SENSITIVITY ANALYSIS AND OPTIMAL COUNTERMEASURES CONTROL OF MODEL OF THE SPREAD OF COVID-19 CO-INFECTION WITH HIV/AIDS

: This paper analyzes and examines the optimal control in the co-infection of COVID-19 with HIV/AIDS by providing preventive and treatment control measures. The population is divided into eight subpopulations. The preventive control of COVID-19 is denoted by u 1 . The preventive control of HIV/AIDS is denoted by u 2 . The treatment control of COVID-19 is denoted by u 3 , and the treatment control of COVID-19 for the subpopulation co-infected with HIV/AIDS is denoted by u 4 . Based on the model analysis, non-endemic and endemic equilibrium points are obtained, along with the basic reproduction number of the COVID-19, HIV/AIDS, and COVID-19-HIV/AIDS sub-models. Numerical simulations reveal that using preventive control u 1 is more effective in reducing the spread of COVID-19 compared to u 3 or u 4 , both individually and together. Preventive control u 2 is more effective in controlling the spread of HIV/AIDS compared to the absence of control. The sensitivity analysis of parameter identifies parameters that significantly affect the reduction or increase in the spread of COVID-19-HIV/


INTRODUCTION
The COVID-19 disease has subsided since the end of 2022, but it has had a significantly adverse impact on health, economy, education, social and cultural progress [1].The pattern of COVID-19 transmission with HIV/AIDS differs, but individuals with HIV may experience a higher prevalence of infection and COVID-19 complications compared to those without HIV [2].Co-infection of COVID-19 with congenital diseases has led to numerous fatalities, including comorbidities with HIV/AIDS.Mathematics plays a pivotal role in modeling, analyzing, predicting, controlling, and optimizing the spread of infectious diseases [1].
The mathematical model of the COVID-19 disease spread in Wuhan was studied and analyzed in 2020 [3].Ahmed (2021) conducted an analysis of the disease spread in the country, considering both symptomatic and asymptomatic cases [4].The majority of COVID-19 fatalities were attributed to individuals with underlying congenital diseases [1].A study examining COVID-19 in the context of hereditary diseases was conducted to identify relevant parameters for disease control [5].In order to optimize the control of COVID-19, preventive measures, isolation, and treatment strategies are implemented [6].The spread of HIV/AIDS can occur through needles and sexual relations [2], [7].Massarvva (2021) conducted a literature review on addressing COVID-19 and HIV co-infection based on previous study findings [8].It is essential to analyze optimal controls to reduce and prevent the spread of HIV, considering non-endemic, endemic, and threshold stability [9].
HIV-infected individuals who contract malaria are at risk of experiencing an increase in HIV virus levels in their bodies, thereby amplifying the chances of HIV transmission to their partners [10], [11].The transmission pattern of malaria co-infection with HIV/AIDS is nearly identical to that of dengue fever co-infection with HIV/AIDS [12].The diseases co-infected with HIV, such as SPREAD OF COVID-19 CO-INFECTION WITH HIV/AIDS tuberculosis, can be studied to analyze the necessary parameters for optimizing the control of such co-infections [13], as well as co-infection studies with Human T-cell leukemia virus (HTLV) [14].
COVID-19 infection can worsen if infected individuals are not screened for congenital diseases [15].The rise in mortality due to COVID-19 infection necessitates investigations into comorbid diseases with SARS [16].
The moderate increase in the risk of death is directly related to COVID-19 infection, with findings indicating that the risk of death for patients with HIV-positive comorbidities is almost double compared to patients with HIV-negative comorbidities [2].The effects of tuberculosis and HIV-1 infection on the dynamics of COVID-19 spread and immune response were investigated, particularly in Africa [13], [17].Elaiw (2022) conducted a global analysis of the HIV and SARS co-infection model, observing the interaction between healthy and latency epithelial cells [16].
HIV-infected patients have a higher likelihood of being infected with COVID-19, and the consequences of HIV disease are independently and positively correlated with increased mortality in patients with COVID-19 [18].The role of digital health and HIV and COVID-19 care management has an impact on the cure rate for HIV, COVID, and co-infection of the two diseases [19].People with HIV have a similar risk of severe COVID-19 infection compared to the general population [20], [21].The COVID-19/AIDS co-infection model was employed to observe the effect on uninfected epithelial cells, infected epithelial cells, and free HIV-1 particles, aiming to reduce viral load in the host [22].Teklu (2023) conducted numerical simulations in the study of the COVID-19/HIV model to identify the parameters that need intervention to decrease the number of infected individuals [23].An evaluation of the effect of COVID-19-HIV-TB co-infection on decreasing people's income was conducted using a combination of protocols and the Burkina Faso method [24].
To optimize the reduction of the spread of COVID-19 or HIV, the government is making efforts to provide prevention and treatment controls [1], [2].Implementing controls to optimally manage COVID-19/HIV co-infection can effectively prevent the spread of both COVID-19 and HIV [25].
A study conducted in southern Africa demonstrated that prevention and care interventions for individuals infected with HIV and COVID-19 had an impact on the severity of COVID-19 infection [26].Numerical simulations yielded results on the problem of optimal control, suggesting the most effective combination of prevention and treatment strategies to minimize the transmission of HIV/AIDS and COVID-19 co-infection in the community.Building upon the model developed by Teklu et al. (2023) [23], the author focused on vaccination parameters and subsequently examined the optimal prevention control for HIV/AIDS, as well as the optimal prevention and treatment control for COVID-19.

MODEL FORMULATION AND ANALYSIS
The population is categorized into eight subpopulations as follows: Susceptible subpopulation   The assumptions of the model studied in this research are as follows: Notation of parameters, description, and value of parameters as shown in Table 1.

COVID-19 Sub-Model
This case is only considered for the spread of COVID-19, so in this case,   =   =   =  =  = 0. Thus, the sub-model for the case of the spread of COVID-19 is given by the following system of differential equations with   =  +   +   .All equilibrium points of System (2) can be obtained by solving the One of the equilibrium points, which is often referred to as the disease-free equilibrium point (  0 ), is obtained when   = 0.The disease-free equilibrium point for the sub-model of the spread of COVID-19, namely The basic reproduction number of System (2) is determined using the next-generation matrix [27].
so that the next-generation matrix is obtained as follows: The basic reproduction number of System (1) is the maximum eigenvalue of the  −1 matrix.So, by using the next-generation matrix, it is obtained the basic reproduction number for the sub-model of the spread of COVID-19 is Suppose that   * = ( * ,   * ,   * ) endemic equilibrium point of system (2), then by solving system (2) is obtained this shows that if   > 1, then   is positive, so   * is also positive.In other words, the endemic equilibrium point for the sub-model of the spread of COVID-19 exists and is single if   > 1.
The local stability of the disease-free equilibrium point of System (1) is determined by the linearization approach.The Jacobian matrix of System (2) at   0 is ).
It is clear that all the eigenvalues of the matrix (  0 ) are negative.Thus, the disease-free equilibrium point   0 is locally asymptotically stable if   < 1. Next, global stability analysis of the endemic equilibrium point   * was performed using the Lyapunov method.The Lyapunov function [28] used to determine the global stability of   * is defined as follows: The Lyapunov function L is a continuously differentiable function and is always positive and zero only at the endemic equilibrium point   * .If the Lyapunov function L [28] is derived with respect to t, then it is obtained Since   > 1, there is a unique endemic equilibrium point   * , and the Lyapunov L function with   < 0 is obtained.Thus, it can be concluded that the endemic equilibrium point   * is globally asymptotically stable.

HIV/AIDS Sub-Model
This case is only concerned with the spread of HIV/AIDS; in this case,   =   =  =  = 0.
Thus, the sub-model for the case of the spread of COVID-19 is given by the following system of differential equations The basic reproduction number of System (3) is determined using the next generation matrix.
So that the next-generation matrix is obtained as follows: The basic reproduction number of System (3) is the maximum eigenvalue of the  −1 .
So, the basic reproduction number for the sub-model of the spread of HIV/AIDS is Suppose that   * = ( * ,   the linearization approach.The Jacobian Matrix of the System (3) at   0 is It is clear that all the eigenvalues of the matrix (  0 ) are negative.Thus, the disease-free equilibrium point   0 is locally asymptotically stable if   < 1.
Next, global stability analysis of the endemic equilibrium point   * was performed using the Lyapunov method.The Lyapunov function used to determine the global stability of   * is defined as follows [28]: (,   ,   ) = < 0. Thus it can be concluded that the endemic equilibrium point   * is globally asymptotically stable.

Model of COVID-19 Co-infection with HIV/AIDS
The co-infection model for COVID-19 and HIV/AIDS is as follows: All equilibrium points of System (1) can be obtained by solving the following system of equations The disease-free equilibrium point of System ( 4) is   0 = ( 0 ,   0 ,   0 , 0,0,0,0,  0 ) with, .

SPREAD OF COVID-19 CO-INFECTION WITH HIV/AIDS
The basic reproduction number of System (1) is determined using the next-generation matrix.
Based on System (1), the F and V matrices are obtained as follows: with  0 =  0 +   0 +   0 +   0 +   0 +   0 +  0 .So that the next-generation matrix is obtained as follows: The basic reproduction number of System (1) is the maximum eigenvalue of the  −1 .So, the basic reproduction number for the co-infection model of COVID-19 and HIV/AIDS is The local stability of the disease-free equilibrium point of System (1) is determined using the Clearly that all eigenvalues of (  0 ) are negative if  44 < 0. It means that if   < 1, then the endemic equilibrium point   0 is locally asymptotically stable.

AN OPTIMAL CONTROL OF MODEL OF THE SPREAD OF COVID-19 CO-INFECTION WITH HIV/AIDS AND NUMERICAL SIMULATION
In the dynamic model of co-infection spread involving both COVID-19 and HIV/AIDS, various controls are implemented.By utilizing the system of equations ( 1) and applying controls u1, u2, u3, and u4, we arrive at the following set of equations.The goal of optimal control and prevention is to minimize the cost of handling and reduce the spread of COVID-19, HIV/AIDS and COVID-19 and HIV/AIDS co-infection.The functional objective of the control review of prevention and treatment of the system is

Teorema
Based on the system of state equations ( 4), objective functional equations ( 5), and the Hamiltonian function ( 6), the co-state function system and optimal control are obtained as follows.In Figure 3(e), it can be observed that the number of individuals in the TH subpopulation increases without control and with control u4.In actions with u2 control, the increase is gradual from the initial time until t = 30 days and from t = 30 days until t = 92 days; it follows a monotonic trend.However, with combined controls u2 and u4, it decreases slowly from the initial time until t = 92 days.In Figure 3(f), the number of individuals in the R subpopulation increases from the (e) (f) initial time until t = 92 days.The combined use of controls u1, u2, and u3 is more effective than using them individually.Additionally, prevention controls u1 and u2 are more effective in increasing the number of individuals in the R subpopulation from the initial time until t = 92 days compared to treatment control u3.

Sensitivity Analysis
To evaluate which parameter of the model has the most proportional impact on the disease spread, it is important to calculate the elasticity index of the basic reproduction number which is defined as follows where  is the parameter and  0 is the basic reproduction number.The advantage of the elasticity index calculation is that we can find another way to control the disease spread by paying attention to the most sensitive parameter.The elasticity index of the COVID-19 basic reproduction number   , the HIV/AIDS reproduction number   , and the co-infection ℜ  is given in Figure 4a-c.In the case of   , it is found that parameter  1 and followed by  2 are the most sensitive.In the case of   , the parameter  2 is the most sensitive and it is followed by  2 .Thus, to gain maximal result in controlling the COVID-19 spread, we should reduce the contact rate  1 , for example by social distancing; while to gain maximal result in reducing the HIV/AIDS spread, we should reduce the contact rate  2 , for example by using condoms when doing sex.In the co-infection case, the simulation shows that the parameters  2 and  are the most sensitive.Thus, in order to reduce the co-infection spread, we should pay attention to the reducing the contact rate of HIV/AIDS patients or increasing the their treatment rate.In Figure 4d-f, we have the sensitivity of parameter  1 on the dynamics of the infected COVID-19   , and the sensitivity of parameter  2 on the dynamics of the infected HIV/AIDS   and co-infected .The result is similar to the elasticity index analysis, that higher contact rate will produce higher numbers of the infected subpopulations.as a function of  1 and other parameters.We want to know the simultaneous influence of contact rate  1 with other parameters on the COVID-19 spread.To do this, we plot the contour of   , and the result is given in Figure 5.All the figures conclude that higher contact rate influences the disease to spread.To control the spread, the parameters ,  2 , , and  should have higher value, or  1 should have very small value.
Second, consider   =  2 ( 2 +) ( 2 ++)(+ 1 +) as a function of contact rate  2 and other parameters.Similar as before, the contour plot of   is plotted, and we present it in Figure 6.The result is similar to the case of COVID-19 spread, that higher value of contact rate will make HIV/AIDS to spread.In order to control the disease's spread, we should have higher value of parameters  1 , , , and , or very small value of  2 .
(S): This group comprises individuals who are still healthy but susceptible to infection with COVID-19 or HIV/AIDS.Vaccination subpopulation (VC): This group consists of individuals who are healthy and have been vaccinated against COVID-19.HIV/AIDS protected subpopulation (PH): Individuals in this group have received cellular immune antibody vaccines against the HIV virus, providing protection.Subpopulation infected with COVID-19 (IC): This subpopulation includes individuals who have already been infected with COVID-19.HIV/AIDS-infected subpopulation (IH): Individuals in this subpopulation are already infected with HIV/AIDS.Subpopulations coinfected with COVID-19 and HIV/AIDS (C): This group comprises individuals who are simultaneously infected with both COVID-19 and HIV/AIDS.Treatment subpopulation (TH): This group includes individuals infected with HIV/AIDS who are undergoing treatment.Recovered subpopulation (R): Individuals in this category are either immune to or have recovered from COVID-19 or are protected against HIV/AIDS.

Figure 1 .
Figure 1.The dynamics of the spread of COVID-19 co-infection with HIV/AIDS

Figure 4 .
Figure 4. (a-c) Elasticity index of   ,   , and ℜ  .(d-f) The sensitivity of parameters  1 and  2 on the dynamics of the infected subpopulations   ,   , and .

Figure 5 .
Figure 5. Contour plots of   as a function of the contact rate  1 and other parameters.

Figure 6 .
Figure 6.Contour plots of   as a function of the contact rate  2 and other parameters.