key: cord-0933513-dq4k3o0y
authors: Tchoumi, S.Y.; Diagne, M.L.; Rwezaura, H.; Tchuenche, J.M.
title: Malaria and COVID-19 co-dynamics: A mathematical model and optimal control
date: 2021-07-02
journal: Appl Math Model
DOI: 10.1016/j.apm.2021.06.016
sha: b6d91c0fe368d7e40b607014e26467804197d7f1
doc_id: 933513
cord_uid: dq4k3o0y

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models’ equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin’s Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

Malaria, a mosquito-borne infectious disease, alone or in combination with other diseases kills millions of people in tropical and subtropical regions, causing an enormous impact on health systems and economies [1, 2] . Humans acquire malaria infection from infected female Anopheles mosquitoes during blood feeding, especially from Plasmodium falciparum. The chain of transmission can be broken through use of mosquito treated nets and anti-malarial drugs as well as other control strategies, see [1, 3] and the references therein. The emergence of malaria drug resistance and the lack of an effective and safe vaccine have increased the complexity of mitigating malaria burden (morbidity and mortality) [4] . Malaria prevention control is mainly based on the use of preventive measures (such as mosquito-reduction strategies and personal protection against mosquito bite).

The outbreak that initially appeared in Wuhan, Hubei Province, China, in December 2019 of the novel severe acute respiratory syndrome corona virus 2 (SARS-CoV-2) known as COVID-19 has spread rapidly around the world, evolving into a pandemic and causing major public health concerns [5, 6, 7] . Its symptoms from mild to severe respiratory infections last 2-14 days include cough, fever, weakness, and difficulty to breath. Coronaviruses can be extremely contagious and spread easily from person to person. COVID-19 transmission routes contain direct transmission, such as close touching and indirect transmission consisting of the air by coughing and sneezing, even if contacting some contaminated surfaces by virus particles [8] . COVID-19 pandemic is having a massive impact on populations and economies around the world, placing an extra burden on health systems around the planet [9] . Countries in sub-Saharan Africa which accounts for more than 90% of global malaria cases and deaths, are facing a double challenge of protecting their citizens to support theoretical results are presented in Section 4 and Appendix A-1, while Section 5 is the conclusion.

Motivated by the empirical observation that compartmental epidemiological models have played a significant role in the development to better understand the mechanism of epidemic transmission and the various preventive strategies used against it [26] , we consider a deterministic mathematical model which uses ordinary differential equations. The human population at time t denoted by N h (t) is divided into sub-populations of susceptible individuals S h (t), individuals exposed to malaria only Em(t), individuals infected with malaria only Im(t), individuals exposed to COVID-19 only Ec(t), infected individuals with COVID-19 only Ic(t), individuals exposed to malaria and COVID-19 Emc(t), individuals infected with malaria and exposed to COVID-19 ImE c (t), individuals infected with COVID-19 and exposed to malaria IcE m (t), and individuals infected with both malaria and COVID-19 Imc(t). The total human population N h (t) is given by N h (t) = S h (t) + Em(t) + Im(t) + Ec(t) + Ic(t) + Emc(t) + EmE c (t) + EcE m (t) + Emc(t).

While anopheles mosquitoes go through four stages in their life cycle: egg, larva, pupa, and adult, in what follows, we consider the adult mosquitoes stage only. Mosquitoes, whatever their status are subject to a natural death (due to their finite life span), which occurs at a rate µv. The total vector (mosquito) population at time t, denoted by Nv(t), is sub-divided into susceptible mosquitoes Sv(t), mosquitoes exposed to the malaria parasite Ev(t), and infectious mosquitoes Iv(t). The total mosquito population Nv(t) is given by

For simplicity of notations, we will subsequently drop the time t in the model variables. Figures 1 and 2 , we derive the following system of nonlinear differential equations                                                                                           Ṡ h = Λ h + ωmIm + ωcIc + ωmcImc − (λm + λc + µ)S h , Em = λmS h − (λc + φm + µ)Em, Ec = λcS h − (λm + φc + µ)Ec, Emc = λcEm + λmEc − (φmc + µ)Emc, Im = φmEm − (δλc + ωm + µ)Im,

where δ,

The parameter b is the biting rate of female mosquitoes and βc, βm, βv represent respectively the probability for a human to be infected after effective contact with corona virus, the probability for a human to be infected with malaria from a bite of an infected mosquito and the probability for a mosquito to be infected after a blood meal from an infected human. The functions λc, λm and λv represent the forces of infection. Because the number of contacts made by an infective per unit time should be limited, or should grow less rapidly as the total population size N h increases, the forces of infection considered are of standard incidence type [27] . The modification parameters δ and (δ, ≥ 1) are the enhancement factors accounting for the relative infectiousness of individuals respectively acquiring COVID-19 following malaria infection or acquiring malaria following COVID-19 infection. It is assumed that personal protection against COVID-19 is adopted in the community. Implementation of these intervention preventive measures could include wearing facial mask, social/physical distancing, hydro alcoholic gel, hand washing with soap and self-isolation. The effect of personal protection is modeled by the factor (1 − κζ), where 0 < ζ < 1 is the efficacy of personal protection strategy adopted,and 0 < κ < 1 is the fraction of the community employing it (compliance), where κ = 1 means 100% compliance and κ = 0 represents no compliance at all. The model parameters, their description, values and sources are provided in Table 1 .

Two sub-models, namely: malaria only and COVID-19 only sub-models are first considered.

By setting Em(t) = Im(t) = Emc(t) = EmE c (t) = Emc(t) = 0, we obtain the following COVID-19 only model.

where λc = βc(1 − κζ) Ic N h is the force of infection and N h = S h (t) + Ec(t) + Ic(t). By adding up all the equations of the system 3, the total human population is given bẏ

The given initial conditions 2 ensure that N h (0) ≥ 0. Thus, the total human population is positive and bounded for all finite time t > 0. Solving the differential equation 4, we have

As t → +∞, we obtain 0 ≤ N h (t) ≤ Λ h µ . From the theory of differential equations [28, 29] , in the region

all solutions of the COVID-19 only model autonomous system 3 starting in Ωc remain in Ωc for all t ≥ 0. This implies that Ωc is positively invariant and attracting [36] . Thus, the model 3 is mathematically and epidemiologically well-posed, and it is sufficient to study its dynamics in Ωc [1, 3] .

The disease-free equilibrium (DFE) of the COVID-19 only model system 3 is obtained by setting each of the system of model system 3 to zero. Also, at the DFE, there are no infections and recovery. Thus, the DFE of the COVID-19 only model 3 is given by

The linear stability of EC0 is established using the next generation operator method on the model system 3. Using the notations of [32] , the matrices F and V for the new infection terms and the remaining transfer terms are given by

The basic reproduction number of the COVID-19 only model 3 is the dominant eigenvalue of the next generation matrix F V −1 given by

The basic reproduction number R0 cov is defined as the expected number of secondary cases generated by one infected individual during its entire period of infectiousness in a fully susceptible population.

From Theorem 2 of [32] , the following result follows.

Lemma 3.1 The disease-free equilibrium EC0 = ( Λ h µ , 0, 0) of the COVID-19 only model system 3 is locally asymptotically stable if R0 c ov < 1 and unstable otherwise.

Proof. The stability of EC0 is obtained from the Routh-Hurwitz criterion for stability, which states that the equilibrium is stable if the roots of the characteristic polynomial are all negative. For EC0, the Jacobian matrix of the system is obtained as

The characteristic polynomial is given by

where a1 = 2µ+φc and a2 = (φc +µ)(wc +µ)−φcβc(1−κζ). After some little algebraic manipulations, one notes that all the eigenvalues of the characteristic equation 8 have negative real parts if R0 cov < 1. Hence, the DFE EC0 of the COVID-19 only model system 3 is locally asymptotically stable when R0 cov < 1.

We now explore the existence of endemic equilibrium of the COVID-19 only model system 3. The endemic equilibrium of system 3 is Ec 1 = (Sc 1 , Ec 1 , Ic 1 ), where

From Theorem 2 of [32] , the endemic equilibrium EC1 is locally asymptotically stable if R0 c ov > 1 and unstable otherwise. Global stability of the model equilibria implies the model does not exhibit the phenomenon of backward bifurcation where co-existence of both a stable disease-free equilibrium with a stable endemic equilibrium cannot hold, [1, 3] . It is important to note that although the disease-free equilibrium interchanges its stability with the endemic equilibrium as the threshold parameter R0 cov is varied, we wish to investigate the direction of the bifurcation (forward/transcritical bifurcation or backward/subcritical bifurcation). We use the centre manifold theory as described in [33] . Let x = (x1, x2, x3) T = (Sc, Ec, Ic) T . The model 3 can be rewritten in the form dx

The Jacobian of the system 9 at the DFE EC0 is given by

Choosing βc as the bifurcation parameter and setting R0 cov = 1, we obtain

The transformed system 9, with βc = β * c , has a non-hyperbolic equilibrium point such that the linear system has a simple eigenvalue with zero real part and all other eigenvalues have negative real parts. Hence, the centre manifold theory [33] can be used to analyze the dynamics of the model 9 near βc = β * c . By using the notation in [33] , we proceed as follows.

The right-eigenvector w = (w1, w2, w3) T associated with the zero eigenvalue of J(Ec 0 ) such that J(Ec 0 ).w = 0 at βc = β * c is given by

Similarly, the left-eigenvector v = (v1, v2, v3) (11) of J(Ec 0 ) such that v.J(Ec 0 ) = 0 associated with the zero eigenvalue is given by

The right-eigenvector w and the left-eigenvector v must satisfy the condition v.w = 1. That is,

To determine the direction of the bifurcation, we compute and determine the sign of the parameters a and b. At the DFE, the bifurcation coefficient a is given by

The bifurcation coefficient b is given by

Because a is negative and since 1 − κζ > 0, the parameter b is positive, and by Theorem 4.1 in [33] , the COVID-19 only model system 3 does not exhibit the phenomenon of backward bifurcation at R0 cov = 1. Hence, the following result. Global stability implies that the classical requirement of having R0 cov < 1, is a necessary and potentially sufficient condition for disease control.

Malaria, caused by mosquito-borne hematoprotozoan parasites of the genus Plasmodium is an important health problem threatening a large proportion of the wold's population. It is transmitted to humans through the bites of infectious female mosquitoes. From the model system 1, the malaria only model which includes critical features of host-vector-parasite interactions is obtained by setting Ec(t) = 0, Ic(t) = 0, Emc(t) = 0, EmE c (t) = 0, EcE m (t) = 0 Emc(t) = 0. The force of infection λm which depends on average number of mosquito bites b and on the transmission probability per bite βm is assumed to be normalized by the total human population [34] and is given as

Thus, the malaria only model is given by the following system of nonlinear ordinary differential equations.

where λv = βvb Im N h and N h = S h + Em + Im. The feasible region for the model system 12 is

which is positively invariant and attracting, that is, solution starting in Ωm will remain in Ωm for all time t ≥ 0. Thus, it is sufficient to consider the dynamics of the model system 12 in Ωm).

The DFE of the malaria only model 12 is given by

Using the next generation matrix operator method in [32] , the associated next generation matrix is given by

and the rate of transfer of individual to the compartments is given by

Hence, the new infection terms F and the remaining transfer terms V are respectively given by

The dominant eigenvalue or spectral radius of the next generation matrix F V −1 which represents the basic reproductive number is given by

Using Theorem 2 of [32] , we establish the following result.

The DFE of the malaria-only model 12 is locally asymptotically stable if R0 m < 1, and unstable otherwise.

Solving the malaria-only model at an arbitrary equilibrium denoted by Em1 = (S h 1 , Em 1 , Im 1 , Sv 1 , Ev 1 , Iv 1 ), we obtain

After some little algebraic manipulations, we obtain

Recall

From equations 17 and 18, we obtain

After some little rearrangements, we obtain the following equation

Equation 19 can be written as

, which can be rearranged as

.

.

So,

Hence, we have established the following result.

The malaria-only model system 12 has precisely one unique endemic equilibrium if R0 m > 1.

Following the same approach as in subsection 3.1.2, we investigate the global stability of the malaria-only model 12 using the center manifold approach [33] .

Let S h (t) = x1, Em(t) = x2, Im(t) = x3, Sv(t) = x4, Ev(t) = x5, and Iv(t) = x6, so that N h = x1 + x2 + x3 and Nv = x4 + x5 + x6. Further, by using vector notation x = (x1, x2, x3, x4, x5, x6) T , the malaria-only model 12 can be written in the form dx

where λm = βmbx6 x1 + x2 + x3 and λv = βvbx3 x1 + x2 + x3 .

Next, we evaluate the Jacobian of the system 22 at the DFE Em0, denoted by J(Em0) given by

Choose βc as the bifurcation parameter, then setting R0 m = 1 gives

It follows that the Jacobian (J(Em0)) of 22 at the DFE, with βm = β * m has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence, the Centre Manifold theory [35] can be used to analyze the dynamics of the malaria-only model 12. Following the approach in [32, 33, 35, 37] , the right-eigenvector w = (w1, w2, w3, w4, w5, w6) T associated with the zero eigenvalue of J(EC0) such that J(EC0).w = 0 at βc = β * c is given by

Similarly, the left-eigenvector v = (v1, v2, v3, v4, v5, v6) of J(EC0) such that v.J(EC0) = 0 associated with the zero eigenvalue is given by

The right-eigenvector w and the left-eigenvector v need to satisfy the condition v.w = 1. The bifurcation coefficient a at the DFE Em0 is given by

The second bifurcation coefficient b is given by

Since the sign of a determines the direction of the bifurcation, it follows from Theorem 4.1 in [33] that the malaria-only model 22 undergoes the phenomenon of backward/subcritical bifurcation at R0 m = 1 whenever both b > 0 and

This result is summarized below.

the malaria-only model 12 will undergo a backward/subcritical bifurcation at R0 m = 1. If a < 0, the the malaria-only model 12 will undergo a forward/transcritical bifurcation at R0 m = 1.

The public health implication of this result when a backward/subcritical bifurcation occurs is that the coexistence of a stable disease-free equilibrium with a stable endemic equilibrium is an epidemiological situation where although necessary, having the basic reproduction number less than unity is no longer sufficient for disease elimination [1, 3] .

The feasible region for system 1 is given by

with Ωc and Ωm are as defined in the previous sections. It can be shown following the approach in [1, 3] that all solutions of the the co-infection malaria-COVID-19 model system 1 with non negative initial conditions remain non negative for all time t ≥ 0. Also, from the theory of permanence [36] , all solutions on the boundary of Ωcm eventually enter the interior of Ωcm. Thus, Ωcm is positively-invariant and attracting under the flow induced by the system 1

The disease-free equilibrium of the malaria-COVID-19 1 is given by E0 = (S h , Em, Ec, Ecm, Im, Ic, ImEc, Imc, IcEm, Sv, Ev, Iv) = Λ h µ , 0, 0, 0, 0, 0, 0, 0, 0, Λv µv , 0, 0 .

Having derived the basic reproduction numbers for the COVID-19 only and malaria only sub-models using the next generation method in [32] , the associated reproduction number for the full model system 1 is given by

The following result follows from Theorem 2 in [32] .

Theorem 3.4 The DFE of the malaria-COVID-19 model 1 is locally asymptotically stable if the threshold parameter R0 cm < 1, and unstable if R0 cm > 1.

Since the malaria-only model 12 may undergo the phenomenon of backward bifurcation, the full malaria-COVID19 model 1 will also undergo backward bifurcation under the same conditions when the bifurcation parameters a > 0 and b > 0 [1, 3] . Using the centre manifold theory described in [33] , the following result holds.

Theorem 3.5 If the bifurcation parameters a > 0 and b > 0, the full model system 1 will undergo the phenomenon of backward/subcritical bifurcation at R0 cm = 1.

For the proof, see [1, 3, 48] . Note that the occurrence of backward bifurcation precludes the global asymptotic stability of the model's equilibria. That is, both the DFE and endemic equilibria can only be locally asymptotically stable, but not globally.

We investigate the impact of implementing non pharmaceutical interventions to prevent mosquitoes bites such as the use of insecticide-treated nets and to protect oneself again corona virus such as facial mask, and hand-washing with soap. We introduce into our proposed malaria-COVID-19 model 1 a set of time-dependent control variables (u1(t), u2(t) where a) u1(t) represents the use of personal protection measures to prevent mosquitoes bites during the day and the night such as the use of insecticide-treated nets, application of repellents to skin or spraying of insecticides, and b) u2(t) represents the use of personal protection measures to protect oneself again corona virus such as facial mask, hydro alcoholic gel, hand-washing with soap.

The malaria-COVID-19 model with optimal control (u1(t), u2(t) consists of the following non-autonomous system of nonlinear ordinary differential equations.

with initial conditions given by

Subject to the model system 27 and suitable initial conditions, we examine an optimal control problem with control (u * 1 , u * 2 ) such that the optimal control function

with U the set of admissible controls, where

The optimal system is then developed to meet the necessary conditions satisfying Pontryagin's Maximum Principle [25] , which determines optimal control of the malaria-COVID-19 co-infection. Thus, to minimize the number of malaria and COVID-19 infections, we consider an optimal control function given by

where in the objective cost functional 29, A1, A2 and A3 are positive constants representing the weight that balance off the infected human population to malaria, infected human population to COVID-19 and co-infected human population to both malaria and COVID-19, respectively, while A4 represents the weight constant of the total vector population, B1 and B2 are respectively weight constants for personal protection measures to prevent mosquitoes bites and personal protection measures to protect oneself again corona virus. The terms B1u 2 (t) and B2u 2 (t) describe the costs associated with the prevention of vector-human contacts and contact with corona virus. We chose a quadratic control function [38] because the positive balancing cost factors transfer the integral into monetary quantity over a finite period of time, while the nonlinearity of the control efforts is chosen for technical reason as this allows the Hamiltonian to attain its minimum over the admissible control set U at a unique point [39] . We note however that the effects of varying the form of the objective functional have been considered in [40, 41] . Next, we prove the existence of an optimal control for system 27 and then derive the optimality system. Theorem 3.6 Consider the objective functional J(u1, u2) given by equation 27, with (u1, u2) ∈ U subject to the constraint state system 1. There exists (u * 1 , u * 2 ) ∈ U such that

To derive the necessary conditions that the two controls and corresponding states must satisfy, we apply Theorem 5.1 (Pontryagin's Maximum Principle [25] ) in Fleming and Rishel [42] to develop the optimal system for which the necessary conditions that must be satisfied by an optimal control and its corresponding states are derived. In fact, Theorem 1 in Agusto [43] which is based on the boundedness of solution of model system 1 without control variables ensures the existence of the optimal control while the existence of an optimal control with a given control pair follows from Fleming and Rishel [42] , and Carathodory's existence Theorem [44] . To convert the state system 27 and objective function 29 into a point-wise minimized problem, we define the Hamiltonian function H for the optimal control system 27 as

where λi, i = 1, . . . , 12 are the adjoint variables or co-state variables. The following result presents the adjoint system and control characterization.

Theorem 3.7 Given an optimal control (u * 1 , u * 2 ), and corresponding state solutions S h , Em, Ec, Emc, Im, Ic, ImE c , IcE m , Imc, Sv, Ev, Iv of the corresponding state system 1, there exist adjoint variables, λi, i = 1, . . . , 12, satisfying

(31) The controls u * 1 and u * 2 satisfy the optimality condition

Proof. The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then, the adjoint system can be written as

with zero final time conditions (transversality) ξi(T ) = 0. Replacing the derivatives of H with respect to S h , Em, Ec, Emc, Im, Ic, ImE c , IcE m , Imc, Sv, Ev, Iv in the above equations, we obtain the optimality condition 32. The optimal conditions for the Hamiltonian are given by ∂H

Thus, u * 1 and u * 2 satisfies 32.

To illustrate some of the analytical results in the previous sections, several numerical simulations are carried out. We consulted a wide range of literature to find relevant model parameter values listed in Table 1 .

Whenever parameter values were not available in the literature, we assume realistic values for the purpose of illustration. Using the default parameter values in Table 1 , numerical simulations are provided to support the analytical results. Our focus is on investigating the effect of optimal non-pharmaceutical control strategies on the transmission dynamics of malaria and COVID-19 co-infection. An iterative fourth-order Runge-Kutta integration scheme is employed to solve the model system 27 and the adjoint system 31. We also assumed, for illustrative purposes that the weight factor values A1 = A2 = A3 = A4 = 1, B1 = 1.5, B2 = 2. The positive constants A1, A2 and A3 represent the weight that balance off the infected human population to malaria, infected human population to COVID-19 and co-infected human population to both malaria and COVID-19, while A4 represents the weight constant of the total vector population, B1 and B2 are respectively the weight constants for personal protection measures to prevent mosquitoes bites and personal protection measures to protect oneself again corona virus. Finally, following the biological interpretation of our system, that is all model and parameter values are non-negative, for illustrative purpose, we consider the following initial conditions S h = 2500, Em = 1, Ec = 1, Ecm = 1, Im = 10, Ic = 20, ImEc = 1, Imc = 3, IcEm = 3, Sv = 10000, Ev = 8, Iv = 10. These assumed values have been intuitively chosen to theoretically investigate the effects of the preventive control measures to mitigate the spread of malaria and COVID-19 co-infection. Because this is a theoretical analysis and not a case study, our model parameter values do not necessarily have a significant meaning attached [45, 46, 47, 48] . However, based on the assumption that high cost could potentially be associated with COVID-19 prevention, B2 has been made slightly greater than B1, while the two control strategies u1, u2 are all constrained between zero and one, that is, 0 ≤ ui(t) ≤ 1, i = 1, 2. For instance, if u1 = 0, it implies no malaria prevention such as insecticide treated bednets are provided, and u1 = 0 implies there is at least some malaria prevention measures being applied in the population. Transmission probability in vectors from infected humans 0.48 [54] 4.1 Simulations when both malaria and COVID-19 preventions are concurrently implemented; u 1 = 0 and u 2 = 0

In this case, the control u1 related to prevention against malaria and the control u2 for prevention against COVID-19 are used to minimize the objective function J(u1, u2). In Figure 3 , there is a depletion of the number of susceptible humans when there is no controls and an increase in the number of susceptible humans when preventive measures are applied. In fact, from Figures 4 -11 , we observe that the controls u1 and u2 result in a decrease of the number of infected and infectious to malaria only, COVID-19 only and malaria and COVID-19 co-infection, while there is an increase in the number of infected and infected humans without control. The controls u1(t) and u2(t) are graphically depicted in Figures 12 and 13 , while Figure 14 depicts the effect of personal COVID-19 protection ζ on the dynamics of the control u2. 

Simulations when there is no malaria prevention u 1 = 0 and prevention against COVID-19 u 2 = 0

In this case, only the prevention against COVID-19 is implemented, i.e., control u2 = 0 is used to minimize the objective function J(u1, u2). In Figure 15 , we observe that controls resulted in an increase in the number of susceptible humans while decrease is observed in the number of susceptible humans when no COVID-19 control measure is implemented in the community. From Figures 16 -20 , we also observe that when there is no malaria control, the COVID-19 control u2 = 0 results in a decrease in the number of exposed and infectious to COVID-19 only, which is also the case on the Figures 21, 22, and 23 where there is a decrease in the number of exposed and infectious to both malaria and COVID-19 simultaneously. This is not the case in Figures 18 and 19 where we can see that the effect of u2 control on the infected with malaria is very temporary. This implies that prevention of COVID-19 would not have in a long run a positive effect on the fight against malaria. The control u2(t) representing prevention against COVID-19 is graphically depicted in Figure 24 . 

A deterministic compartmental model for the transmission dynamics of malaria and COVID-19 is proposed and analyzed. Theoretical results obtained are as follows. For the malaria-only and COVID-19 only models, it is shown that the DFE of each sub-model is locally asymptotically stable when their associated basic reproduction numbers R0 cov and R0 m are less that unity, and unstable otherwise. Whenever R0 cov > 1 and R0 m > 1, the malaria-only and COVID-19 only sub-systems respectively have a locally asymptotically stable endemic equilibrium. However, by using the center manifold theory, under certain conditions when the bifurcation parameters a < 0 and b > 0, the disease-free equilibrium of the COVID-19 only model 3.1 interchanges its stability with the endemic equilibrium as the threshold parameter R0 cov and a forward or transcritical bifurcation occurs. This guarantees that the equilibria of the COVID-19 model system 3.1 are globally asymptotically stable.

Similarly, it is shown that when both bifurcation parameters a > 0 and b > 0, the malaria-only model exhibits the phenomenon of backward or subcritical bifurcation [1, 3, 48] , where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number R0 m less than unity. Consequently, its equilibria are only locally stable. Consequently, under these conditions, the co-infection model 1 will undergo a backward bifurcation at R0 cm = 1, inherited from the the malaria-only model 3.2 [1, 3, 55, 56] . Thus, for the malaria and co-infection models, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for disease elimination [1, 3] . In fact, the occurrence of a backward bifurcation further complicates malaria eradication efforts. The two diseases will co-exist whenever the reproduction number of each of the two diseases exceed unity (regardless of which number is larger).

Next, we incorporated time-dependent controls u1(t) and u2(t) into model system 1 and applied the Pontryagin's Maximum Principle to determine the optimal prevention strategy for controlling the co-infection of COVID-19 and malaria. It is implicitly assumed that individuals adhere to the prevention measures for each disease. We analytically derived the optimality conditions for disease eradication.

Numerical simulations of the optimal control of the full model are carried out using a set of model parameter values mostly obtained from literature and others assumed within realistic range for the purpose of illustration to assess the impact of co-infection of the two diseases. Thus, the following are observed:

i) The use of the two controls reduces the number of malaria and COVID-19 infected individuals as well as those co-infected with both diseases.

ii) The exclusive use of COVID-19 prevention control measure u2 reduces the number of COVID-19 infected individuals, and also the number of co-infected individuals, but does not prevent the increase of malaria infections.

iii) The exclusive use of malaria prevention control u1 aimed at reducing contact between humans and mosquitoes reduces the number of malaria infected individuals, and also slightly reduces the number of individuals infected with COVID-19 and co-infected individuals.

In summary, we formulated and analyzed a deterministic compartmental model for the transmission dynamics of COVID-19 and malaria and their co-infection in a given community. The COVID-19 only, malaria only and the co-infection models were qualitatively examined. By applying optimal control theory to the full model system 1, and using Pontryagin's Maximum Principle, existence of the optimal control problem which satisfied the necessary conditions was established.

This study provides the first in-depth mathematical analysis of a comprehensive study for the transmission dynamics of COVID-19 and malaria. While efforts were made to incorporate several basic epidemiological features of the two diseases and their control using non therapeutic measures, the proposed study can be extended. Future studies could incorporate the following limitations i) therapeutic strategies for COVID-19 (treatment and vaccination and malaria (treatment and prophylactic drugs, vector-reduction strategies)

ii) acquisition of malaria immunity for adults in malaria endemic settings, following repeated exposure [3] iii) treatment efficacy of both diseases. With the emergence of new COVID-19 strains, resistance to treatment should not be ignored [4] .

iv) cost and cost-effectiveness of the proposed control strategies [41, 43, 55, 57] .

A-1 Simulations when malaria prevention u 1 = 0 and no prevention against COVID-19 u 2 = 0

At the onset of the COVID-19 pandemic, when the world was struggling to understand what is happening and finding how to deal with the disease, there was basically no prevention measures against COVID-19. Figures 25 -34 below depict this situation. In Figure 25 , we observe that malaria control could result in an increase in the number of susceptible individuals. From Figures 28 -29 , we observe that the malaria prevention control u1 = 0 lead to a decrease in the number of infected and infectious to malaria only. This is also the case in the Figures 27 , 30 , 31, 32, and 33 where we observe that the control resulted in a decrease in the number of infected and infectious COVID-19 only and malaria and COVID-19 co-infection. That is, prevention of malaria only could potentially have a minimal positive effect on the fight against COVID-19.

There is as expected an increase in the number of exposed and infectious individuals when there is no control being implemented in the community. 

A mathematical analysis of malaria and tuberculosis co-dynamics

Fractional model for malaria transmission under control strategies

Mathematical analysis of a model for HIVmalaria co-infection

A mathematical model for antimalarial drug resistance

Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19

Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study

Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia

Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK

WHO (2020) Malaria & COVID-19

Effects of disruption from COVID-19 on antimalarial strategies

Malaria and parasitic neglected tropical diseases: Potential syndemics with COVID-19?

Understanding unreported cases in the COVID-19 epidemic outbreak in Wuhan, China, and the importance of major health interventions

Predicting the cumulative number of cases for COVID-19 in China from early data

A COVID-19 epidemic model with latency period

Modelling the anti-COVID19 individual or collective containment strategies in Cameroon

Indirect effects of the COVID-19 pandemic on malaria intervention coverage, morbidity, and mortality in Africa: a geospatial modelling analysis, The Lancet Inf

The potential public health consequences of COVID-19 on malaria in Africa

Dynamic analysis of a mathematical model with health care capacity for COVID-19 pandemic

Identifying and combating the impacts of COVID-19 on malaria

Potential impact of the COVID-19 pandemic on HIV, tuberculosis, and malaria in low-income and middle-income countries: a modelling study

Co-infection of dengue and COVID-19: A case report

Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour la prevenir

Daniel Bernoulli's epidemiological model revisited

Conditions for a Second Wave of COVID-19 Due to Interactions Between Disease Dynamics and Social Processes

The mathematical theory of optimal control process

Dynamical behavior of a vector-host epidemic model with demographic structure

Global dynamics of an SEIR epidemic model with saturating contact rate

Differential Equations and Dynamical Systems

Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

A contribution to the mathematical theory of epidemics

Infectious disease of humans: dynamics and control

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Dynamical models of tuberculosis and their applications

A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria

Applications Centre Manifold Theory

Permanence and the dynamics of biological systems

Backwards bifurcations and catastrophe in simple models of fatal diseases

Mathematical modeling of Zika virus

Optimal control analysis applied to a two-path model for Guinea worm disease

Optimal control methods applied to disease models

Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model

Differential Equation Classical to Controlled Mathematical in Science and Engineering

Optimal control of the chemotherapy of HIV

Dynamic multidrug Therapies for HIV: Optimal and STI approaches

Optimal control analysis of malaria in the presence of non-linear incidence rate

Analysis of a malaria model with mosquito host choice and bed-net control

Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: a statistical analysis of publicly available case data

Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

Propagation du COVID19 au Burkina Faso, Modelisation Bayesienne et Quantification des incertitudes : premiere approche, preprint

Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

A basic general model of vector-borne diseases

Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection