Fuzzy Global Stability Analysis of the Dynamics of Malaria with Fuzzy Transmission and Recovery Rates

In this paper, fuzzy techniques have been used to track the problem of malaria transmission dynamics. The fuzzy equilibrium of the proposed model was discussed for different amounts of parasites in the body. We proved that when the amounts of parasites are less than the minimum amounts required for disease transmission ( min ν ν≤ ), we reach the model disease-free equilibrium. Using Choquet integral, the fuzzy basic reproduction number through the expected value of fuzzy variable was introduced for the fuzzy Susceptible, Exposed, Infected, Recovered, susceptible-Susceptible, Exposed and Infected (SEIRS-SEI) malaria model. The fuzzy global stabilities were introduced and discussed. The disease-free equilibrium 0 Y is globally asymptotically stable if min ν ν≤ or if the basic reproduction number is less than one ( ( ) 0 1 ν ≤  ). When min ν ν> and ( ) 0 1 ν >  , there exists a co-existing endemic equilibrium which is globally asymptotically stable in the interior of feasible set Ω . Finally, the numerical simulation has been done for showing the effectiveness of our


Introduction
Malaria, a mosquito-borne disease, is one of the oldest diseases studied in mathematical approaches. It is one the public health problem across the world. In 2020, WHO observed a real decreasing of malaria cases. The total number of malaria cases has fallen from 238 million in 2000 to 229 million in 2019 [1].
During that period, the population of Sub-Saharan Africa, which comprises more than 90% of the global burden of malaria had increased from 665 million to over a billion people [1]. Malaria is transmitted during a bloodmeal by an infected female anophele mosquito. Malaria is developed in two stages [2]. Firstly in mosquitoes (vectors), the parasite enters during a bloodmeal of a susceptible mosquito to an infected human. After the parasite is developed, the mosquito becomes contagious between 10 to 16 days [3]. Secondly in humans (hosts), the parasite enters during a bloodmeal of an infected mosquito to susceptible human.
The mathematical study of infectious diseases is important and a more realistic tool for a better understanding of their evolution, stability, control and for identifying the influential parameters in the spread of the disease [4] [5]. Since the first mathematical model of malaria in 1910 with Ronald Ross [6], many studies have been done in malaria modeling. Some of them included the environmental, climatic, seasonal and periodic aspects [7] [8] [9] [10] [11]. Others are interesting in vertical transmission and relapse [12] [13] [14].
However, most of these models used a deterministic approach with constant parameters. For example, they assumed that the contact transmission and recovery rates are constant. These assumptions are not realistic in the context of malaria transmission. The parameters such as contact and recovery rates are uncertain in reality. They depend on the amount of parasites (parasitic viral load) in the blood cells [15] [16]. In addition, the classical deterministic mathematical models which describe malaria do not incorporate the high degree of subjectivity involved in modeling. In order to describe the transmission dynamics of an infectious disease (malaria for this study) in a more realistic way, several models that incorporate some subjectivity and heterogeneity by the use of fuzzy theory introduced by Zadeh [17], have been studied and proposed. L. C. de Barros et al. in 2003 studied an SI epidemiological model with a fuzzy transmission parameter. In this model, they consider different degrees of infectivity of contact rate. Madhu Jain et al. [18] studied the transmission dynamics of malaria in human host using a neuro-fuzzy approach. Hassan Zarei et al. [19] used fuzzy theory in the modeling and control of HIV infection. They proposed a fuzzy mathematical model of HIV infection consisting of a linear fuzzy differential equations (FDEs) system describing the ambiguous immune cells level and the viral load which are due to the intrinsic fuzziness. Mondal et al. [20] studied the SIS epidemiological model by considering the disease transmission parameter and treatment control parameter as fuzzy number. Belinda O. Emokhare and Emadomi M. Igbape [21] used a fuzzy logic approach of Ebola Hemorrhagic Fever. Renu Verma et al. [22] proposed a fuzzy SIR epidemic model. In their work, they assumed that the transmission rate, the recovery and disease-induced rate are fuzzy numbers. More recently in 2020, G. Bhuju et al. used a fuzzy approach to track the problem of Dengue transmission dynamics in Nepal using the SEIR-SEI scheme. In their work, they assumed that the transmission and the recovery rates

Preliminaries
Let X be a nonempty set. A fuzzy set ξ of X is defined as a set of all pairs  [26]. The fuzzy set ξ will be a fuzzy number if: • X is a set of real numbers and there exists at least 0 x ∈  such that ( ) • µ is upper semi-continuous on  and the support of ξ is a compact set, • ξ is a convex fuzzy set. That is, Note that the credibility measure is the average of possibility and necessity measures introduced by L. Zadeh [27], defined respectively by: and None of these two measures is self-dual. That reason justified the introduction of the credibility measure by B. Liu and Y.-K. Liu [23] [28] [29]. The self-duality property is very important in order to control the behavior of a system. From here a fuzzy number can be considered as a particular case of a fuzzy variable. In order to use the defuzzification process to obtain some crisps values (real numbers) which represents a fuzzy number, we use the Expected Value of Fuzzy Variable introduced by B. Liu provided that at least one of these integrals is finite. The expected value of a trapezoidal fuzzy variable denoted by and the expected value of a triangular fuzzy variable denoted by ξ is a real, in this paper it will be used to defuzzify some of the fuzzy parameters which will be used.

The Fuzzy SEIRS-SEI Model for Malaria Transmission
In this model, we use two classical schemes known in the Literature. For human population we use the classical SEIRS scheme and for mosquitoes population, we use the SEI scheme. The model describes fuzzy interaction between individuals (humans and mosquitoes) in these two populations.
We consider the different degrees of susceptibility and infectivity of malaria among populations. Considering the heterogeneity of population, in this work, we assume that the contact transmission rate h β (chance that in one contact between a susceptible human and an infected mosquito results to an infection of susceptible human) and recovery rate h γ are fuzzy variables. To describe the amount of parasites (parasitic viral load) ν in the transmission rate we use the membership function introduced firstly by L.C. de Barros [5] and used by G. Bhuju et al. [4] and Renu Verma et al. [22] defined as: where 0 0 1 γ < < is the human recovery rate when we reach a maximum amount of parasites in the blood cells.
The amount of parasites is different to each group of humans. Therefore, making the model more realistic, we consider only the human individuals in a given group V. With classification (weak, medium and strong) given by an expert, V can be seen as a linguistic variable with membership function ( ) ν Γ defined by L. C. Barros [5] as: The parameter ν is a central value and δ gives the dispersion of each one of the fuzzy sets assumed by V. Its graphic representation is given on Figure 2 which has a triangular sharp.
The fuzzy incidence rate of mosquitoes to human is given by   Figure 3 is given by the following seven ordered system of ordinary differential equations (ODEs) with fuzzy parameters.
The system (10)- (16) is solved under the non negative initial conditions: The total population of human at each time is given by: and the total population of mosquitoes at each time is managed by: We assume equal the recruitment and natural death rates of human population at the rate µ . The recruitment and natural death rates of mosquitoes are also assumed to be equal to m µ . Parameters h σ and m σ represent the latent rates of both humans and mosquitoes respectively. Parameters δ and m δ denote the malaria-induced death rate and the parasite-induced death rate respectively. Below, we prove the well-posedness of model, we prove the existence, non negativity and the boundness of solution of the system (10)- (16 , , , , , , g Y t g g g g g g g = , where: Theorem 1. Under the initial condition given in (17), the system (10)- (16) Proof. All functions i g of system (21) are continuous functions on Ω .
Hence, the vector function g is differentiable. This means that all partial derivatives exist and are continuous on Ω . Therefore, by the existence and uniqueness theorem [30], there exists a unique solution of system (21) for the initial condition (17). Additionally, suppose that ( ) Y t is a solution of system (10) (10)-(16) is well-posed and biologically meaningful. Finally, the solution ( ) Y t of system (10)-(16) is bounded because the limit to infinity of the total humans population is less than the initial human population, that is and the limit to infinity of the total mosquitoes population is less than the initial mosquitoes population, that is

Analytical Analysis
In this section, we introduce the fuzzy equilibrium of a fuzzy model. We introduce and compute the fuzzy basic reproduction number of fuzzy SEIRS-SEI model for malaria using the expected value of a fuzzy variable introduced by Baoding Liu and Y.-K. Liu [23] and finally we introduce the fuzzy global stabilities of the fuzzy SEIRS-SEI model for malaria.

Fuzzy Equilibrium of the Model
Theorem 2. Let given the system (10)-(16) with initial conditions (17) 1  1  1  1  1  1  1 , , , , , , , The second one is given and noted by: From Equation (25), we have: From Equation (24), we obtain: Putting Equation (29) into Equation (22), we get: From Equation (26) The substitution of Equation (32) into (33) gives: The substitution of Equation (35) into (34) gives: Putting Equation (36) into Equation (31) gives: The equilibrium points are expressed as implicit functions in terms of infectious humans h I , where:  36) and (37), we obtain respectively: . Therefore, we obtain: In this case, the equilibrium points are given by: where the vector's components are given by the implicit-solutions functions (37), (30), (38), (29), (32), (35) and (36) We obtain the equilibrium point The equilibrium points obtained in cases 2 and 3 are called "endemic equilibriums", they occur when malaria persists in the population. Thus, when the amount of parasites is greater than the minimum amount required, malaria occurs and persists in the population. 

Fuzzy Basic Reproduction Number
The basic reproduction number is given by: denotes the expected value of a fuzzy number defined in Formula (6).

Fuzzy Global Stability of the Disease-Free Equilibrium
Equating Equations (47) . Therefore, by Theorem 2.1 of [24], the Lyapunov function L is constructed as follows: is a Lyapunov function for the system (10)- (16). The differentiation of L with respect to t gives:    (3), we use the classical Lyapunov function theory to derive this fuzzy global asymptotic stability. We use the classical graph-theoretical method introduced by Zhisheng Shuai and Pauline van den Driessche [24]. For the construction of a Lyapunov function, let define:   17 14 14 ln ln

Fuzzy Global Stability of the Endemic Equilibrium
respectively. Therefore, the weighted associated digraph ( ) , G A is shown on Figure 4.
From this digraph, the first condition of Theorem 3.5 of [24] is satisfied, but no its second one. Thus, we cannot formulate a Lyapunov function of system (10)- (16) Hence, the largest invariant set for system (10)

Numeric Results and Discussion
In this section, we have done some numerical simulations to confirm our analytical analysis. . The malaria reported data of DRC, estimates the malaria-induced death rate to be 285 × 10 −5 [33]. The fuzzy parameters ( ) h β ν and ( ) h γ ν depend on the amount of parasites ν . The amount of parasites (parasitic viral load) in the blood is determined and expressed as "para-sites per microliter (μl) of blood" from the parasite density formula (DP) [16] below:

× =
The minimum amount of parasites in the blood for malaria transmission is estimated under 1000 parasites/μl, the medium amount is estimated between 1000 to 10,000 parasites/μl and the high parasitemia is estimated to over 10,000 parasites/μl [16]. In this paper, for a particular group of individuals chosen, we take  [12]. From there, the parasite-induced death rate should be less than m µ , we therefore assumed that m δ is in order of 10 −6 .
When the maximum amounts of parasites are reached in blood cells, the recover becomes uncertain and can be assumed in order 10 −3 . Due to the fact that all recovered individuals form malaria become susceptible, we assume the progression rate h θ equal to 0.138. Table 1  Situation on Figure 6 occurs when the amount of parasites ν is less than the minimum amount required for malaria transmission min ν . This assures the δ malaria-induced death rate 285 × 10 −5 [33] h σ latent rate of humans 0.0588 [35] m β contact transmission rate of humans to mosquitoes 0.03525 [35] global stability of the disease-free equilibrium of the model (See Theorem (3)). Figure 7 and Figure 8 show the behaviors of the model when the amount of parasites ν is greater than the minimum amount required for malaria transmis- ). In addition, the basic reproduction number is greater than the unity in either case. These confirm the assumptions of Theorem (4). Either case, the endemic equilibrium is reached. See Theorem (4). Figure 9 shows the trends of susceptible humans for different amounts of parasites ν . When the amounts of parasites in the body are less than the minimum amounts required, the population of susceptible humans is almost unchanged. When the amounts of parasites are greater than the minimum amounts required, the population of susceptible humans is decreasing. Figure 6. Distribution of individuals in each compartment for the parameter values given in Table 1     When the amounts of parasites approaches the maximum amount, the susceptible humans population goes to zero; which is normal because the transmission rate is approaching one. The case of exposed humans is quite different, (see Figure 10). Figure 10 shows the behavior of exposed humans for different amounts of parasites ν . When the amounts of parasites is less than the minimum amounts required min ν , which implies ( ) 0 h β ν = , the population of exposed humans goes to zero. That means there is no contamination. Just when the amount of parasites exceeds the minimum amounts required, the population of exposed humans grows. Figure 11 shows the profiles of fuzzy contact rate µ and max 50000 parasites l ν = µ ; and show that when the amounts of parasites increases, the susceptible humans and the exposed humans populations decrease and increase respectively. Regarding in the analysis of this paper, the fuzzy models are more realistic, flexible, general than classical models because many of the analysis in the classical models can be derived from the fuzzy analysis. For instance, we obtain the disease-free equilibrium by the fuzzy analysis of the equilibrium of fuzzy model. That is, by analyzing the amounts of parasites in the body. In addition, in the classical model, the basic reproduction number is only a function of parameters, whereas in fuzzy environment, it's a function of the amounts of parasites. The fuzzy parameters used in this paper play an important role in the analysis of malaria transmission dynamics. We may extend this model by considering other parameters as fuzzy variables. One may consider, the mosquitoes population as a fuzzy variable.

Data Availability
The data used are available in the literature.