Global Stability of a Reaction–Diffusion Malaria/COVID-19 Coinfection Dynamics Model

: Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a new virus which infects the respiratory system and causes the coronavirus disease 2019 (COVID-19). The coinfection between malaria and COVID-19 has been registered in many countries. This has risen an urgent need to understand the dynamics of coinfection. In this paper, we construct a reaction–diffusion in-host malaria/COVID-19 model. The model includes seven-dimensional partial differential equations that explore the interactions between seven compartments, healthy red blood cells (RBCs), infected RBCs, free merozoites, healthy epithelial cells (ECs), infected ECs, free SARS-CoV-2 particles


Introduction
The coronavirus disease 2019 (COVID-19) is a viral disease that appeared in China at the end of 2019 and spread to most countries of the world. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the cause of COVID-19. Malaria-endemic regions face a great challenge due to the possibility of coinfection between malaria and other viral diseases. Indeed, malaria/COVID-19 coinfection has been founded in several countries [1]. This has increased the necessity to understand the dynamics of the coinfection and its effect on the patient. SARS-CoV-2 is an RNA virus and belongs to the family Coronaviridae [2]. It uses the angiotensin-converting enzyme 2 (ACE2) receptor to step into the ECs [3]. Such receptor is expressed in kidney, heart, gastrointestinal tract, blood vessels, and other organs [4]. The human-to-human transmission of SARS-CoV-2 occurs via respiratory droplets containing viruses [5]. Eleven vaccines for COVID-19 were approved by the World Health Organization (WHO) for emergency use. These include Novavax/Nuvaxovid, Bharat Biotech/Covaxin, CanSino/Convidecia, Pfizer/BioNTech/Comirnaty, Moderna/Spikevax, Serum Institute of India COVOVAX (Novavax formulation), Janssen (Johnson & Johnson)/Jcovden, Oxford/AstraZeneca/Vaxzevria, Serum Institute of India Covishield (Oxford/AstraZeneca formulation), Sinopharm (Beijing)/Covilo, and Sinovac/CoronaVac [6].
to effectively treat coinfected patients and save their lives. Mathematical modeling can support these studies and reduce the number of experiments needed to test hypotheses. We noted that a diffusive malaria/COVID-19 coinfection model has not yet been considered. In this paper, we formulate a reaction-diffusion malaria/COVID-19 model. This model considers the interactions between healthy RBCs, infected RBCs, free merozoites, healthy ECs, infected ECs, free SARS-CoV-2 particles, and antibodies. For this model, we (i) validate the boundedness and nonnegativity of solutions, (ii) calculate all model's equilibria and extract the conditions of their existence, (iii) show the global stability of equilibria, and (iv) enhance the analytical results by executing some numerical simulations.
The paper is written as follows: Section 2 gives a description for the proposed model. Section 3 shows the properties of the model's solutions. Furthermore, it calculates all models' equilibria. Section 4 introduces the Lyapunov method to establish the global stability of all model's equilibria. Section 5 is devoted for numerical simulations. Finally, the results are discussed and some future research points are suggested in Section 6.

Reaction-Diffusion Malaria/COVID-19 Model with Immune Response
In this section, we give a detailed description of the proposed model. We construct the malaria/COVID-19 coinfection model as a system of seven PDEs: ∂V(x, t) ∂t = D V ∆V(x, t) + eN(x, t) − q 2 V(x, t)Z(x, t) − d 6 V(x, t), for t > 0 and x ∈ Γ, where U(x, t), I(x, t), M(x, t), Y(x, t), N(x, t), V(x, t), and Z(x, t) stand for the concentrations of healthy RBCs, infected RBCs, free merozoites, healthy ECs, infected ECs, free SARS-CoV-2 particles, and antibodies. Healthy RBCs are generated at a constant rate σ 1 , get infected by merozoites at rate β m U M, and die at rate d 1 U. Infected RBCs die at rate d 2 I and burst to generate η merozoites per infected cell. Free merozoites die at rate d 3 M and are cleared by antibodies at rate q 1 MZ. Healthy ECs are recruited from its source at rate σ 2 , get infected by SARS-CoV-2 at rate β v YV and die at rate d 4 Y.
Infected ECs die at rate d 5 N and release SARS-CoV-2 at rate eN. SARS-CoV-2 particles are eliminated by antibodies at rate q 2 VZ and die at rate d 6 V. Antibodies die at a natural death rate d 7 Z and are stimulated to target malaria merozoites and SARS-CoV-2 at rates p 1 MZ and p 2 VZ, respectively. The spatial domain Γ is continuous, bounded and its boundary ∂Γ is smooth. ∆ = ∂ 2 ∂x 2 is the Laplacian operator. We assume that each component C(x, t) of the model diffused in the domain with a diffusion coefficient D C . The initial conditions (ICs) of model (1) are defined as the following: The boundary conditions are given by the following Neumann boundary conditions (NBCs): where ∂ ∂ υ is the outward normal derivative on ∂Γ. This type of boundary condition simulates a natural barrier that prevents cells and viruses from crossing the boundary.

Properties of Solutions
In this section, we verify the basic properties of model (1) including the existence, nonnegativity, and boundedness of the solutions. Furthermore, we evaluate all possible equilibrium points with their conditions of existence.
Let H = C b Γ , R 7 be the set of all bounded and continuous functions fromΓ to R 7 , and H + = C b Γ , R 7 + ⊂ H. The positive cone H + induces a partial order on H. Let φ H = sup x∈Γ |φ(x)|, where | · | is the Euclidean norm on R 7 . This reveals that (H, · H ) is a Banach lattice [42,43].
The components are given by , . We see that U 2 , I 2 , M 2 and Y 2 are always positive, while Z 2 > 0 when R 1m > 1. Therefore, E 2 exists if R 1m > 1. R 1m is a threshold parameter which sets the initiation of antibody immune response against malaria merozoites. (4) The SARS-CoV-2 single-infection without immunity equilibrium is defined as E 3 = (U 3 , 0, 0, Y 3 , N 3 , V 3 , 0). The components are given by . Notably, U 3 and Y 3 are always positive, while N 3 and V 3 are positive when R 0v > 1. Here, R 0v is a threshold parameter which determines the establishment of SARS-CoV-2 infection. (5) The SARS-CoV-2 single-infection with immunity is given by . We see that U 4 , Y 4 , N 4 and V 4 are always positive, The threshold parameter R 1v marks the establishment of antibody immunity against SARS-CoV-2 infection. (6) The malaria/SARS-CoV-2 coinfection without immunity equilibrium is given by The components U 5 and Y 5 are always positive. I 5 and M 5 are positive when R 0m > 1, while N 5 and V 5 are positive when R 0v > 1. Consequently, E 5 exists when R 0m > 1 and R 0v > 1. (7) The malaria/SARS-CoV-2 coinfection with immunity equilibrium is given by E 6 = (U 6 , I 6 , M 6 , Y 6 , N 6 , V 6 , Z 6 ), where By substituting Y 6 in the fourth equation of model (1), we obtain Thus, V 6 fulfills the following equation Let us define a function G(V) as follows: By computing the value of G(V) at V = 0, we obtain .
In addition, we find that This implies that there exists a root 0 because E 2 coexists with E 6 when R 1m > 1, but it will not be stable as can be concluded from Theorem 4), we have U 6 > 0, I 6 > 0, M 6 > 0, Y 6 > 0, N 6 > 0 and Z 6 > 0. Similarly, to find the third existence condition of E 6 , we form a function of Z and extract the conditions at which there is a positive root. This will give It follows that E 6 exists if conditions (5), (6), and (7) are met.

Global Stability of Equilibria
This section confirms the global stability of all equilibrium points by building appropriate Lyapunov functionals. Define a Lyapunov functional Theorem 2. The uninfected equilibrium E 0 is globally asymptotically stable (GAS) when R 0m ≤ 1 and R 0v ≤ 1.

Proof.
Define Then, we have By calculating the time derivative of ∆ 0 (t), we have Depending on the Divergence theorem and NBCs, we have This implies that By applying (9) to (8), we obtain We note that d∆ 0 dt ≤ 0 when R 0m ≤ 1 and R 0v ≤ 1. In addition,  (1) imply that I = N = 0. Then, K 0 = {E 0 } and thus LaSalle's invariance principle (LIP) [47] assures the global asymptotic stability of E 0 when R 0m ≤ 1 and R 0v ≤ 1.
Then, we obtain The equilibrium conditions at E 1 are By utilizing (11) to collect terms of Equation (10), we obtain By computing d∆ 1 dt , we obtain Thus, we see that The solutions tend to K 1 , which has V = 0 and then dV dt = 0. From the sixth equation of (1), we obtain N = 0. Hence, Accordingly, LIP proves the global asymptotic stability of E 1 if R 0m > 1, R 0v ≤ 1 and R 1m ≤ 1.

Proof. Consider
Then, we obtain The equilibrium conditions at E 2 are computed as After using the equilibrium conditions to collect terms of Equation (12), we obtain By using the values of Y 2 and Z 2 , we have .
Accordingly, d∆ 2 dt is given by We observe that . In addition, d∆ 2 dt = 0 when U = U 2 , I = I 2 , M = M 2 , Y = Y 2 , and V = 0. We can prove that the elements of K 2 satisfy N = 0 and Z = Z 2 . Consequently, K 2 = {E 2 }. Therefore, the global asymptotic stability of E 2 is followed by LIP when R 1m > 1 and R p ≤ 1 + ηβ m σ 1 p 1 q 2 q 1 d 6 (p 1 d 1 + β m d 7 ) .

Proof.
Define Then, we obtain By using the equilibrium conditions at E 3 the partial derivative in (13) is transformed to Accordingly, d∆ 3 dt is given by This implies that d∆ 3 dt ≤ 0 if R 0m ≤ 1 and R 1v ≤ 1. In addition, one can show that As a result, LIP insures the global asymptotic stability of E 3 when R 0v > 1, R 0m ≤ 1 and R 1v ≤ 1.

Theorem 6.
Assume that R 1v > 1. Then, the SARS-CoV-2 single-infection with immunity Then, we have The equilibrium conditions at E 4 can be written as By utilizing Equation (15) to collect terms of Equation (14), we obtain By using Equation (9), d∆ 4 dt is computed as Theorem 7. Assume that R 0m > 1 and R 0v > 1. Then, the malaria/SARS-CoV-2 coinfection Proof. Define Then, we obtain The equilibrium conditions at E 5 can be written as By using the above conditions, the derivative in (16) becomes To evaluate the fifth term in (17), we calculate Accordingly, d∆ 5 dt is provided as Hence, we have In addition, Then, the malaria/SARS-CoV-2 coinfection with immunity equilibrium E 6 is GAS.

Sensitivity Analysis
Sensitivity analysis evaluates a relative change in a variable when a parameter changes. We execute sensitivity analysis for R 0m and R 0v as they are the main determinants for the stability of the uninfected equilibrium E 0 . The normalized forward sensitivity index of a differentiable function θ with respect to a parameter p is defined as The normalized forward sensitivity index of R 0m is given by We calculate the sensitivity indices of R 0m with respect to each parameter using the values provided in Table 1. The results are listed in Table 2. We note that the sensitivity indices of R 0m do not depend on any parameters. For instance, the sensitivity index of R 0m with respect to η is Therefore, it is useful to justify the sign of the sensitivity indices of R 0m . According to Table 2, the number of merozoites produced per infected cell, η, the infection rate of RBCs, β m , and the recruitment rate of healthy RBCs, σ 1 , are the parameters that increase malaria infection in the body. Conversely, the death rate of uninfected RBCs, d 1 , and the death rate of merozoites, d 3 , are the parameters that have a crucial role in eliminating malaria infection from the body. Table 2. Sensitivity indices of R 0m .

Parameter
Sensitivity Index The normalized forward sensitivity index of R 0v is given by As for R 0m , we calculate the sensitivity index of each parameter in R 0v using the values given in Table 1. The results are presented in Table 3. We see that, when one of the parameters with a positive index (e, β v , or σ 2 ) is increased while the other parameters remain constant, this raises the value of R 0v . In other words, these parameters lead to the growth of SARS-CoV-2. Conversely, the parameters with negative indices have a role in eliminating SARS-CoV-2 infection from the body.

Parameter
Sensitivity Index

Results and Discussion
Malaria/COVID-19 coinfection represents a true concern especially in malaria-endemic regions. Therefore, there is an urgent need to understand the dynamics of this coinfection within a human body. In this paper, we develop a reaction-diffusion in-host malaria/COVID-19 coinfection model. This model considers the interactions between healthy RBCs, infected RBCs, free merozoites, healthy ECs, infected ECs, free SARS-CoV-2 particles and antibodies. We show that the system admits seven equilibrium points and we prove the following: (1) The uninfected equilibrium E 0 always exists. Moreover, E 0 is GAS if R 0m ≤ 1 and R 0v ≤ 1. This situation represents an individual who recovered from both malaria and SARS-CoV-2 infections. (2) The malaria single-infection without immunity equilibrium E 1 exists if R 0m > 1. In addition, E 1 is GAS if R 0v ≤ 1 and R 1m ≤ 1. This simulates the situation of malaria mono-infection patient with inactive immunity.
. At this point, the antibody immune response is activated to eradicate malaria merozoites. (4) The SARS-CoV-2 single-infection without immunity equilibrium E 3 exists if R 0v > 1.
In addition, E 3 is GAS if R 0m ≤ 1 and R 1v ≤ 1. This point simulates the situation of a patient who is only infected by SARS-CoV-2 and the immune response is inactive. (5) The SARS-CoV-2 single-infection with immunity equilibrium E 4 exists if R 1v > 1.
The numerical results agree with the analytical results. Based on our results, we assume that the malaria/COVID-19 coinfection can be protective as the shared antibody immune response works on clearing SARS-CoV-2. This can decrease the severity of COVID-19. This result comes in agreement with some studies that reported the positive impact of the shared antibody immune response [4,[17][18][19]. However, other studies suggested that there is an increased risk of death in malaria patients with SARS-CoV-2 infection [1,13]. Therefore, more studies are required to investigate the impact of coinfection between malaria and COVID-19, to evaluate the effect of the immune system during the coinfection, and to find the suitable ways for treating the coinfected patients. The main limitation of this research work is that we did not estimate the values of the model's parameters using real data. The reasons are as follows: (1) The data on malaria/COVID-19 coinfection are still very limited; (2) Comparing our results with a small number of real studies may not be very precise; (3) Collecting real data from patient coinfected with malaria and SARS-CoV-2 is not an easy process; (4) Working on experiments to obtain data is beyond the scope of this paper. Thus, the theoretical results obtained in this paper need to be tested against empirical findings when real data become available.

Conclusions
Malaria/COVID-19 coinfection has been reported in many countries. In this paper, we formulated a reaction-diffusion in-host model to study the coinfection between malaria and COVID-19. We assumed that the shared antibody immune response decreases the concentrations of SARS-CoV-2 and malaria merozoites. This can reduce the severity of SARS-CoV-2 in coinfected patients. The principal limitation of this paper is that we did not use real data to estimate the values of parameters or to compare the results due to the scarcity of data. Therefore, our results need to be validated when real data become available.

Future Works
The model developed in this work can be improved by (i) using real data to find a good estimation of the parameters' values, (ii) examining the influence of time delays that may occur during infection or production of SARS-CoV-2 particles and malaria merozoites, (iii) considering viral mutations [41,50,51], (iv) considering the effect of treatments on the progression of both diseases, (v) incorporating the role of CTLs in killing infected RBCs or ECs, and (vi) considering an age-structured model to account for the age structure in the infected cells compartments, which can lead to important observations. Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/math10224390/s1, File S1.