J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 Journal of the Nigerian Society of Physical Sciences Stability and Sensitivity Analysis of Dengue-Malaria Co-Infection Model in Endemic Stage Solomon Akyenyi Ayubaa,∗, Imam Akeyedea, Adeyemi Sunday Olagunjua,b aDepartment of Mathematics Federal University of Lafia, Nigeria bDepartment of Mathematical Science Bingham University, Karu, Nigeria Abstract In this study, a deterministic co-infection model of dengue virus and malaria fever is proposed. The disease free equilibrium point (DFEP) and the Basic Reproduction Number is derived using the next generation matrix method. Local and global stability of DFEP are analyzed. The results show that the DFEP is locally stable if R0dm < 1 but may not be asymptotically stable. From the analysis of secondary data sourced from Kenyan region, the value of R0dm computed is 19.70 greater than unity; this implies that dengue virus and malaria fever are endemic in the region. To identify the dominant parameter for the spread and control of the diseases and their co-infection, sensitivity analysis is investigated. From the numerical simulation using Maple 17, increase in the rate of recovery for co-infected individual contributes greatly in reducing dengue and malaria infections in the region. Decreasing either dengue or malaria contact rate also play a significant role in controlling the co-infection of dengue and malaria in the population. Therefore, the center for disease control and policy makers are expected to set out preventive measures in reducing the spread of both diseases and increase the approach of recovery for the co-infected individuals. DOI:10.46481/jnsps.2021.196 Keywords: Co-infection, Dengue, Malaria, Stability Analysis, Sensitivity Analysis, Simulation. Article History : Received: 9 April 2021 Received in revised form: 30 April 2021 Accepted for publication: 5 May 2021 Published: 29 May 2021 c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. Introduction The spread of mosquitoes borne diseases has gained con- cern globally in recent decades because of their recurring out- breaks. Millions of people die every year as a result of these infectious diseases and their control has increasingly become a complex issue [1]. Dengue virus and Malaria fever are common mosquitoes-borne diseases that have become a public health threat in the last few decades with high morbidity and mortality for many patients in various part of the world [2]. The world ∗Corresponding author tel. no: +234(0)8130254488 Email address: ayubasolomona@gmail.com (Solomon Akyenyi Ayuba) malaria report [3], estimated 229 million cases of malaria in 2019 compared to 228 millions cases in 2018, with 409 000 deaths. 94% of the cases and deaths are reported from sub- Saharan Africa. Dengue is currently common in tropical and subtropical regions. The virus have four distinct stereotypes and are transmitted to human through bite of infected Aedes mosquitoes (aegyptic & albopictus) [4]. Dengue cases reported increased over 8 fold in the last two decades from 505430 cases in the year 2000 to 2.4 million in 2010 and to 4.2 million in 2019 [5]. While dengue is causing devastating impacts on the tropical and subtropical communities, malaria fever is endemic in some of these dengue affected regions there by drastically increases public health burden among the people in tropical 96 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 97 communities living at risk of contracting both diseases concur- rently. The two pathogens share similar geographical areas, and clinical distinction between them is difficult due to their over- lapping symptoms. The work in [6], the researchers proposed a mathematical model to study the transmission dynamics of Zika and Malaria in malaria-endemic area. In Ref. [7] devel- oped a novel mathematical model describing the co-infection dynamics of malaria and typhoid fever. [8] formulated A deter- ministic co-infection model between malaria and HIV in human population. Ref. [9] developed and analyzed the stability of dis- ease free equilibrium point (DFEP) of a co-infection model be- tween dengue virus and chikungunya in closed population. [10] developed a mathematical model for dengue-zika co-infection and carried out their synergistic relationship in the presence of prevention and treatment. Ref. [11] proposed a co-infection of altered vector infectivity and antibody- dependent enhancement of dengue-zika interplay. Ref. [12] formulated and analyzed a co-infection model of dengue fever and leptospirosis diseases. In [13], a deterministic model for dengue, malaria and typhoid triple co-infection was developed but limited only to the sta- bility (Local and global) analysis. The authors in Ref. [14], developed a SEIR co-infection model of dengue and malaria but only established the local and global stability. In this study, we propose a SIR-SI deterministic model of dengue virus and malaria co-infection and determine the stability anal- ysis, sensitivity analysis and carryout numerical simulation for the co-infection model. The remainder of this paper is arranged as follows: In section 2, model descriptions, flow diagram (de- picting the co-infection interactions) and the model formulation are presented. Section 3 is devoted to results and analysis; In- variant region, Disease free equilibrium point, Basic reproduc- tion number, stability analysis, parameters estimation, sensitiv- ity analysis and numerical simulation. Discussion of findings is presented in section 4. Finally, conclusions are drawn in section 5 and some possible directions for future studies are presented. 2. Model Formulation 1 The data used in this study are secondarily sourced from [7, 15]. In accordance with previous studies on mathemati- cal model of dengue virus [10, 16, 17, 15] and malaria model [19, 20, 21, 22], we formulate a SIR-SI deterministic model of dengue and malaria co-infection. In this model, the total hu- man population Nh is partitioned into seven classes; susceptible human S h, infected human with dengue virus Ihd , infected hu- man with malaria Ihm, infected human with both dengue virus and malaria Idm, recovery of infected human from dengue virus ,malaria fever and co-infected individuals are Rhd , Rhm, Rdm re- spectively. The vectors population are subdivided into; suscep- tible dengue vector S vd , dengue carrier vector Ivd , susceptible malaria vector S vm and malaria carrier vector Ivm. The recruit- ment rates for human, dengue and malaria vectors respectively, are Λh, Λd and Λm. The recovery rate from dengue and malaria 1Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model are σ,α, transmission rate of dengue and malaria vectors to hu- man per unit time are ηd, ηm, probability of dengue and malaria vectors to be infected are denoted by ηvd, ηvm respectively. Re- covered human from malaria become susceptible at γ and ac- quired immunity ρ rate. The co-infected individuals recover at the rate ψ; but those individuals either recover only from dengue and join Rhd with probability of qψ, or recover only from malaria and join Rhm with probability of ψl(1 − q), or re- cover from both diseases and join Rdm with the probability of ψ(1 − l)(1 − q). The human natural death rate denote µh while dengue and malaria vectors death rate are µd ,µm respectively. τ, δ are dengue and malaria induced death rates while φ, θ are dengue and malaria related death rates. The following assump- tions are made to formulate the co-infection model: the total population is not constant, the susceptible rates are recruited through birth or immigration and the number increases from malaria recovered and co-infectious recovered individuals by losing their temporal immunity. Recovered individuals from dengue virus is permanent. Figure 2 shown the flow diagram for the interactions between dengue and malaria co-infection model in human population. The time dependent dynamical Figure 1. Flow diagram depicting Dengue virus and Malaria co-infection dy- namics system associated with the parameters interaction is shown as follows. S ′h = Λh + γRhm + πRdm − (ηd Ivd +ηm Ivm ) Nh S h −µhS h I′hd = ηd Ivd Nh S h − ηm Ivm Nh Ihd − (σ + τ + µh + φ)Ihd I′hm = ηm Ivm Nh S h − ηd Ivd Nh Ihm − (α + ρ + δ + µh + θ)Ihm I′dm = ηm Ivm Nh Ihd + ηd Ivd Nh Ihm − (ψ + µh + θ + φ)Idm R′hd = σIhd + qψIdm −µhRhd R′hm = αIhm + ψl(1 − q)Idm − (γ + µh)Rhm R′dm = ψ(1 − l)(1 − q)Idm − (π + µh)Rdm S ′vd = Λd − ηvd (Ihd +Idm ) Nh S vd −µd S vd I′vd = ηvd Ihd Nh S vd + ηvd Idm Nh S vd −µd Ivd S ′vm = Λm − ηvm (Ihm +Idm ) Nh S vm −µmS vm I′vm = ηvm Ihm Nh S vm + ηvm Idm Nh S vm −µm Ivm (1) 97 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 98 Table 1. Parameters description of Dengue and Malaria co-infection Model Parameters Description Λh Recruitment rate of Human Population Λd Recruitment rate of Dengue Vectors Λm Recruitment rate of Malaria Vectors ρ Rate of human acquired immunity from Malaria α Rate of Human recovery from Malaria σ Rate of Human recovery from Dengue ψ Rate of human recovery from both Dengue and Malaria γ Rate of Immunity warning for Rhm to become susceptible ηd Transmission rate of Dengue vectors to human per unit time ηm Transmission rate of Malaria vectors to human per unit time ηvd Probability for Dengue Vectors to be infected ηvm Probability for Malaria parasite Vectors to be infected qψ Proportion of co-infected human recovery from Dengue only ψl(1 − q) Proportion of co-infected human recovery from Malaria only π Rate at which Rdm become susceptible τ Disease induced death rate for human infected with Dengue δ Disease induced death rate for human infected with Malaria φ Dengue related death rate θ Malaria related death rate µh Natural death rate of humans µd Natural death rate of Dengue vectors µd Natural death rate of Malaria vectors 3. Results and Analysis 3.1. Invariant regions In this section, we obtain the bounded region of solution for the dengue-malaria model. The total human population is given by Nh = S h + Ihd + Ihm + Idm + Rhd + Rhm + Rdm, then N′h = S ′ h + I ′ hd + I ′ hm + R ′ hd + I ′ dm + R ′ hm + R ′ dm (2) =⇒ N′ = Λh −µh Nh (3) Solving equation (3) as t →∞ yields Dh = {(S h, Ihd, Ihm, Idm, Rhd, Rhm, Rdm) ∈< 7; 0 ≤ N ≤ Λh µh } For the dengue vector population, if there is no spread of infec- tion, then N′d = Λd −µd Nd (4) Dd = {(S vd, Ivd ) ∈< 2; Nd ≤ Λd µd } Similarly, for malaria vector population, we obtain N′m = Λm −µm Nm (5) Dm = {(S vm, Ivm) ∈< 2; Nm ≤ Λm µm } Therefore, the feasible solution of dengue-malaria model is given by D = {(Dh × Dd × Dm)< 11 + } Thus, the solution of dengue-malaria model is bounded in D. Theorem 3.1. If at t = 0 and {S h(0), Ihd (0), Ihm(0), Idm(0), Rhd (0), Rhm(0), Rdm(0) , S vd (0), Ivd (0), S vm(0), Ivm(0)} ≥ 0, then the solution of dengue-malaria model are nonnegative at t > 0. 3.2. Existence of Disease Free Equilibrium Point 2 To investigate the condition of existence of the disease free equilibrium point and also the asymptotic behaviour of the dengue-malaria co-infection model in this section, we will in- vestigate whether the diseases die out or become endemic. This can only be addressed through the asymptotic behaviour of the diseases. This behaviour depends largely on the equilibrium point, that is time-independent solutions of the system. Since these solutions are independent of time, we set the left hand side of system (1) to zero. S ′h = I ′ hd = I ′ hm = I ′ dm = R ′ hd = R ′ hm = R′dm = 0 and S ′ vd = I ′ vd = S vm = I ′ vm = 0. 2Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model 98 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 99 Thus, the equilibrium point is given by E0dm = [S h(0), Ihd (0), Ihm(0), Idm(0), Rhd (0), Rhm(0), Rdm(0), S vd (0), Ivd (0), S vm(0), Ivm(0)] = [ Λh µh , 0, 0, 0, 0, 0, 0, Λd µd , 0, Λm µm , 0 ] (6) 3.3. Basic Reproduction Number R0dm The linear stability of the equilibrium point E0dm is established using next generation matrix method on system (1) to obtain the threshold behavior R0dm. Hence, we introduce two matrices; matrix A for rates of new infection and B is the transfer rate of in or out of a compartment. Taking the partial derivative of the right hand side of (1) at DFEP with respect to Ihd, Ihm, Idm, Ivd , Ivm, we obtain A =  0 0 0 ηdΛh µh Nh 0 0 0 0 0 ηmΛh µh Nh 0 0 0 0 0 ηvdΛd µd Nh 0 ηvdΛd µd Nh 0 0 0 ηvmΛm µm Nh ηvmΛm µm Nh 0 0  B =  −κ1 0 0 0 0 0 −DT 0 0 0 0 0 −κ2 0 0 0 0 0 −µd 0 0 0 0 0 −µm  ∴ B−1 =  − 1 κ1 0 0 0 0 0 − 1DT 0 0 0 0 0 − 1 κ2 0 0 0 0 0 − 1 µd 0 0 0 0 0 − 1 µm  where κ1 = (σ + τ + µh + φ), κ2 = (ψ + µh + θ + φ),DT = (α + ρ + δ + µh + θ) and β = ψ(1− l)(1−q) from equation (1). The basic reproduction number R0dm of dengue-malaria co-infection model is the number of secondary infections of dengue or malaria in the population due to a single dengue or malaria infective individual. The reproduction number is the spectral radius of AB−1 defined as R0dm := p(AB−1), and is given by R0dm = max  √ ηdηvdΛdΛh µhµ 2 dκ1 N 2 h , √ ηmηvmΛmΛh µ2mµh DT N 2 h  (7) 3.3.1. Local stability of disease free equilibrium point 3 The Jacobian matrix J0dm of dengue-malaria model (1) at E0dm is obtained as seen in matrix (8). −µh 0 0 0 0 γ π 0 −ηdΛh µh Nh 0 −ηmΛh µh Nh 0 −κ1 0 0 0 0 0 0 ηdΛh µh Nh 0 0 0 0 −DT 0 0 0 0 0 0 0 ηmΛh µh Nh 0 0 0 −κ2 0 0 0 0 0 0 0 0 σ 0 qψ −µh 0 0 0 0 0 0 0 0 ρ ψl(1 − q) 0 (−γ−µh ) 0 0 0 0 0 0 0 0 β 0 0 (−π−µh ) 0 0 0 0 0 −ηvdΛd µd Nh 0 −ηvdΛd µd Nh 0 0 0 −µd 0 0 0 0 ηvdΛd µd Nh 0 ηvdΛd µd Nh 0 0 0 0 −µd 0 0 0 0 −ηvmΛm µm Nh ηvmΛm µm Nh 0 0 0 0 0 −µm 0 0 0 ηvmΛm µm Nh ηvmΛm µm Nh 0 0 0 0 0 0 −µm  (8) Theorem 3.2. The disease free equilibrium E0dm is locally asymptotically stable if R0dm < 1 and unstable if R0dm > 1. 3Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model 99 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 100 Proof 3.1. . The local stability of E0dm is establish by the Jaco- bian matrix (8) at E0dm. The characteristic polynomial of J0dm is determine by det(J0dm − tI) = (−µh − t) ×(−µh − t) × (−γ−µh − t) ×(−π−µh − t)(−µd − t) ×(−µm − t) × det(Ĵ0dm − tI) = 0 where Ĵ0dm is given by Ĵ0dm =  −κ1 0 0 ηd Λh µh Nh 0 0 −DT 0 0 ηmΛh µh Nh 0 0 −κ2 0 0 ηv dΛd µd Nh 0 ηvdΛd µd Nh −µd 0 0 ηv mΛm µm Nh ηv mΛm µm Nh 0 −µm  Using the properties of determinant, we obtain det(Ĵ0dm−It) = det  −DT − t 0 ηmΛh µh Nh 0 0 0 −κ2 − t 0 0 0 ηvmΛm µh Nh ηvmΛm µh Nh −µm − t 0 0 0 ηvdΛd µh Nh 0 −µd − t ηvdΛd µh Nh 0 0 0 −ηdΛh µh Nh − t −κ1 − t  det  −DT − t 0 ηmΛh µh Nh 0 −κ2 − t 0 ηvmΛm µm Nh ηvmΛm µm Nh −µm − t  × det −µd − t −ηvdΛdµd NhηdΛh µh Nh −κ1 − t  = 0 (9) The five eigenvalues of J0dm are (−µh − t) × (−µh − t) × (−γ − µh − t)×(−µd − t)×(−µm − t) = 0 and the other five eigenvalues are obtained from the solution of matrix equation (9) by det  −DT − t 0 ηmΛh µh Nh 0 −κ2 − t 0 ηvmΛm µm Nh ηvmΛm µm Nh −µm − t  = 0 det −µd − t −ηvdΛdµd NhηdΛh µh Nh −κ1 − t  = 0 The above determinant becomes t3 − (DT + κ2 + µm)t2 − ( κ2(DT + (DT + κ2) + (1 − R20m)DT µm ) t +(κ2 − R20m)DT µm = 0 (10) t2 + (µd + κ1)t + (1 − R 2 0d )µdκ1 = 0 (11) The above eigenvalues of equation (10) and (11) are also neg- ative. Therefore, the disease free equilibrium point are locally asymptotically stable iff R0d < 1 and R0m < 1. 3.3.2. Global stability of disease free equilibrium point 4 The global asymptotic stability of the DFEP is investi- gated using Carlos Castillo-Chavez conditions as described in [23]. From the co-infection model (1), we define the time de- pendent derivatives by X′ = F(X, Z) (12) Z′ = G(X, Z), G(X, 0) = 0 (13) Where X = (S h, Rhd, Rhm, Rdm, S vd, S vm) and Z = (Ihd, Ihm, Idm, Ivd, Ivm) denote uninfected and infected populations respectively. To guarantee the global asymptotic stability, the following condi- tions must be satisfied. (a) X′ = F(X, 0); X∗ is globally stable (b) G(X, Z) = DzG(X∗, 0)Z − Ĝ(X, Z), Ĝ(X, Z) ≥ 0 ∀ X, Z ∈ Ω Theorem 3.3. The equilibrium point E0dm = (X∗, 0) of system (1) is globally asymptotically stable if R0dm ≤ 1 and the condi- tions (a), (b) are satisfied. Proof: F(X, Z) and G(X, Z) is given by F(X, Z) =  Λh + γRhm + πRdm − ηd Ivd +ηm Ivm Nh S h −µhS h σRhd + qψIdm −µhRhd ρRhm + (1 − qψ)Ihm − (γ + µh)Rhm βIdm − (π + µh)Rdm Λd − ηvd (Ihd +Idm ) Nh S vd −µd S vd Λm − ηvm (Ihm +Idm ) Nh S vm −µmS vm  G(X, Z) =  ηd Ivd Nh S h − ηm Ivm Nh Ihd − (σ + τ + µh + φ)Ihd ηm Ivm Nh S h − ηd Ivd Nh Ihm − (α + ρ + δ + µh + θ)Ihm ηm Ivm Nh Ihd + ηd Ivd Nh Ihm − (ψ + µh + θ + φ)Idm ηvd Ihd Nh S vd + ηvd Idm Nh S vd −µd Ivd ηvm Ihm Nh S vm + ηvm Idm Nh S vm −µm Ivm  For X′ = F(X, 0), system (1) is reduced to X′ =  S ′h = Λh + πRdm + γRhm −µhS h S vd′ = Λd −µd S vd S vm = Λm −µmS vm with X∗ = ( Λh µh , Λd µd , Λm µm ) (14) Given G(X, Z) = DzG(X∗, 0)Z − Ĝ(X, Z), Ĝ(X, Z) ≥ 0 G(X∗, 0) =  −κ1 0 0 ηdΛh µh Nh 0 0 −DT 0 0 ηmΛh µh Nh 0 0 −κ2 0 0 ηvdΛd µd Nh 0 ηvdΛd µd Nh −µd 0 0 ηvmΛm µm Nh ηvmΛm µm Nh 0 −µm  (15) 4Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model 100 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 101 DzG(X ∗, 0)Z =  −κ1 0 0 ηdΛh µh Nh 0 0 −DT 0 0 ηmΛh µh Nh 0 0 −κ2 0 0 ηvdΛd µd Nh 0 ηvdΛd µd Nh −µd 0 0 ηvmΛm µm Nh ηvmΛm µm Nh 0 −µm  ×  Ihd Ihm Idm Ivd Ivm  =  −κ1 Ihd + ηdΛh µh Nh Ivd −DT Ihm + ηmΛh µh Nh Ivm −κ2 Idm ηvdΛd µd Nh Ihd + ηvdΛd µd Nh Idm −µd Ivd ηvmΛm µm Nh Ihm + ηvmΛm µm Nh Idm −µd Ivm  Ĝ(X, Z) =  Ĝ1(X, Z) Ĝ2(X, Z) Ĝ3(X, Z) Ĝ4(X, Z) Ĝ5(X, Z)  =  ηd Ivd Nh ( Λh µh − S h) + ηm Ivm Ihd Nh ηm Ivm Nh ( Λh µh − S h) + ηd Ivd Ihm Nh − ( ηm Ivm Ihd Nh + ηd Ivd Ihm Nh ) ηvd Nh ( Λd µd − S vd )(Ihd + Idm) ηvm Nh ( Λm µm − S vm)(Ihm + Idm)  (16) Since Ĝ3(X, Z) < 0 in equation (16) and condition (b) re- quires Ĝ(X, Z) ≥ 0. Hence, condition (b) is not met as Ĝ(X, Z) < 0 for all X, Z ∈ Ω. Thus, it implies that the DEFP may not be globally asymptotically stable if R0dm < 1. Therefore, the en- demic equilibrium exist with DFEP if Rodm < 1. Whence, we can deduced that the dengue-malaria model exhibits backward bifurcation when the basic reproduction number R0dm = 1. 3.4. Parameters Estimation and Sensitivity Analysis 3.4.1. Parameters estimation and initial value 5 The parameters in Table 2 are obtained (or estimated) in line with the work of [7, 15], from Kenyan region where malaria and dengue virus are said to be endemic. Conservatively, the following initial values are estimated. The total human pop- ulation is estimated to be 52, 000, 000 and the susceptible hu- man are assumed to be 25, 000, 000 which is about half of the population at the onset of the diseases. For vectors population, 10, 000, 000 is assumed to be susceptible malaria mosquitoes with 2, 000, 000 malaria carrier mosquitoes. Dengue suscep- tible mosquitoes are estimated to 5, 000, 000 and 100, 000 for dengue carrier mosquitoes. Therefore, the initial infected hu- man with malaria is estimated to be 10, 000 and infected human with dengue estimate is 5000. 3.5. Sensitivity analysis of the model In order to identify the dominant parameter for the spread and control of dengue and malaria infections in the population, we performed the sensitivity analysis. As described in Carlos 5Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model Table 2. Parameters values of dengue-malaria co-infection model Parameter Value/day Source Λh 467 [7] µh 0.00004 calculated Λd 221056.75 estimated ηd 0.000451 estimated ηvd 0.13502 estimated σ 0.035 estimated π 0.003 estimated τ 0.0245 estimated φ 0.00023 estimated µd 0.00005 calculated ηm 0.000408 [7] ηvm 0.15096 [7] γ 0.06 [0,1] [7] α 0.038 [7] ρ 0.37 [7, 15] δ 0.0019 [7] θ 0.00025 estimated µm 0.00005 calculated Castillo-Chavez [23], the sensitivity index of R0dm with a pa- rameter say β is expressed as Υ R0dm β = ∂Rodm ∂β × β R0dm (17) Since Rodm is defined by R0dm = { √ ηdηvdΛdΛh µhµ 2 dκ1 N 2 h , √ ηmηvmΛmΛh µ2mµh DT N 2 h } Therefore, we evaluate the sensitivity index of R0d and Rom sep- arately as follows: 6 Υ R0d ηd = ∂R0d ∂ηd × ηd R0d = 1 2 > 0 Υ R0d ηvd = ∂R0d ∂ηvd × ηvd R0d = 1 2 > 0 Υ R0d σ = ∂R0d ∂σ × σ R0d = − σ 2κ1 < 0 Υ R0d τ = ∂R0d ∂τ × τ R0d = − τ 2κ1 < 0 Υ R0d φ = ∂R0d ∂φ × φ R0d = − φ 2κ1 < 0 Υ R0d µd = ∂R0d ∂µd × µd R0d = −1 < 0 Υ R0d µh = ∂R0d ∂µh × µh R0d = − σ + τ + 2µh + φ 2κ1 < 0 6Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model 101 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 102 Υ R0m ηm = ∂R0m ∂ηm × ηm R0m = 1 2 > 0 Υ R0m ηvm = ∂R0m ∂ηvm × ηvm R0m = 1 2 > 0 Υ R0m α = ∂R0m ∂α × α R0m = − α 2DT < 0 Υ R0m ρ = ∂R0m ∂ρ × ρ R0m = − ρ 2DT < 0 Υ R0m δ = ∂R0m ∂δ × δ R0m = − δ 2DT < 0 Υ R0m θ = ∂R0m ∂θ × θ R0m = − θ 2DT < 0 Υ R0m µm = ∂R0m ∂µm × µm R0m = − α + ρ + δ + 2µm + θ 2DT < 0 The parameters with positive sensitivity indices are ηd,ηvd,ηm,ηvm and the negative indices includes σ,τ,φ,µd,α,ρ,δ,θ,µm. The positive sign parameters have great influence in the spread of the diseases and their co-infection in the region. Whereas, the parameters with negative sign have potential influence on the control of the spread of dengue, malaria and their co-infection. Hence, the center for disease control is expected to make poli- cies and control measures in this regard to combat dengue, malaria and their co-infection in an endemic region. 3.6. Numerical Simulations 3.6.1. Effect of malaria recovery rate (α) on infectious (Ihm) population As seen in Figure 2, it is shown that α plays a significant influence in decreasing malaria infection. When the value of α increases from 0.038 to 1, the infectious population due to malaria decreased, where the contact rate ηm is kept constant. Figure 2. Effect of malaria recovery rate on infectious population 3.6.2. Effect of dengue recovery rate (σ) on infectious (Ihd ) population In Figure 3, as the value of σ varies from 0.035 to 0.99, the number of dengue infection decreases when the contact rate ηd is kept constant. Hence, this can be use by policy makers to combat the disease. Figure 3. Effect of dengue recovery rate on infectious population 3.6.3. Effect of dengue contact rate (ηd ) on co-infectious (Idm) population In Figure 4, the contact rate of dengue ηd varies from 0.000451 to 0.040451, the number of co-infectious population increases as the recovery rate is kept constant. Thus, the center for dis- ease control and policy makers are expected to apply vector control measures and mechanism to reduce the expansion of co-infection in the region. Figure 4. Effect of dengue contact rate on co-infectious population 3.6.4. Effect of dengue-malaria recovery rate (ψ) on co-infectious (Idm) population The recovery rate described in dengue-malaria model is ei- ther the individual recovery from dengue only, recovery from malaria only or both dengue and malaria infections. As shown in Figure 5, increasing ψ play a significant role in reducing both dengue and malaria infections in the region. 102 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 103 Table 3. Parameters value and sensitivity indices Parameter Sensitivity indice Sensitivity index <0d Basic reproduction number of dengue - µh -ve -0.001787 ηd +ve +0.5 ηvd +ve +0.5 σ -ve -0.001046 τ -ve -0.000732 φ -ve -0.000007 µd -ve -1 <0m Basic reproduction number of malaria - ηm +ve +0.5 ηvm +ve +0.5 α -ve -0.007794 ρ -ve 0.075885 δ -ve -0.00049 θ -ve -0.00005 µm -ve -1 Figure 5. Effect of recovery rate on co-infection population 4. Discussion In this paper, we develop a deterministic mathematical model that studies the dynamics of dengue virus and malaria fever in an endemic stage. Base on the qualitative and numerical anal- ysis of the data sourced from [7, 15] with conservative esti- mates, the results depict some interesting insights into the un- derlying relationship between dengue virus and malaria fever and provide information that are useful to combat the diseases. The qualitatively analysis of the model shows that there is a bounded invariant region where the model is mathematical and epidemiological well posed. The basic reproduction number of the model was derived using the next generation matrix method. Stability and sensitivity analysis of the disease free equilibrium point (DFEP) were established. The result shows that the DFEP is locally stable if R0dm < 1 but may not be asymptotically sta- ble. Therefore, the endemic equilibrium exist when Rodm < 1 with DFEP and this implies that the model undergoes backward bifurcation. We demonstrated numerically using Maple 17 , the effects of basic parameters for the spread and control of dengue and malaria co-infection. From the results, we conclude that an increase in dengue and malaria recovery rates plays a great role in reducing dengue and malaria infections respectively, in the region. Similarly, the recovery rate for co-infectious individu- als also contributes greatly to reducing the co-infection in the population if its value increases as seen in Figure 5. Another findings obtained is that, increasing dengue vectors contact rate has a great influence on spreading the co-infection in the pop- ulation. We computed the R0dm = 19.70 > 1, indicating that dengue virus and malaria fever are endemic in the area. Thus, we recommend that center for disease control set out preventive measures in reducing the spread of both diseases and increase the measures on recovery co-infected individuals. 5. Conclusion and Recommendation As demonstrated in this study, the co-infection between dengue virus and malaria fever may have devastating impacts in the tropical/subtropical communities. The model helps in iden- tifying distinct features and underlying relationships between dengue and malaria co-infection. This will be of help to policy makers to devise strategies for controlling the diseases. For fu- ture studies, we recommend a formulation with optimal control parameters to determine the strategies for mitigating the spread and control of dengue and malaria co-infection. Data and materials. The data used for this co-infection model are from previous articles published. Acknowledgment The authors are grateful for the data gathered from the previ- ously published articles. 103 Ayuba et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 96–104 104 References [1] P. Yongzhen, L. Shgaoying, L. Shuping, & L Changguo, “A delay seiqr epidemic model with impulse vaccination and the quarantine measure”, Computers and Mathematics with Applications 58 (2009) 135. [2] W. Viroj, “Concurrent malaria and dengue infection: a brief summary and comment”, Asian Pacific Journal of Tropical Biomedicine 1 (2011) 326. [3] World Health Orginization. Fact sheets/detail/malaria report. 2020. [4] N. F. B. Simo, J. J. Bigna, S. Kenmoe, S. M. Ndangang, E. Temfack, P. F. Moundipa, & M. Demanou, “Dengue virus infection in people residing in africa: a systematic review and meta-analysis of prevalence studies”, Scientific Reports: Nature Research 9 (2019) 3626. [5] World Health Organization. Dengue cases report. 2019. [6] J. A. Mensah, K. I. Dontwi, & E. Bonyah, “Stability analysis of multi- infections (malaria, zikka dna elephantiasis)”, Journal of Advances in Mathematics and Computer Science 30 (2018) 2. [7] M. J. Mutua, F. B. Wang, & K N. Vaidya, “Modeling malaria and typhoid fever co-infection dynamics”, Journal of Mathematical Bioscience 264 (2015) 128. [8] A. B. Gumel, Z. Mukandavire, W. Garira, & J. M. Tchuenche, “Mathe- matical analysis of a model for hiv-malaria co-infection”, Mathematical Biosciences and Engineering 6 (2009) 333. doi:10.3934/mbe.2009.6.333 [9] D. Aldila & M. Agustin, “A mathematical model of dengue-chikungunya co-infection in aclosed population”, Journal of physics: Conf. Series 974 (2018) 012001. [10] E. Bonyah, M. A. Khan, K. O. Okosun, & J. F. G. Aguilar, “On the co- infection of dengue fever and zika virus”, Optimal and Control App Meth 40 (2018) 394. [11] O. Olawoyin & C. Kribs, “Co-infection, altered vector infectivity, and antibody-dependent enhancement: The dengue-zika interplay”, Society for mathematical biology 83 (2020) 13. [12] H. T. Alemmeh, “A co-infection model of dengue and leptospirosis dis- eases”, Advances in Difference Equations 2020 (2020) 664. [13] T. J. Oluwafemi, E. Azuaba, & Y. M. Kura, “Stablity analysis of disease free equilibrium of malaria, dengue and typhoid triple infection model”, Asian Research Journal of Mathematics, 16 (2020) 11. [14] T. J. Oluwafemi, N. I. Akinwande, R. O. Olayiwola, & A. F. Akuta, “Co-infection model formulation to evaluate the transmission dynamics of malaria and dengue fever virus”, J. Appl. Sci. Environ. Manage 24 (2020) 7. [15] O. Gabriel, K. J. Koske, & M. MutisoJohn, “Transmission dynamics and optimal control of malaria in kenya”, Discrete Dynamis in Nature and Society 2016 (2016) 1. [16] A. Aurelio delos Reyes & J. M. L. Escaner. “Dengue in the philippines: model and analysis of parameters affecting transmission”, Journal of Bi- ological Dynamics 12 (2018) 894. [17] M. Derouich & A. Boutayeb, “Dengue fever: Mathematical modelling and computer simulation”, Applied Mathematics and Computation 177 (2006) 528. [18] S. Syafruddin & N. Lyapunov, “function of sir and seir model for trans- mission of dengue fever disease”, Int. J. Simulation and Process Mod- elling, 8 (2013) 177. [19] U. D. P. Fatmawati & J. Nainggolan, “Parameter estimation and sensi- tivity analysis of malaria model”, Journal of physics Conference Series DOI: 10.1088/1742- 6596/1490/012039 1490 (2020) 012039. [20] S. Olaniyi & O. S. Obabiyi, “Mathematical model for malaria transmis- sion dynamics in human and mosquito populations with nonlinear forces of infection”, International Journal of Pure and Applied Mathematics 88 (2013) 125. [21] Y. Xing, Z. Guo, & J. Liu, “Backward bifurcation in malaria transmission model”, Journal of Biological Dynamics 1 (2020) 14. [22] H. Yin, C. Yang, & X. Zhang, “Dynamics of malaria transmission model with sterile mosquitoes”, J. Bio. Dny. 12 (2018) 577. [23] C. C. Chavez, Z. Feng, & W. Huang, “On the computation of r and its role on global stability”, //www.researchgate.net/profile/Carlos Castillo- chavez2/publication/228915276, Biometric Unit Technical Report M- 1553 (2001) 1. 104