Stability and Sensitivity Analysis of Dengue-Malaria Co-Infection Model in Endemic Stage

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


Introduction
The spread of mosquitoes borne diseases has gained concern globally in recent decades because of their recurring outbreaks. 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 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 communities living at risk of contracting both diseases concurrently. The two pathogens share similar geographical areas, and clinical distinction between them is difficult due to their overlapping 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] developed a novel mathematical model describing the co-infection dynamics of malaria and typhoid fever. [8] formulated A deterministic co-infection model between malaria and HIV in human population. Ref. [9] developed and analyzed the stability of disease free equilibrium point (DFEP) of a co-infection model between 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 stability (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 analysis, 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 (depicting the co-infection interactions) and the model formulation are presented. Section 3 is devoted to results and analysis; Invariant region, Disease free equilibrium point, Basic reproduction number, stability analysis, parameters estimation, sensitivity 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.

Model Formulation
1 The data used in this study are secondarily sourced from [7,15]. In accordance with previous studies on mathematical 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 human population N h is partitioned into seven classes; susceptible human S h , infected human with dengue virus I hd , infected human with malaria I hm , infected human with both dengue virus and malaria I dm , recovery of infected human from dengue virus ,malaria fever and co-infected individuals are R hd , R hm , R dm respectively. The vectors population are subdivided into; susceptible dengue vector S vd , dengue carrier vector I vd , susceptible malaria vector S vm and malaria carrier vector I vm . The recruitment rates for human, dengue and malaria vectors respectively, are Λ h , Λ d and Λ m . The recovery rate from dengue and malaria 1 Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model are σ , α, transmission rate of dengue and malaria vectors to human per unit time are η d , η m , probability of dengue and malaria vectors to be infected are denoted by η vd , η vm respectively. Recovered human from malaria become susceptible at γ and acquired immunity ρ rate. The co-infected individuals recover at the rate ψ; but those individuals either recover only from dengue and join R hd with probability of qψ, or recover only from malaria and join R hm with probability of ψl(1 − q), or recover from both diseases and join R dm with the probability of ψ(1 − l)(1 − q). The human natural death rate denote µ h while dengue and malaria vectors death rate are µ d , µ m respectively. τ, δ are dengue and malaria induced death rates while φ, θ are dengue and malaria related death rates. The following assumptions 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

Invariant regions
In this section, we obtain the bounded region of solution for the dengue-malaria model. The total human population is given by N h = S h + I hd + I hm + I dm + R hd + R hm + R dm , then Solving equation (3) as t → ∞ yields For the dengue vector population, if there is no spread of infection, then Similarly, for malaria vector population, we obtain Therefore, the feasible solution of dengue-malaria model is given by then the solution of dengue-malaria model are nonnegative at t > 0.

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 investigate 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.
Thus, the equilibrium point is given by

Basic Reproduction Number R 0dm
The linear stability of the equilibrium point E 0dm is established using next generation matrix method on system (1) to obtain the threshold behavior R 0dm . 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 I hd , I hm , I dm , I vd , I vm , we obtain (1). The basic reproduction number R 0dm 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 R 0dm := p(AB −1 ), and is given by 3.3.1. Local stability of disease free equilibrium point 3 The Jacobian matrix J 0dm of dengue-malaria model (1) at E 0dm is obtained as seen in matrix (8).
Theorem 3.2. The disease free equilibrium E 0 dm is locally asymptotically stable if R 0dm < 1 and unstable if R 0dm > 1. 3 Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model Proof 3.1. . The local stability of E 0dm is establish by the Jacobian matrix (8) at E 0dm . The characteristic polynomial of J 0dm is determine by whereĴ 0dm is given bŷ Using the properties of determinant, we obtain The above determinant becomes The above eigenvalues of equation (10) and (11) are also negative. Therefore, the disease free equilibrium point are locally asymptotically stable iff R 0d < 1 and R 0m < 1.

3.3.2.
Global stability of disease free equilibrium point 4 The global asymptotic stability of the DFEP is investigated using Carlos Castillo-Chavez conditions as described in [23]. From the co-infection model (1), we define the time dependent derivatives by Where X = (S h , R hd , R hm , R dm , S vd , S vm ) and Z = (I hd , I hm , I dm , I vd , I vm ) denote uninfected and infected populations respectively. To guarantee the global asymptotic stability, the following conditions must be satisfied.
Proof: F(X, Z) and G(X, Z) is given by Given G(X, Z) = D z G(X * , 0)Z −Ĝ(X, Z),Ĝ(X, Z) ≥ 0 4 Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model SinceĜ 3 (X, Z) < 0 in equation (16) and condition (b) re-quiresĜ(X, Z) ≥ 0. Hence, condition (b) is not met asĜ(X, Z) < 0 for all X, Z ∈ Ω. Thus, it implies that the DEFP may not be globally asymptotically stable if R 0dm < 1. Therefore, the endemic equilibrium exist with DFEP if R odm < 1. Whence, we can deduced that the dengue-malaria model exhibits backward bifurcation when the basic reproduction number R 0dm = 1. 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 population is estimated to be 52, 000, 000 and the susceptible human 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 susceptible mosquitoes are estimated to 5, 000, 000 and 100, 000 for dengue carrier mosquitoes. Therefore, the initial infected human with malaria is estimated to be 10, 000 and infected human with dengue estimate is 5000.

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 5 Stability and Sensitivity Analysis of Dengue-Malaria Co-infection Model  [23], the sensitivity index of R 0dm with a parameter say β is expressed as Since R odm is defined by Therefore, we evaluate the sensitivity index of R 0d and R om separately as follows: 6 The parameters with positive sensitivity indices are η d , η vd , η m , η vm and the negative indices includes σ, τ, φ, µ d , α, ρ, δ, θ, µ m . 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 policies 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 (I hm ) 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 η m is kept constant.  population In Figure 4, the contact rate of dengue η d 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 disease control and policy makers are expected to apply vector control measures and mechanism to reduce the expansion of co-infection in the region. 3.6.4. Effect of dengue-malaria recovery rate (ψ) on co-infectious (I dm ) population The recovery rate described in dengue-malaria model is either 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.

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 analysis of the data sourced from [7,15] with conservative estimates, the results depict some interesting insights into the underlying 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 R 0dm < 1 but may not be asymptotically stable. Therefore, the endemic equilibrium exist when R odm < 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 individuals 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 population. We computed the R 0dm = 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.

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 identifying 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 future 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.