Modeling and optimal control of COVID-19 and malaria co-infection based on vaccination

1.   Introduction

Malaria is a vector-borne infection caused by the bite of a mosquito carrying the plasmodium parasite. Of all infectious diseases, malaria remains one of the largest contributors to the global burden and continues to receive worldwide attention in terms of suffering and death (see ^[1]). Efforts have been made to curb malaria transmission (see the works^[1,2]), for example, through the use of effective antimalarial drugs ^[3] and infecting the mosquito population with Wolbachia ^[4]. Also, COVID-19 was caused by the outbreak of SARS-CoV-2 at the end of 2019, which spreads rapidly around the world and poses an unprecedented challenge to global health. The clinical manifestations of COVID-19 include fever, difficulty breathing, dry cough, and a range of symptoms that overlap those of malaria ^[5]. This makes diagnosis and treatment challenging. In addition, malaria cases are mainly distributed in African countries, which have a heavy burden of infectious diseases and weak public health infrastructure. At the same time, most of these countries have limited health budgets ^[6], and controlling the spread of COVID-19 will have an impact on the intervention level of other infectious diseases ^[7]. This provides a geographical advantage for co-infection of COVID-19 and malaria. Hence, Gutman et al. ^[8] pointed out that co-infection of malaria and COVID-19 can occur in those countries where malaria is endemic. Many scientists also believed COVID-19 would hit Africa hard ^[9,10,11]. Moreover, there are also concerns about the broader health, economic, and social impacts of measures to mitigate the spread of COVID-19, not just about the disease itself. From the aforementioned reasons, co-infection is a concern, and it is important to theoretically investigate the impact of control measures on their long-term dynamics.

Mathematical models are important tools for predicting and simulating the spread of epidemics, and can provide timely information for decision-making ^{[12,13,14,15]}. At the beginning of the COVID-19 outbreak, a large number of papers about COVID-19 infection modeling have been produced. Scholars have made many contributions to COVID-19 research from the perspective of micro ^[16,17], macro (see ^[18,19]), or using fractional order models (see, for example, ^[19,20,21]). However, reports of co-infection of COVID-19 and malaria are rare, and the researchers are focused on studying the complex dynamics and possible control of this global infection. Tchoumi et al. ^[13] constructed a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. They showed that applying both COVID-19 and malaria protective strategies could help reduce their spread in comparison to applying each preventive measure singly, but only control strategies for non-pharmaceutical interventions were analyzed. Ojo et al. analyzed the impact of COVID-19 on a malaria-dominated region and investigated the impact of threshold and co-infection transmission rate on the synergistic relationship between the two diseases^[14]. The co-infection of two diseases can have devastating consequences globally, so research on the co-infection of COVID-19 and other diseases is booming. For example, COVID-19 and dengue ^[22], COVID-19 and HIV ^[23,24], COVID-19 and cholera ^[25] and other co-infections ^{[26,27,28,29,30]}.

Some researchers developed dynamic models of epidemiology for infectious disease based on preventive measures (social distancing, wearing masks, and lockdown) ^[21,31,32]. Couras et al. ^[33] introduced two control functions in the SEIR model representing vaccination and plasma transfusion, and the results showed that the strategies may have a real application for the COVID-19 pandemic. So, among the measures to control the spread of disease, vaccination is an effective way to cut the chain of transmission, prevent, and reduce virus infection ^[34]. There are also some studies on the impact of vaccination on the spread of COVID-19. For example, see ^{[15,18,35,36,37]} and the references therein. Despite these control measures, controlling the spread of COVID-19 remains a challenge due to inadequate vaccine supply, vaccine uptake hesitancy, vaccine efficacy, vaccine waning, non-adherence to public health orders, and virus mutation ^[38].

In addition to vaccination, asymptomatic carriers also play an important role in infectious disease modeling ^[39], neither of which has been mentioned in literature ^[13]. Currently, there are insufficient studies to analyze the impact of vaccination on co-infection with COVID-19 and malaria. To fill this gap, we consider and improve the limitations in ^[13] in modeling and formulate a COVID-19 and malaria co-infection model to study the effects of basic reproduction numbers and vaccination on the COVID-19-malaria co-infection. This study is seemingly the first of its kind to theoretically detailly investigate the effects of vaccination on the co-dynamics of COVID-19 and malaria, as well as the key prevention and therapeutic measures that are incorporated into the control system. Then, we discuss numerical simulation depicts results.

The structure of this paper is as follows: In Section 2, the co-infection model of malaria and COVID-19 is established. In Section 3, the sub-models are first analyzed, and then the co-infection model is studied. To mitigate the spread of these two diseases, four control measures are incorporated into the co-infection model, and the optimal control problem is explored in Section 4. In Section 5, numerical simulations are carried out. This includes data fitting, sensitivity analysis, stability analysis, the impact of vaccination parameters on disease transmission and control, and the implementation of control strategies. Section 6 is the summary part.

2.   Model formulation

The human population at time $t$, denoted by $N(t)$, is divided into susceptible individuals $S(t)$, individuals vaccinated against COVID-19 $V(t)$, asymptomatic infectious individuals $A(t)$, symptomatic infectious individuals $I(t)$, individuals infected with malaria $E(t)$, individuals infected with both malaria and COVID-19 $I_{E}(t)$, and recovered individuals $R(t)$. So that,

The mosquito population at time $t$, denoted by $N_v(t)$, is divided into susceptible mosquitoes $S_v(t)$, mosquitoes exposed to the malaria parasite $E_v(t)$, and infectious mosquitoes $I_v(t)$. So that,

The model has the following assumptions:

(ⅰ) Co-infected individuals can not transmit the mixed infections at the same time;

(ⅱ) Co-infected individuals can recover either from COVID-19 or malaria but not from the mixed infection at the same time.

The changes that occur in each compartment in disease transmission can be interpreted by Figures 1 and 2.

The system of differential equations related to the transmission diagram Figures 1 and 2 is as follows:

where,

and

All variables and parameters of model (2.1) are non-negative at time

In this paper, as in article ^[13], the incidence of the disease is considered as the standard incidence type. The descriptions and values of other associated parameters are provided in Tables 1 and 2.

3.   Model analysis

Before analyzing the dynamics of the full model, analyze the sub-models first, i.e., COVID-19-only model and malaria-only model.

3.1.   COVID-19-only model

The COVID-19-only sub-model is not related to malaria, so the COVID-19-only model is obtained by setting

in (2.1). We have

where

is the force of infection and

By adding up all the equations of the model (3.1), the total human population is given by

Solving the differential Eq (3.2), we have

Thereby, $0 \le N\left(t \right) \le \frac{\Lambda }{\mu }$ as $t \to + \infty$. In the region

all solutions of the model (3.1) starting in $\Gamma_C$ remain in $\Gamma_C$ for all $t\geq0$. Thus, we will consider the dynamics of model (3.1) in $\Gamma_C$.

3.1.1.   Stability of the disease-free equilibrium

The disease-free equilibrium of model (3.1) is given by

According to the next generation matrix method ^[41], the matrices $F$ and $V$ are calculated by

and

Then,

and

Thus, $R_{C0}$ is given by

From Theorem 2 of ^[41], the result follows:

Lemma 3.1. The disease-free equilibrium $E_{C0}$ of the COVID-19-only model (3.1) is locally asymptotically stable if $R_{C0} < 1$, and unstable if $R_{C0} > 1$.

Theorem 3.1. The disease-free equilibrium $E_{C0}$ of the COVID-19-only model (3.1) is globally asymptotically stable if $R_{C0} < 1$.

Proof. Consider a Lyapunov function $L_1(t)$ as follows

From model (3.1), as

Since $S \le {S_0}$ and $V \le {V_0}$, the following inequalities which will be used subsequently hold:

and

Thus,

It can be verified that $\dot{L_1}\leq0$ for $R_{C0} < 1.$ Thus, by LaSalle's invariance principle ^[42], the disease-free equilibrium $E_{C0}$ of model (3.1) is global asymptotically stable when $R_{C0} < 1$. □

3.1.2.   Existence and stability of the endemic equilibrium

Solving the COVID-19-only sub-model (3.1) at an arbitrary equilibrium denoted by

yields

where

Note that

From Eq (3.9), we obtain

From Eqs (3.10) and (3.11) and after some little rearrangements, we obtain the following polynomial

and

Hence, the following result:

Theorem 3.2. The endemic equilibrium

of the COVID-19-only model (3.1) has one unique endemic equilibrium if $R_{C0} > 1$.

Theorem 3.3. The endemic equilibrium

of the COVID-19-only model (3.1) is globally asymptotically stable if $R_{C0} > 1$.

Proof. The Lyapunov function $L_2(t)$ is given by

The derivative of $L_2(t)$ along the solutions of model (3.1) is as follows:

Let

we obtain

Because the arithmetic mean is greater than or equal to the geometric mean,

It can be verified that ${\dot L_2} \le 0$ for $R_{C0} > 1$. Hence, by the LaSalle's invariance principle ^[42], the endemic equilibrium $E_{C1}$ of model (3.1) is global asymptotically stable when $R_{C0} > 1$. □

3.2.   Malaria-only model

The malaria-only model is obtained by setting

in (2.1). We have

where,

and

Consider the region

It can be shown that the region $\Gamma_M$ is positively invariant.

3.2.1.   Stability of the disease-free equilibrium

The disease-free equilibrium of the malaria-only model (3.19) is given by

According to the next generation matrix method ^[41], the basic reproduction number is calculated as follows:

Thus,

The basic reproduction number $R_{M0}$ of the model (3.19) is the spectral radius of matrix $FV^{-1}$. Thus, $R_{M0}$ is given by

Lemma 3.2. [43, Theorem A.1] Given $a > 0, b > 0, c > 0$ and $d > 0,$ all roots of the function

are negative or have negative real parts if and only if $abcd > e$.

Theorem 3.4. The disease-free equilibrium $E_{M0}$ of the malaria-only model (3.19) is locally asymptotically stable if $R_{M0} < 1$, and unstable if $R_{M0} > 1$.

Proof. The Jacobian matrix of model (3.19) at $E_{M0}$ is given by

The eigenvalues of the Jacobin matrix $J(E_{M0})$ include $-\mu$, $-\mu_v$. The other eigenvalues are the roots of the following equation:

Due to the $R_{M0} < 1$, then

According to Lemma 3.2, all the roots of the characteristic equation have negative real parts if $R_{M0} <$1. Hence, the disease-free equilibrium $E_{M0}$ of the malaria-only model (3.19) is locally asymptotically stable when $R_{M0} <$1. □

3.2.2.   Existence of the endemic equilibrium of the malaria-only model

By setting each of the equations of model (3.19) to zero, the endemic equilibrium is given by

where

and

From expressions (3.22), we obtain

After some little algebraic manipulations, we obtain

Substituting (3.23) and (3.24) into the expression for $\lambda _m^*$, we obtain

After some lengthy algebraic manipulations, the endemic equilibria of the malaria-only model (3.19) satisfy the following polynomial in terms of $\lambda _m^*$ given by

where,

It is worth noting that the coefficient A is always positive and C is positive if $R_{M0} < 1$, and negative if $R_{M0} > 1$. Hence, we have established the following result:

Theorem 3.5. The malaria-only model (3.19) has:

(i) Precisely one unique endemic equilibrium if $C < 0 \Leftrightarrow {R_{M0}} > 1$;

(ii) Precisely one unique endemic equilibrium if $B < 0$ and $C = 0$ or ${B^2} - 4AC = 0$;

(iii) Precisely two endemic equilibria if $C > 0, B < 0$ and ${B^2} - 4AC > 0$;

(iv) No endemic equilibrium otherwise.

Case (ⅲ) indicates the possibility of backward bifurcation in the model, and the disease may spread even though $R_{M0} <$1.

3.3.   Malaria-COVID-19 model

The feasible region for model (2.1) is given by

where $\Gamma_C$ and $\Gamma_M$ are defined in (3.1) and (3.19), respectively.

The disease-free equilibrium of model (2.1) is given by

From the basic reproduction number of the COVID-19-only and malaria-only sub-models, the basic reproduction number of the full model is given as

Following Theorem 2 in ^[41], the model (2.1) has the following results:

Theorem 3.6. The disease-free equilibrium of the full model (2.1) is locally asymptotically stable if $R_{0} < 1$, and unstable if $R_{0} > 1$.

4.   Optimal control model

To investigate the impact of intervention measures, we incorporate the following four controls into the full model (2.1):

$u_1$:Control against incident COVID-19 infection, such as vaccination;

$u_2$:Control against incident malaria infection, such as insecticide treatment of mosquito nets;

$u_3$: COVID-19 treatment control;

$u_4$: Malaria treatment control.

The controls $u_1$ and $u_2$ satisfy

following the general efficacy of the COVID-19 vaccine ^[15],

The COVID-19 and malaria treatment controls $u_3$ and $u_4$ are bounded as follows:

The model (2.1) now reads

with initial conditions

The following objective function is considered.

where $T$ is the final time, $C_i, i = 1, \cdots, 5$ are positive weight constants, and $w_i, i = 1, \cdots, 4$ are weight constants for the strategies and treatments against proliferation of the COVID-19 and malaria. The goal is to find an optimal control, $u_1^*$–$u_4^*$, such that

where

such that $u_1^*$–$u_4^*$ are measurable with

for $t \in \left[ {0, T} \right]$ is the control set. The Hamiltonian is given by

where ${\lambda _i}, i = 1, \cdots, 10$ are the adjoint variables. Pontryagins maximum principle ^[44] is applied, such that there exist adjoint variables satisfying:

Writing (4.6) in details gives

The necessary and sufficient optimality conditions are

Therefore, the optimal controls are given

5.   Numerical simulations

To verify the results in the previous analysis, a numerical simulation of model (2.1) is carried out. The parameter values in Table 2 refer to a large number of literatures. In the absence of reference values, we make assumptions based on the actual situation. The initial value conditions of the model (2.1) are as follows:

We search the data of confirmed cases in South Africa in the early stage of the outbreak of COVID-19, and use Matlab to model fit the data. Data from the WHO website.

The period of the fitting covered 100 days, from March 6, 2020 to June 16, 2020. The "20" in Figure 3 represents the 20th day from March 6, 2020, and so on. The results are presented in Figure 3, from which it can be intuitively seen that our model fits well to the data set.

5.1.   Sensitivity analysis

Understanding the relative importance of parameters can help to develop effective intervention strategies to control disease spread. Forward sensitivity analysis plays an important role in determining the relative importance of each parameter in disease epidemics and quantifies the impact of parameter variations. When $R_{C0}$ and $R_{M0}$ are response functions, we study the impact of parameters on $R_{C0}$ and $R_{M0}$, respectively. In this section, we use the formula

for analysis.

As can be seen from Table 3, when the COVID-19 associated with the basic reproduction number $R_{C0}$ is used as a response function, the effective contact rate for the COVID-19 transmission ($\beta_c$, positively correlated) and the vaccine efficacy against COVID-19 ($\theta$, negatively correlated), as well as the COVID-19 vaccination rate ($\tau$, negatively correlated), dominate the disease dynamics. Furthermore, parameters $\delta_2, {\varphi _{\rm{2}}}, {\varphi _{\rm{3}}}$ (negatively correlated) and $\varepsilon_1$, $\rho$ (positively correlated) also dominate the disease dynamics. These index values further indicate that if $\beta_c, \mu, \varepsilon_1, \rho$ increase or decrease by 10$\%$, $R_{C0}$ increases or decreases by 10$\%$, 0.073$\%$, 3.461$\%$, and 0.708$\%$, respectively. But on the other hand, the index for parameters $\varphi_2, \varphi_3, \delta_2, \theta$, and $\tau$ illustrates that increasing their values by 10$\%$ will decrease the values of reproduction number $R_{C0}$ by 3.884$\%$, 4.165$\%$, 2.648$\%$, 39.578$\%$, and 0.084$\%$, respectively.

Therefore, in order to effectively mitigate the transmission of COVID-19 in the population, the COVID-19 infection rate should be reduced first. In conjunction with the relevant policies, the populace should take appropriate protective measures to reduce the risk of infection. At the same time, effective vaccination can provide the public with physical immunity. This makes people less likely to get sick, which in turn reduces the infection rate $\beta_c$.

In Table 4, using the malaria associated with the basic reproduction number $R_{M0}$ as a response function, the parameters $\Lambda_v, \beta_m, \beta_v, b, \alpha_v$ and $\mu$ have a positive effect on $R_{M0}$, which describe that the growth or decay of these parameters, say by 10$\%$, will increase or decrease $R_{M0}$ by 5$\%$, 5$\%$, 5$\%$, 10$\%$, 1.241$\%$, and 4.995$\%$, respectively. But the index for parameters $\Lambda, \mu_v, \varphi_1$ and $\delta_1$ illustrates that increasing their values by 10$\%$ will decrease the values of $R_{M0}$ by 5$\%$, 11.241$\%$, 4.757$\%$, and 0.238$\%$, respectively. These need to be achieved by comprehensive preventive measures and adequate medical resources. Second, mosquito control cannot be ignored. This requires spraying insecticides and hanging mosquito nets to reduce mosquito bites.

The analysis of the basic reproduction number is crucial as the $R_0$ value is highly significant in determining the state of an epidemic. We study the effect of two parameters on the basic reproduction number when

Figure 4a shows that with the increase in the infection rate $\beta_c$, the $R_0$ value increases and decreases with the increase in vaccination rate $\tau$. In Figure 4b, we clearly witness the decrease in the $R_{0}$ value with the decrease in the infection rate $\beta_c$ and rate of onset of symptoms $\rho$ values. Figure 4c shows that $R_0$ decreases with increase in vaccination rate $\tau$ and decrease in symptom onset rate $\rho$. The results indicate that vaccination provides protection to the population, resulting in a reduced probability of contracting the disease and a shorter recovery time for infected people. Thus, the reduction of $\beta_c$ and $\rho$ values is achieved. The decrease in the $R_{0}$ value also suggests that vaccination reduces the number of co-infections and the burden of health care. Additionally, we learn the range of these parameters when the $R_{0}$ value with below 1, which can indicate a direction for disease control.

5.2.   Stability of equilibrium points

In Figures 5–8, we plot the time series of sub-models and full model for parameter values given in Table 2, and verify the stability of equilibria. Figure 5 simulates the stability of the disease-free equilibrium of model (3.1). Other values of parameters are the same as in Table 2 except $\beta_c = 0.18,$ then

satisfying the condition of Theorem 3.1. It can be seen from Figure 5 that when $t$ tends to infinity, the number of asymptomatic infectious individuals $A$ and symptomatic infectious individuals $I$ both tend to zero, indicating the spread of the disease is under control, which is consistent with the conclusion of Theorem 3.1.

Figure 6 simulates the stability of the endemic equilibrium of model (3.1). There

and other parameters are the same as in Table 2. Then, the

satisfies the conditions of Theorem 3.3. As shown in Figure 6, when $t$ tends to infinity, the numbers of $A$ and $I$ are both greater than zero, indicating that humans will continue to be attacked by disease, which is also consistent with the conclusion of Theorem 3.3.

In a similar way, we simulate the stability of the disease-free equilibrium of models (2.1) and (3.19) in Figures 7 and 8, respectively, which verifies the correctness of Theorems 3.4 and 3.6.

5.3.   Impact of several parameters on disease prevalence

The impacts of key parameters $\tau, \theta$, and $\rho$ on variation in the co-infected population are shown in Figure 9. In Figure 9a, the co-infected population has been plotted for different vaccination rates $\tau$, and it is clearly shown that the number of co-infected population decreases with the increase of vaccination rate. Set

in drawing ^[37]. Figure 9b shows the trajectories of the co-infected population for different values of the vaccine efficacy $\theta$. The larger the value of $\theta$, the smaller the size of the co-infected group. Therefore, improving the vaccination rate and efficacy of vaccines can reduce the number of co-infections. In Figure 9c, the co-infected population has been illustrated for different values of symptom occurrence rates $\rho$. It is obvious from the figure that the co-infected population increases with the increase in symptom occurrence rate, so we should pay attention to the asymptomatic infected class and improve their recovery rate.

5.4.   Impacts of vaccination rate and efficacy of vaccine on co-infection

Co-infected individuals have been presented in Figure 10 by varying the vaccination rate and vaccine efficacy at the same time. Figure 10 is drawn under the condition of low recovery rates. Let

then

If we increase the vaccination rate together with the vaccine efficacy, the number of co-infected people will gradually decrease. This suggests that active vaccination with effective vaccines can reduce the scale of an outbreak during an epidemic when the recovery rate is reduced due to the shortage of medical resources.

In Figure 11, we hypothesize the effects of the following three situations on: co-infected population:

(1) Low vaccination rate and high vaccine efficacy;

(2) High vaccination rate and low vaccine efficacy;

(3) Low vaccination rate and low vaccine efficacy.

Other values of parameters are the same as in Table 2, except $\tau$ and $\theta$. Through the two graphs in Figure 11, we can find that the curves of co-infected people basically coincide in the cases of (1) and (2), while in the case of (3), the number of co-infected population is much larger than the previous two cases.

5.5.   Optimal control model simulation

To support the analytical results, the optimal control model (4.1) is simulated using the model parameter values in Table 2. The balancing factors are assumed as follows: ${C_1} = {C_2} = {C_3} = 2, {C_4} = {C_5} = 1$. The weight constants are set as follows: ${w_1} = 1500, {w_2} = 1000, {w_3} = 1200,$ and ${w_4} = 900$.

5.5.1.   Strategy A: COVID-19 prevention and treatment (${u_1} \ne 0, {u_3} \ne 0, u_2=u_4=0$)

The control model (4.1) is simulated when strategy A is implemented. The results of this strategy are shown in Figure 12a–c, respectively. When this intervention strategy is implemented, the number of symptomatic individuals $I$ drastically decreases (Figure 12a). Interestingly, strategy A increases the number of individuals infected with malaria (Figure 12b). In other words, the implementation of strategy A has prevented the progress of activities to control the spread of malaria, which conforms to the actual situation during the COVID-19 epidemic ^[7]. Many key activities are not effectively implemented, such as insecticide treatment of mosquito nets, chemical prevention for pregnant women and young children, and reduced access to effective antimalarial drugs. This means that tracking, treating, and controlling malaria are much more difficult than before. Also, it is worth noting that strategy A also has a positive population level impact on the number of individuals co-infected with COVID-19 and malaria (Figure 12c). The control profile depicted in Figure 12d shows that prevention is at optimal from the onset of the implementation and drops at around 100 days, while COVID-19 treatment drops at around 110 days.

5.5.2.   Strategy B: malaria prevention and treatment (${u_2} \ne 0, {u_4} \ne 0, u_1=u_3=0$)

Optimal control simulations of strategy B for model (4.1) are implemented. This strategy B reduces new cases of COVID-19, malaria, and co-infections. It is worth noting that strategy B delays the emergence of the peak of COVID-19 infection and lengthens the entire infection cycle. Moreover, this strategy also decreases cases of vector infections, thereby significantly reducing the infectious vector population (Figure 13d). The control profile for this control strategy is given in Figure 13e. The control profile shows that malaria treatment is at its peak within 10 to 80 days, drops and then rises again, and reaches about 0.07 and then drops slowly for the remaining days of the simulation, while prevention do not start to work until about 19 days, and then there is a little bump at about 200 days.

6.   Conclusions

The burden of malaria on human health has persisted for many years, especially in endemic areas. The symptoms of COVID-19 are similar to those of malaria, and there is extensive geographic overlap. Besides, the COVID-19 pandemic has reversed progress on malaria. Thus, co-infection is a double whammy. In order to reduce the human burden of malaria and COVID-19, research on the co-infection of these diseases is particularly needed. In this work, a co-infection model of COVID-19 and malaria is constructed, which includes asymptomatic individuals infected with COVID-19. Vaccination and its timeliness are included in the model, as vaccination is an important measure to protect people from infectious diseases. We investigate the influence of vaccination on the co-infection of malaria and COVID-19, which creates a novel and more epidemiologically realistic model. This study seems to be the first of its kind to study the effect of vaccination on COVID-19 and malaria co-dynamics in detail. The theoretical analysis results are as follows:

(1) The disease-free equilibria of the sub-models are shown to be locally asymptotically stable when the respective reproduction numbers are below unity.

(2) When $R_{C0} > 1$, the COVID-19-only sub-model has a globally asymptotically stable endemic equilibrium.

(3) Compared with literature ^[13], the number of endemic equilibrium of the malaria sub-model is theoretically discussed. There is a possibility of backward bifurcation in this model, and although $R_{M0} < 1$ is necessary, it is not sufficient to eliminate the disease.

(4) The disease-free equilibrium of the co-infection model is locally asymptotically stable whenever the reproduction number

is less than unity.

The simulations of the stability of the disease-free equilibrium and endemic equilibrium verify the correctness of the theoretical analysis. The influence of important parameters on the basic reproduction number is numerically analyzed. The effects of different vaccination rates, vaccine efficacy and symptoms onset rates on co-infected individuals are also analyzed. In addition, the question of how the vaccination rate and efficacy affect the number of co-infected people is explored. The simulation results are as follows:

(1) The key parameters that dominate the disease dynamics are $\beta_c, {\beta _m}, {\beta _v}, \theta, {\varphi _2}, {\varphi _3}, {\delta _2}, {\varphi _1}, {\delta _1}, b, \tau$ (see Table 1 for parameter description).

(2) From the numerical results of sensitivity analysis, it is necessary to reduce the spread of these two diseases and increase effective vaccine management in order to curb the spread of the epidemic.

(3) The scale of the co-infected population decreases with the increase of vaccination rate and efficacy and increases with the increase of symptoms onset rate. Therefore, it is important to detect and treat asymptomatic individuals before they become symptomatic.

(4) When the recovery rate is low, that is, the medical resources are relatively tight, the influence of accepting effective vaccines actively on the co-infected population is simulated. Actively accepting effective vaccines (high vaccination rate and effectiveness) is helpful to reduce the number of co-infections and control the spread of diseases.

(5) A simple simulation of the three hypothetical situations in the paper aims to demonstrate that lack of effective vaccine vaccination will be detrimental to the control of co-infection of COVID-19 and malaria.

Next, based on the direction of future studies pointed out in literature ^[13], this paper adopts strategies different from those in literature ^[13]. After simulation of combination strategies, the results show that:

(1) Each strategy can reduce the burden of co-infection. Strategy B is more effective than strategy A.

(2) The implementation of strategy A reduces the number of individuals infected with COVID-19 but increases the number of malaria cases. In contrast, the use of strategy B results in a reduction in the number of individuals infected with both malaria and COVID-19.

(3) Strategy B reduces the peak value of COVID-19 infection but delays the occurrence of the peak value, indicating a trend toward a longer infection cycle.

(4) The use of strategy B reduces the number of vectors.

The findings suggest that if prevention and treatment for malaria infection are neglected, the disease will spread more widely. Therefore, other infectious diseases must not be neglected during the COVID-19 pandemic, and appropriate preventive measures and treatment services must be taken. These results provide insight into the impact of the implementation of one strategy on the other.

While this study provides further analysis for the transmission dynamics of malaria and COVID-19, this study can be extended. We explored the impact of vaccination on COVID-19-malaria co-infection, but the model did not take into account people's different attitudes towards vaccination. The co-infection model can be analyzed in the future based on different vaccination attitudes, or other extensions can be made, such as incorporating virus variation and stratifying the population by age.

Acknowledgments

This work was supported by The National Natural Science Foundation of China (11971278) and the Fund for Shanxi 1331KIRT.

Conflict of interest

The authors declare that they have no competing interests in this paper.