh = S∗ h, Eh = E∗ h, Ih = I∗h, Sv = S∗ v , Ev = E∗ v , Iv = I∗v . In the meantime, Rh → R∗ h as time, t→ ∞ and so [9] the endemic equilibrium (E1) is globally asymptotically stable whenever R0 > 1. 3.4. Numerical Simulation We conduct a numerical simulation using the developed model to explore the dynamics of malaria within the human population. Our simulation utilizes the Runge-Kutta 4 (R-K 4) scheme integrated into Maple 2016, with parameter values specified in Table 4.2 and initial conditions as described in reference [2]. Figures 3.2(a) to 3.2(g) shows the disease Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 98 A.I. ABIOYE, ET. AL trend in each population. The results indicate an active malaria outbreak. The decreasing human classes reflect the progression of the disease within the human population, while the increasing mosquito classes suggest that mosquitoes are becoming more infectious, increasing the risk of further transmission to humans. Effective control measures, such as mosquito control and treatment of infected individuals, may be needed to mitigate the outbreak. 4. SENSITIVITY ANALYSIS 4.1. Sensitivity Analysis of Basic Reproduction Number R0 parameters This model is of absolute imperative to examine the sensitivity analysis of the parameter standards. Sensitivity analysis can be used to determine which of the parameters from basic reproduction number R0 has more influence on the disease spread. According to [7–9,15,16], sensitivity index (S.I) analysis is an essential methodology for virtually all other sensitivity analysis approaches. This is also known as elasticity index. This can be done by using partial derivatives as soon as the variable is a differentiable function of the parameter. The normalized forward sensitivity index (S.I) of a variable or quantity R0 to a parameter α is a ratio of the relative change in the variable or quantity R0 to the relative change in the parameter α. Theorem 4.1: The normalized forward sensitivity index (S.I) of a variable or quantity R0 that depends differentiable on a parameter, α, is defined as: XR0 θ = ∂R0 ∂θ × θ R0 (4.17) Sensitivity index of the parameters in eq. (3.4) is to use the formula in eq. (3.5). Therefore, the sensitivity index (S.I) for the parameters in eq. (3.4) can be written as follows: ∂R0 ∂Λh × Λh R0 = 1 2 , ∂R0 ∂Λv × Λv R0 = 1 2 ; ∂R0 ∂δh × δh R0 = − δh 2(ω + µ+ δh − ψ) ; ∂R0 ∂δv × δv R0 = − δv 2(δv + η) ; ∂R0 ∂ψ × ψ R0 = ψ 2(ω + µ+ δh − ψ) ; ∂R0 ∂υ × υ R0 = µ 2(υ + µ) ; ∂R0 ∂µ × µ R0 = 1 2 (2µ (ψ − ω − υ − δh) + υ (ψ − ω − δh)− 3µ2) (υ + µ) (ω + µ+ δh − ψ) ; ∂R0 ∂ξ × ξ R0 = 1; ∂R0 ∂σh × σh R0 = 1 2 ; ∂R0 ∂σv × σv R0 = 1 2 ; ∂R0 ∂α × α R0 = η 2(α + η) ; ∂R0 ∂ω × ω R0 = − ω 2(ω + µ+ δh − ψ) ; ∂R0 ∂η × η R0 = 1 2 ( −3 η2 − 2η (α + δv)− α δv (α + η) (δv + η) ) . (4.18) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 99 (a) The graph of Susceptible human against time (b) The graph of Exposed human against time (c) The graph of Infected human against time (d) The graph of Recovered human against time (e) The graph of Susceptible mosquito against time (f) The graph of Exposed mosquito against time (g) The graph of Infected mosquito against time Fig. 3.2. Disease dynamics Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 100 A.I. ABIOYE, ET. AL 4.2. Tables of the Model and the Sensitivity Index Table 4.1. The Model Variables and Descriptions Variables Description Sh(t) Number of susceptible humans at time t Eh(t) Number of exposed humans at time t Ih(t) Number of infectious humans at time t Rh(t) Number of recovered humans at time t Sv(t) Number of susceptible mosquitoes at time t Ev(t) Number of exposed mosquitoes at time t Iv(t) Number of infectious mosquitoes at time t Nh(t) Total number of humans population at time t Nv(t) Total number of mosquitoes population at time t Table 4.2. Parameters, Descriptions Values and Sources Parameter Description Value Source Λh Recruitment rate of humans 1.2 [2] Λv Recruitment rate of mosquitoes 0.7 [2] δh Disease-induced death rate of humans 0.068 [2] δv Disease-induced death rate of mosquitoes 0.001 [2] ψ Rate of the newborn birth with infection humans 0.003 [2] υ Developing rate of exposed humans 0.05 [2] µ Natural Death of humans 0.01146 [2] ξ Biting rate of mosquitoes 0.12 [2] α Developing rate of exposed mosquitoes 0.083 [2] η Natural death rate of mosquitoes 0.00083 [2] εh Proportion of an antibody produced by human in response to the incidence of infection caused by mosquito 1.0 [2] εv Proportion of an antibody produced by mosquito in response to the incidence of infection caused by human 1.0 [2] σh Probability of transmission of infection from an infectious mosquitoes to a susceptible humans 0.1 [2] σv Probability of transmission of infection from an infectious humans to a susceptible mosquitoes 0.09 [2] ω Recovery rate of humans 0.0035 [2] γ1 Rate of loss of immunity in humans 0.00017 [2] γ2 Relapse rate of humans 0.04 [2] 4.3. Graphical Representation The diagram below shows the graphical representation of the sensitivity analysis of each parameter present in the basic reproduction number. 4.4. Discussion of the Results Table 4.3 shows the sensitivity index of the parameters by using basic reproduction number (R0) in eq. (3.4) and at the baseline parameter values in Table 4.2. The negative sign of sensitivity index shows that, by increasing the value of the parameters, this will reduce the value of R0 but for the positive sign of sensitivity index, it indicates that as parameter increases so also basic reproduction number R0. The most sensitive parameters with respect to the transmission of malaria from Table 4.3 is the biting rate of mosquitoes (ξ) with the sensitivity index equals to 1.0000 (S.I = 1.0000). Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) GLOBAL STABILITY AND SENSITIVITY ANALYSIS OF BASIC REPRODUCTION NUMBER ... 101 Table 4.3. Parameters, Descriptions and Sensitivity indexes of R0 Parameter Description Sensitivity Index Λh Recruitment rate of humans 0.5000 Λv Recruitment rate of mosquitoes 0.5000 δh Disease-induced death of humans - 0.4252 δv Disease-induced death of mosquitoes - 0.2732 ψ Rate of the newborn birth with infection humans 0.0188 υ Exposed rate of humans to infectious class 0.0932 µ Natural Death of humans - 0.6649 ξ Biting rate of mosquitoes 1.0000 α Exposed rate of mosquitoes to infectious class 0.00495 η Natural death rate of mosquitoes - 0.7318 σh Probability of transmission of infection from an infectious mosquitoes to a susceptible humans 0.5000 σv Probability of transmission of infection from an infectious humans to a susceptible mosquitoes 0.5000 ω Recovery rate of humans -0.0219 Fig. 4.3. Sensitivity Index against parameters This shows that if there is 10% increase in the biting rate of mosquitoes, this will increase basic reproduction number (R0) by 10.00% as the respective index for the parameter is one. Next parameter to the most sensitive parameter due to the spread of malaria from Table 4.3 is the natural death rate of mosquitoes with the sensitivity index equals to - 0.7318 (S.I = - 0.7318). This shows that increasing the natural death rate of mosquitoes by 10% will result in increases in the value of R0 by 7.318%. Similarly, Table 4.3 is the natural death rate of humans with the sensitivity index equals to - 0.6649 (S.I = - 0.6649). This shows that increasing (or decreasing), the natural death rate of humans by 10% will increase or decrease R0 by 6.649%. Moreover, the recruitment rates of humans (Λh) and mosquitoes (Λv) show that probabilities of transmission of infections from infectious mosquitoes to susceptible humans (σh), and from infectious humans to susceptible mosquitoes (σv) with the sensitivity index equal to 0.5000 (S.I = 0.5000). This shows that increasing or decreasing in each of these parameters by 10% increase or decrease in R0 in each of these parameters by 5%. Similarly, Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 102 A.I. ABIOYE, ET. AL Table 4.3 is the disease-induced death rate of humans (δh ) with the sensitivity index equals to - 0.4252 (S.I = - 0.4252). This illustrates that increasing or decreasing the disease-induced death rate of humans by 10% will result in increase or decrease in R0 by 4.252%. Likewise, the disease-induced death rate of mosquitoes δv with the sensitivity index is equal to - 0.2732 (S.I = - 0.2732). This show that increasing or decreasing, disease-induced death rate of mosquitoes by 10% will increase or decrease the value of R0 by 2.732%. On the other hand, the other parameters sensitivity indices are very small which show that they have no impact on R0. 5. CONCLUSION In conclusion, this study delved into a complex mathematical model of malaria, incorporating the crucial basic reproduction number (R0) and employing Lyapunov functions to assess the global stability of disease-free and endemic equilibria. Sensitivity analysis of key parameters provided insights into their role in disease control. The findings reveal an ongoing malaria outbreak characterized by declining human populations and increasing mosquito populations, signifying elevated transmission risk. The imperative response lies in implementing robust control measures, including mosquito management and prompt treatment of infected individuals, to effectively curb the outbreak and safeguard public health. REFERENCES 1. 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Adv Syst Sci Appl (2023) Introduction Model formulation Flow Chart of the Model Differential Equations of the Model Existence of Equilibrium Points Basic Reproduction Number (R0) Global Stability Analysis of Disease-Free Equilibrium Global Stability Analysis of Endemic Equilibrium Numerical Simulation Sensitivity Analysis Sensitivity Analysis of Basic Reproduction Number R0 parameters Tables of the Model and the Sensitivity Index Graphical Representation Discussion of the Results Conclusion