https://orcid.org/0000-0003-4879-0921
Figures
Abstract
HIV/AIDS and COVID-19 co-infection is a common global health and socio-economic problem. In this paper, a mathematical model for the transmission dynamics of HIV/AIDS and COVID-19 co-infection that incorporates protection and treatment for the infected (and infectious) groups is formulated and analyzed. Firstly, we proved the non-negativity and boundedness of the co-infection model solutions, analyzed the single infection models steady states, calculated the basic reproduction numbers using next generation matrix approach and then investigated the existence and local stabilities of equilibriums using Routh-Hurwiz stability criteria. Then using the Center Manifold criteria to investigate the proposed model exhibited the phenomenon of backward bifurcation whenever its effective reproduction number is less than unity. Secondly, we incorporate time dependent optimal control strategies, using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. Finally, we carried out numerical simulations for both the deterministic model and the model incorporating optimal controls and we found the results that the model solutions are converging to the model endemic equilibrium point whenever the model effective reproduction number is greater than unity, and also from numerical simulations of the optimal control problem applying the combinations of all the possible protection and treatment strategies together is the most effective strategy to drastically minimizing the transmission of the HIV/AIDS and COVID-19 co-infection in the community under consideration of the study.
Citation: Kotola BS, Teklu SW, Abebaw YF (2023) Bifurcation and optimal control analysis of HIV/AIDS and COVID-19 co-infection model with numerical simulation. PLoS ONE 18(5): e0284759. https://doi.org/10.1371/journal.pone.0284759
Editor: Oluwole Daniel Makinde, Stellenbosch University, SOUTH AFRICA
Received: January 22, 2023; Accepted: April 8, 2023; Published: May 5, 2023
Copyright: © 2023 Kotola et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Infectious diseases are diagnostically proven illnesses caused by tiny microorganisms such as viruses, bacteria, fungi, and parasites and have been the leading causes of death throughout the world, for example; viruses cause both COVID-19 and HIV/AIDS infections [1–3].
Human immunodeficiency virus (HIV) is one of the most dangerous viruses that is spreading around the world. AIDS, or acquired immunodeficiency syndrome, is one of the most devastating epidemics in history, caused by HIV, which has been a worldwide epidemic since 1981 [4–10]. It remains a significant world health issue that impacts almost seventy million people worldwide and has been a significant cause of morbidity and mortality [11,12]. HIV is transmissible through sexual contact, needle sharing, and direct contact with virus-infected blood or other body fluids, as well as from mother to child during giving birth [10,13–15].
In early December 2019, a coronavirus called COVID-19 was reported in Wuhan, China, with symptoms similar to pneumonia. According to reports, it is one of the most devastating infectious diseases caused by the novel coronavirus SARS-CoV-2, which has been a significant impact on the health, social, and economic integration of communities worldwide [16–29]. On March 11, 2020, the World Health Organization (WHO) confirmed it as a global pandemic, and on July 25, 2020, the world total number of COVID-19 infected individuals was 15,762,007, with 640,276 deaths [25,28,29]. It was suspected to be pneumonia or a common cold-like illness, with symptoms such as fatigue, alter in taste, fever, muscular pains, shortness of breath, ironical cough, and sore throat [25,27,30]. Despite massive efforts to reduce the virus’s transmission and survivability, the death rate from COVID-19 remains high [15]. COVID-19 can be transmitted through sneezing or coughing droplets expelled from the human lungs, as well as when humans come into contact with contaminated dispatched materials [17,26,31]. Among the unfortunate aspects of the COVID-19 pandemic is that patients over the age of 60 are more likely to be infected than anyone below the age of 60 [31]. It is an extremely infectious contagious agent that has spread throughout most of the world’s nations and has a significant impact on the global economy and public health [24,32]. COVID-19 infection may be more common in people with compromised immunity from other infections such as tuberculosis, HIV, pneumonia, and cholera [1,25,33–37]. WHO unanimously implemented vaccination, quarantine, wearing face masks, hand washing with alcohol, and significant discrepancies as possible prevention and control strategies [26,27,31]. Symptomless and pre-symptomatic transmission, a low incidence or lack of dominant systemic symptoms such as fever, airborne transmission that may require a high infectious dose and super-spread events are the essential aspects of COVID-19 spreading that make it challenging to handle [16].
A co-infection is the infection of a single individual with two or more different pathogens or different strains of the same pathogens, leading to co-existence of strains (pathogens) at population level [10]. Co-infection of two or more diseases in one individual is a regular occurrence in today’s society [2,14]. Different researchers have investigated that COVID-19 infection could be high in people living with other infections like TB, HIV, and cholera who have compromised immunity [1,8,21,25,30,33–44].
Mathematical modelling approaches have been crucial to provide basic frameworks in order to understand the transmission dynamics of infectious diseases [37]. Many scholars throughout the world have been formulated and analyzed mathematical models to investigate the transmission dynamics of different infectious diseases using ordinary differential equations approach like [2,9,15,17,19,22,23,26–29,31,32,45–47] using stochastic approach like [48], and using fractional order derivative approach like [1,5,49,50]. In the structure of this study, we have reviewed research papers that have been done on the transmission dynamics of different infectious diseases especially co-infections of HIV/AIDS and other infectious diseases. Teklu and Rao [14] constructed and examined HIV/AIDS and pneumonia co-infection model with control measures such as pneumonia vaccination and treatments of pneumonia and HIV/AIDS infections. Hezam et al. [40], formulated a mathematical model for cholera and COVID-19 co-infection which describes the transmission dynamics of COVID-19 and cholera in Yemen. The model analysis examined four controlling measures such as social distancing, lockdown, the number of test kits to control the COVID-19 outbreak, and the number of susceptible individuals who can get CWTs for water purification. Anwar et al. [15], constructed a mathematical model on COVID-19 with the isolation controlling measure on the COVID-19 infected individuals throughout the community. Ahmed et al. [1] formulated and analyzed HIV and COVID-19 co-infection model with ABC-fractional operator approach to investigate an epidemic prediction of a combined HIV-COVID-19 co-infection model. Numerical simulations were carried out to justify that the disease will stabilize at a later stage when enough protection strategies are taken. Teklu and Terefe [3] analyze COVID-19 and syphilis co-dynamics model to investigate the impacts of intervention measures on the disease transmission.
Similarly, various Scholars have formulated and analyzed mathematical models with optimal control strategies to investigate the effect of prevention and control measures on HIV/AIDS, COVID-19, HIV/AIDS and COVID-19 co-infection and other various infectious diseases transmission throughout nations in the world. For instance, Tchoumi et al. [37] proposed and investigated the co-dynamics of malaria and COVID-19 co-dynamics: with optimal control strategies. The numerical simulation results verifies the theoretical optimal control analysis and illustrates that using malaria and COVID-19 protection measures concurrently can help mitigate there transmission compared with applying single infections protection measures. Omame et al. [25] investigated a mathematical model for the dynamics of COVID-19 infection in order to assess the impacts of prior comorbidity on COVID-19 complications and COVID-19 reinfection with optimal control strategies. The authors recommended that the strategy that prevents COVID-19 infection by comorbid susceptible is the best cost-effective of all the other control strategies for the prevention of COVID-19. Ringa et al. [43] formulated and analyzed a mathematical model on HIV and COVID-19 co-infection with optimal control strategies. Their analysis suggested that COVID19 only prevention strategy is the most effective strategy and it averted about 10,500 new co-infection cases. Keno et al. [51] investigated an optimal control and cost effectiveness analysis of SIRS malaria disease model with temperature variability facto. Their result suggested that the combination of treatment of infected humans and insecticide spraying was proved to be the best efficient and least costly strategy to eradicate the disease. Keno et al. [52] investigated a mathematical model with optimal control strategies for malaria transmission with role of climate variability. Their result suggested that the combination of treated bed net and treatment is the most optimal and least-cost strategy to minimize the malaria. Goudiaby et al. [39] formulated and analyzed a COVID-19 and tuberculosis co-dynamics model with optimal control strategies. They suggested that COVID-19 prevention, treatment and control of co-infection yields a better outcome in terms of the number of COVID-19 cases prevented at a lower percentage of the total cost of this strategy. Asamoah et al. [53] constructed a mathematical model on COVID-19 to investigate optimal control strategies and comprehensive cost-effectiveness. Okosun et al. [54] formulated a mathematical model on HIV/AIDS to investigate the impact of optimal control on the treatment of HIV/AIDS and screening of unaware invectives. Their analysis recommended that the combination of all the control strategies is the most cost-effective strategy. Furthermore, notice that optimal control modeling and cost-effectiveness analysis model have been applied in recent infectious diseases models like [55,56].
As we observed from review of literatures done by various epidemiology and medical scholars, HIV/AIDS and COVID-19 co-infection is a public health concern especially in developing nations of the world. The main purpose of this paper is to investigate the impacts of COVID-19 protection with quarantine, COVID-19 treatment, HIV protection and HIV treatment prevention and controlling strategies on the transmission dynamics of HIV/AIDS and COVID-19 co-infection in the community with mathematical modelling approach. We have reviewed literatures [1,43] invested much effort in studying HIV/AIDS and COVID-19 co-infection, but did not considered COVID-19 protection with quarantine, COVID-19 treatment, HIV/AIDS protection, and HIV/AIDS treatment as prevention and control strategies simultaneously in a single model formulation which motivates us to undertake this study and fill the gap.
2. Mathematical model construction
2.1. Basic frameworks of the model
In this paper, we partitioned the total human population at a given time t denoted by N(t), into eleven mutually-exclusive classes depending on their infection status: susceptible class to both COVID-19 and HIV S(t)), COVID-19 protection by quarantine class (Cq(t)), HIV protected (such as by using condom, limit sexual partners, creating awareness etc.) class (Hp(t)), COVID-19 protection by vaccination class (Cv(t)), COVID-19 mono-infection class (Ci(t)), HIV unaware mono-infection class (Hu(t)), HIV aware mono-infection class (Ha(t)), HIV unaware and COVID-19 co-infection class (Mu(t)), HIV aware and COVID-19 co-infection class (Ma(t)), COVID-19 recovery class (R(t)), and HIV aware treatment class (Ht(t)) so that;
Since HIV is a chronic infectious disease the susceptible individuals acquires HIV infection at the standard incidence rate given by
(1)
where ρ3 ≥ ρ2 ≥ ρ1 ≥ 1 are the modification parameters that increase infectivity and β1 is the HIV transmission rate. Since COVID-19 is a very acute infection the susceptible individuals acquires COVID-19 infection at the mass action incidence rate as stated in [50,51,54].
(2)
where ω2 > ω1 > 1 are the modification parameters that increase infectivity and β2 is the COVID-19 transmission rate.
Additional model assumptions
- k1, k2, k3, and k4 where k4 = 1 − k1 − k2 − k3 are portions of the number of recruited individuals those are entering to the susceptible class, the COVID-19 protected class, the HIV protected class and the COVID-19 vaccination class respectively.
- The susceptible class is increased by individuals from the COVID-19 vaccinated class in which those individuals who are vaccinated against COVID-19 but did not respond to vaccination with waning rate of ρ and from COVID-19 recovery with treatment class who develop their temporary immunity by the rate η.
- COVID-19 vaccine is may not be 100% efficient, so vaccinated individuals also have a chance of being infected with portion ε of the serotype not covered by the vaccine where 0 ≤ ε < 1.
- 0 < υ ≤ 1 is the modification parameter such that COVID-19 infected individual is less susceptible to HIV infection than a susceptible individuals due to morbidity.
- There is screening and testing mechanisms for the previous and current status in each class.
- The human population distribution is homogeneous in each class.
- HIV treated individuals do not transmit infection to others due to awareness.
- Population of human being is variable.
- There is no dual-infection transmission simultaneously.
- No vertical HIV transmission.
- No permanent immunity for COVID-19 infection.
In this section using parameters given in Table 1, model variables given in Table 2, and the model basic frame work, and assumptions given in (2.1), the schematic diagram for the transmission dynamics of HIV/AIDS and COVID-19 co-infection is given by Fig 1.
Now using Fig 1 the system of differential equations of the HIV/AIDS and COVID-19 co-infection is given by
(3)
with the corresponding initial conditions
(4)
The sum of all the differential equations in (3) is
(5)
2.2. The basic qualitative properties of the model (3)
The COVID-19 and HIV/AIDS co-infection model given in Eq (3) is both biologically and mathematically meaningful if and only if all the model solutions (state variables) are non-negative and bounded in the invariant region
(6)
Theorem 1 (Positivity of the model solutions)
Let us given the initial data in Eq (4) then the solutions S(t), Hp(t), Cv(t), Ci(t), Mu(t), Hu(t), Ha(t), Ma(t), R(t), Cq(t), and Ht(t) of the COVID-19 and HIV/AIDS co-infection model (3) are nonnegative for all time t > 0.
Proof: Let us consider S(0) > 0, Cq(0) > 0, Hp(0) > 0, Cv(0) > 0, Ci(0) > 0, Hu(0) > 0, Ha(0) > 0, Mu(0) > 0, Ma(0) > 0, R(0) > 0, and Ht(0) > 0 then for all t > 0.
We have to show that S(t) > 0, Cq(t) > 0, Hp(t) > 0, Cv(t) > 0, Ci(t) > 0, Hu(t) > 0, Ha(t) > 0, Mu(t) > 0, Ma(t) > 0, R(0) > 0, and Ht(t) > 0.
Define: τ = sup{S(t) > 0, Cq(t) > 0, Hp(t) > 0, Cv(t) > 0, Ci(t) > 0, Hu(t) > 0, Ha(t) > 0, Mu(t) > 0, Ma(t) > 0, R(0) > 0, and Ht(t) > 0}. Now since the entire co-infection model state variables are positive and all the state variables are continuous, we can justify that τ > 0. If τ = +∞, then non-negativity holds. But, if 0 < τ < +∞ we will have S(τ) = 0 or Cq(τ) = 0 or Hp(τ) = 0 or Cv(τ) = 0 or Ci(τ) = 0 or Hu(τ) = or Ha(τ) = 0 or Mu(τ) = 0 or Ma(τ) = 0 or R(τ) = 0 or Ht(τ) = 0.
Here from the first equation of the COVID-19 and HIV/AIDS co-infection model (3) we have got
and integrate using method of integrating factor we have determined the constant value
where
and from the meaning of τ, the solutions Cq(t) > 0, Hp(t) > 0, Cv(t) > 0, R(t) > 0. Moreover, the exponential function is always positive, then the solution S(τ) > 0 hence S(τ) ≠ 0. Thus following the same procedure for τ = +∞, all the solutions of the COVID-19 and HIV/AIDS co-infection system (3) are non-negative.
Theorem 2 (The invariant region): All the feasible positive solutions of the co-infection model (3) are bounded in the region (6).
Proof: Let is an arbitrary non-negative solution of the system (3) with initial conditions given in Eq (4). Now adding all the differential equations given in Eq (3) we have got the derivative of the total population N which is given in Eq (5) as
Then by ignoring the infections we have determined that and using separation of variables whenever t → ∞, we have obtained that
. Hence, all the positive feasible solutions of the co-infection model (3) entering in to the region given in Eq (6).
Note: Since the model (3) solutions are both positive and bounded in the region (6) the HIV/AIDS and COVID-19 co-infection model (3) is both mathematically and biologically meaning full [45,47,57], then we can consider the two mono-infection models, namely; HIV mono-infection and COVID-19 mono-infection models. This is fundamental for the analysis of the COVID-19 and HIV/AIDS co-infection model.
3. Analytical result of the models
Before analyzing the HIV/AIDS and COVID-19 co-infection model given in Eq (3), it is very crucial to gain some basic backgrounds about the COVID-19 and HIV/AIDS mono-infection models.
3.1. Mathematical analysis of HIV/AIDS mono-infection model
In this subsection we assume there is no COVID-19 infection in the community i.e. Cq =, Cq = Ci = Mu = Ma = R = 0 in (3) then the HIV/AIDS sub-model is given by
(7)
where the total population N1(t) = S(t) + Hp(t) + Hu(t) + Ha(t) + Ht(t), and the HIV sub-model force of infection given by
and initial conditions S(0) > 0, Hp(0) ≥ 0, Ha(0) ≥ 0, Hu(0) ≥ 0 and Ht(0) ≥ 0. In a similar manner of the full co-infection model (3) in the region
it is sufficient to consider the dynamics of the sub-model (7) in Ω1 as biologically and mathematically well-posed.
3.2. Disease-free equilibrium point of HIV mono-infection model (7) local stability
The disease-free equilibrium point of the HIV mono-infection system in (7) is obtained by making its right-hand side is equal to zero and setting the infected classes and treatment class to zero as Hu = Ha = Ht = 0 which yields, . Hence the disease-free equilibrium point is given by
.
The local stability of the HIV mono-infection model (7) disease-free equilibrium point is examined by its effective reproduction number denoted by , which is calculated by using the next generation operator method determined by Van den Driesch and Warmouth stated in [2]. Applying the method stated in [29], the transmission matrix F and the transition matrix V i.e., for the new infection and the remaining transfer respectively, are given by
After some computations we have determined that
and
Then, the effective reproduction number of the HIV mono-infection model (7) is defined as the largest eigenvalue in magnitude of the next generation matrix, FV−1 given by
The value is defined as the total average number of secondary HIV unaware and HIV aware infection cases acquired from a typical HIV unaware or HIV aware individual during his/her effective infectious period in a susceptible population. The threshold result
is the effective reproduction number for HIV mono-infection.
Theorem 3: The disease-free equilibrium point of the HIV mono-infection model given in Eq (7) is locally asymptotically stable (LAS) if , and it is unstable if
.
Proof: The local stability of the disease-free equilibrium point of HIV mono-infection model (7) is evaluated by applying the Routh-Hurwitz stability criteria stated in [52].
The Jacobian matrix of the HIV mono-infection model given in Eq (7) at the disease-free equilibrium point is given by
Then the corresponding characteristic equation of the Jacobian matrix is given by
Finally we have determined
where
, and
.
Then we have got λ1 = −μ < 0 or λ2 = −(α2 + μ) < 0 or λ3 = −μ < 0 or
(8)
On Eq (8) we applied Routh-Hurwitz stability criteria stated in [47] and we have determined that both eigenvalues are negative if . Furthermore, we can conclude that the disease-free equilibrium point of the model (7) is locally asymptotically stable whenever
since all the eigenvalues are negative when
. The biological meaning of Theorem 3 can be stated as HIV infection can be eradicated from the population (whenever
) if the initial size of the sub-populations of the HIV mono-infection model given in Eq (7) is in the basin of attraction of the disease-free equilibrium point
.
3.3. Existence of HIV mono-infection endemic equilibrium point(s)
Let be an arbitrary endemic equilibrium point of the HIV mono-infection model (7) which can be determined by making the right hand side of Eq (7) as zero. The after a number of steps of computations we have got
(9)
where m1 = (α2 + μ), m2 = (θ + μ + d2), and m3 = (γ + d3 + μ).
Now substitute and
given in Eq (9) in to the HIV/AIDS force of infection
Then we have the result
(10)
where m4 = β1k1Δm1m3m3μ + β1α2k3Δm3m3μ + β1ρ1k1Δθm1m3μ + β1ρ1α2k3Δθm3μ, m5 = k1Δm1m2m3μ + α2k3Δm2m3μ + k3Δm2m3μμ, m6 = k3Δm2m3μ + k1Δm1m3μ + α2k3Δm3μ + k1Δθm1μ + α2k3Δθμ + k1Δθγm1 + α2k3Δθγ.
Then the non-zero solution of (10) is . Therefore, the required non-zero solution (force of infection is obtained as
. Then we have got
whenever
. Thus, the HIV/AIDS mono-infection model (7) has a unique positive endemic equilibrium point if and only if
.
Theorem 4: The HIV/AIDS mono-infection model given in (7) has a unique endemic equilibrium point if and only if .
3.4. COVID-19 sub-model analysis
The corresponding COVID-19 sub-model of the system (3) is determined by making Hp = Ha = Hu = Mu = Ma = Ht = 0, and it is given by
(11)
with COVID-19 infection initial conditions S(0) > 0, Cq(0) ≥ 0, Cv(0) ≥ 0, Ci(0) ≥ 0, R(0) ≥ 0, total population N2(t) = S(t) + Cq(t) + Cv(t) + Ci(t) + R(t), and COVID-19 force of infection given by λC = β2Ci(t). Here like the full model (3) and the HIV/AIDS sub-model (7) in the region
, it is sufficient to consider the dynamics of model (11) in Ω2 be both biologically and mathematically meaningful.
3.4.1. Local stability of COVID-19 mono-infection model (11) Disease-free equilibrium.
Disease-free equilibrium point of the COVID-19 mono-infection model (11) is obtained by making its right-hand side as zero and setting the infected class and recovered with treatment class to zero as Ci = R = 0 and after some simple steps of calculations we have determined that ,
, and
. Hence the COVID-19 mono-infection model (11) disease-free equilibrium point is given by
Here we are applying the Van Den Driesch and Warmouth next-generation matrix approach stated in [2] to determine the COVID-19 mono-infection model (11) effective reproduction number . After long computations, we have determined the transmission matrix given by
and the transition matrix given by
Then using Mathematica we have determined as
The characteristic equation of the matrix FV−1 is .
Then the spectral radius (effective reproduction number ) of FV−1 of the COVID-19 mono-infection model (11) is
.
Theorem 5: The Disease-free equilibrium point of the COVID-19 mono-infection model (11) is locally asymptotically stable if
otherwise unstable.
Proof: The local stability of the disease-free equilibrium of the system (11) at point can be studied from its Jacobian matrix and Routh-Hurwitz stability criteria. The Jacobian matrix of the dynamical system at the disease-free equilibrium point is given by
Then the characteristic equation of the above Jacobian matrix is given by
where
and after some steps of computations we have got λ1 = −μ < 0 or λ2 = −(α1 + μ) < 0 or λ3 = −(ρ + μ) < 0 or
if
or λ5 = −(μ + η) < 0.
Therefore, since all the eigenvalues of the characteristics polynomials of the system (11) are negative if the disease-free equilibrium point of the COVID-19 mono-infection model (11) is locally asymptotically stable.
3.4.2. Existence of endemic equilibrium point (s) of the COVID-19 mono-infection model.
Before checking the global stability of the disease-free equilibrium point of the COVID-19 mono-infection model (11), we shall find the possible number of endemic equilibrium point(s) of the model (11). Let be the endemic equilibrium point of COVID-19 mono-infection and
be the COVID-19 mono-infection mass action incidence rate (“force of infection”) at the equilibrium point. To find equilibrium point(s) for which COVID-19 mono-infection is endemic in the population, the equations are solved in terms of
at an endemic equilibrium point. Now setting the right-hand sides of the equations of the model to zero (at steady state) gives
and
where b1 = α1 + μ, b2 = ρ + μ, b3 = μ + d1 + κ, b4 = μ + η, b5 = k1Δb1b3b4, b6 = α1k2Δb3b4, b7 = ρk4Δb1b3b4, b8 = k4Δb1ηκε, b9 = b1b3b4k4Δε, b10 = b2b1ηκk4Δε, b11 = b1k4Δεηκε, b12 = b1b3b3b4, b13 = b1b3ηκ.
Then we have substituted in the COVID-19 force of infection given by
we have got the non-zero solution of
is obtained from the cubic equation
(12)
where
(13)
It can be seen from and (13) that c3 > 0 (since the entire model parameters are nonnegative). Furthermore, c0 > 0 whenever . Thus, the number of possible positive real roots the polynomial (12) can have depends on the signs of c1, and c2. This can be analyzed using the Descartes’ rule of signs on the cubic f(x) = c3x3 + c2x2 + c1x + c0 (with
). Hence, the following results are established.
Theorem 6: The COVID-19 mono-infection model (11) could have
- (a). a unique endemic equilibrium point if
either of the following holds.
- (i) c1 > 0 and c2 > 0.
- (ii) c1 < 0 and c2 < 0.
- (b). more than one endemic equilibrium point if
either of the following holds.
- (i) c1 > 0 and c2 < 0.
- (ii) c1 < 0 and c2 > 0.
- (c). two endemic equilibrium points if
and c2 < 0.
Here, item (c) shows the happening of the backward bifurcation in the model (11) i.e., the locally asymptotically stable disease-free equilibrium point co-exists with a locally asymptotically stable endemic equilibrium point if ; examples of the existence of backward bifurcation phenomenon in mathematical epidemiological models, and the causes, can be seen in [8,17,26,31,58–60]. The epidemiological consequence is that the classical epidemiological requirement of having the reproduction number
to be less than one, even though necessary, is not sufficient for the effective control of the disease. The existence of the backward bifurcation phenomenon in sub-model (11) is now explored.
Theorem 7: The COVID-19 mono-infection model (11) exhibits backward bifurcation at whenever the inequality D2 > D1 holds, where
and
.
In this section, we have used the center manifold theory stated in [60] to ascertain the local asymptotic stability of the endemic equilibrium due to the convolution of the first approach (eigenvalues of the Jacobian). To make use of the center manifold theory, the following change of variables is made by symbolizing S = x1, Cp = x2, Cv = x3, Ci = x4 and R = x5 such that N2 = x1 + x2 + x3 + x4 + x5. Furthermore, by using vector notation X = (x1, x2, x3, x4, x5)T, the COVID-19 mono-infection model (11) can be written in the form with F = (f1, f2, f3, f4, f5)T, as follows
(14)
with λC = β2x4 then the method entails evaluating the Jacobian of the system (14) at the DFE point
, denoted by
and this gives us
Consider, and suppose that β2 = β* is chosen as a bifurcation parameter. From
as
.
Solving for β2 we have got .
After some steps of the calculation we have determined the eigenvalues of Jβ* as λ1 = −μ, λ2 = −(α1 + μ) or or λ3 = −(ρ + μ) or λ4 = 0 or λ5 = −(μ + η). It follows that the Jacobian of Eq (14) at the disease-free equilibrium with β2 = β*, denoted by Jβ*, has a simple zero eigenvalue with all the remaining eigenvalues have negative real part. Hence, Theorem 2 of Castillo-Chavez and Song stated in [60] can be used to analyze the dynamics of the model to show that the model (11) undergoes backward bifurcation at
.
Eigenvectors of Jβ*: For the case , it can be shown that the Jacobian of the system (14) at β2 = β* (denoted by Jβ*) has a right eigenvectors associated with the zero eigenvalue given by u = (u1, u2, u3, u4, u5)T as
(15)
Then solving Eq (15) the right eigenvectors associated with the zero eigenvalue are given by
Similarly, the left eigenvector associated with the zero eigenvalues at β2 = β* given by v = (v1, v2, v3, v4, v5)T as
(16)
where
.
Then solving Eq (16) the left eigenvectors associated with the zero eigenvalue are given by v1 = v2 = v3 = v4 = 0 and v4 = v4 > 0. After long steps of calculations the bifurcation coefficients a and b are obtained as
where
, and
. Thus, the bifurcation coefficient a is positive whenever D2 > D1.
Hence, from the theory of Castillo-Chavez and Song stated in [60] the COVID-19 mono-infection model (11) exhibits a phenomenon of backward bifurcation at and whenever D2 > D1.
The diagram representation of this bifurcation is given in Fig 2 below.
Fig 2 shows the appearance of backward bifurcation, which results in the coexistence of several equilibrium points. In such a case, the common conditions of disease eradication such as making RC < 1 will not work, and the initial number of infected persons also plays a crucial role.
3.5. Analytical result of HIV/AIDS and COVID-19 co-infection model
3.5.1. Disease-free equilibrium point.
The disease free equilibrium point of the dynamical system (3) when the state variable Ci = Hu = Ha = Mu = Ma = 0 is given by
3.5.2. Effective reproduction number of the co-infection model.
The effective reproduction number of the dynamical system (3) by applying the next generation operator method is the largest (dominant) eigenvalue (spectral radius) of the matrix: , where
is the rate of appearance of new infection in compartment i, vi is the transfer of infections from one compartment i to another, and E0 is the disease-free equilibrium point. After some steps of calculations we have determined that
where
, and
Applying Mathematica we have determined as
After some computations and simplifications we have determined the dominant eigenvalue in magnitude of the matrix FV−1 which is the HIV/AIDS and COVID-19 co-infection effective reproduction number given by
where
is the COVID-19 effective reproduction number and
is the HIV/AIDS effective reproduction number.
3.5.3. Locally asymptotically stability of the disease-free equilibrium point.
The Jacobian matrix of the system (3) at disease free equilibrium point is given as
where
and
Then the eigenvalues of the matrix J(E0) are λ1 = −μ < 0 or λ2 = −(α1 + μ) < 0 or λ3 = −(α2 + μ) < 0 or λ4 = −(ρ + μ) < 0 or λ5 = −μ < 0 or λ6 = −(μ + η) < 0 or or λ8 = −(μ + d4 + δ + θ1) < 0 or λ9 = −(μ + d5 + Θ2) < 0 or λ2 + [(γ + d3 + μ) + (θ + μ + d2) − ℵ8]λ − [(ℵ8 − (θ + μ + d2))(γ + d3 + μ) + θρ1ℵ8] = 0.
Then after some calculations we have got the last two eigenvalues of the quadratic equation as λ10 < 0 and λ11 < 0 whenever . Thus, since all the eigenvalues are negative, the disease-free equilibrium point of the full model (3) is locally asymptotically stable whenever
.
3.5.4. Global asymptotic stability of disease-free equilibrium point.
In this sub-section we have used the method derived by Castillo-Chavez et al. and stated in reference [61] to look into the global asymptotic stability (GAS) of the co-infection model (3) disease-free equilibrium point. We mention two requirements that, if satisfied, also ensure the disease-free equilibrium is globally asymptotically stable. Then the new system (3) is rewritten as:
where
denotes the number of uninfected components and
denotes the number of infected components. Π0 = (Ψ0, 0), denotes the disease-free equilibrium point of the system. The following requirements must be satisfied to ensure the globally asymptotic stability:
- (H1) For
, Π0 is globally asymptotically stable.
- (H2)
, for (Ψ, Υ) ∈ Ω, where A = DΥG(Ψ0, 0) is a Metzler matrix (the off diagonal elements of A are nonnegative) and Ω is the region where the model makes biological sense.
Theorem 8: The fixed point Π0 = (Ψ0, 0) is a globally asymptotically stable equilibrium point of system (3) provided and the assumptions (H1) and (H2) are satisfied otherwise unstable.
Proof: The system (1) is rewritten as
where Ψ represents the number of non-infectious compartments and Υ represents the number of infectious compartments.
Then , where,
where
and Σ4 = −(γ + d3 + μ), so that
It is clear from the above discussion, that, . Hence by the same reason given by results in reference [38], the disease-free equilibrium point may not be globally asymptotically stable.
4. Analysis of the optimal control strategy
In this section, we provide a thorough qualitative analysis of the time-dependent HIV/AIDS and COVID-19 co-infection model (3). The Pontryagin’s Maximum Principle stated in literatures [25,43,51,52,55] is used to describe this analysis, with the aim of minimizing the HIV/AIDS infection aware individuals denoted by Ha, the COVID-19 infected individuals denoted by Ci and the total HIV/AIDS and COVID-19 co-infected individuals denoted by Mu + Ma. In the case of time-dependent optimal control, we employ Pontryagin’s Maximum Principle to derive the necessary conditions for diseases control mechanisms. After incorporating the controls into the HIV/AIDS and COVID-19 co-infection transmission model (3), the optimal control problem is as follows:
(17)
with the corresponding initial conditions
(18)
and
represents HIV/AIDS infection protective control,
represents the COVID-19 infections protective control using quarantine,
represents the COVID-19 infection treatment control, and
represents the HIV/AIDS treatment control.
The objective is to find the optimal control values of the controls
such that the associated state trajectories
are solution of the optimal control system (17) in the intervention time interval [0, Tf] with initial conditions as given in (18) and minimize the objective functional given by
(19)
where the coefficients
and
are positive weight constants and
and
are the measure of relative costs of interventions associated with the controls
and
, respectively, and also balances the units of integrand. In the cost functional, the term
refer to the cost related to COVID-19 infected class, the term
refer to the cost related to individuals mono-infected with HIV and aware, the term
refer to the cost related to co-infected individuals unaware of HIV infection and the term
refer to the cost related to co-infected individuals aware of HIV infection.
, measures the current cost at time t. The set of admissible Lebesgue measurable control functions is defined by
(20)
More precisely, we seek an optimal control pair
(21)
Theorem 9 (Existence Theorem): There exists an optimal control in
and a corresponding solution vector
to the optimal control dynamical system (17) with the initial values (18) such that
.
Note: We utilize Pontryagin’s Maximal principle stated in literatures [51,52,55], to determine the prerequisites for the optimal control model (17). The optimal control problem (17) and (19) defined Hamiltonian (H) function is expressed as
(22)
where
stands for the ith state variable equation and λ1(t), λ2(t), λ3(t), λ4(t), λ5(t), λ6(t), λ7(t), λ8(t), λ9(t), λ10(t) and λ11(t) are adjoint variables. Similarly to obtain the co-state variables by using Pontryagin’s Maximum Principle stated in literatures [51,52,55], with the existence result the following theorem is stated:
Theorem 10: Let be the optimal control and
be the associated unique optimal solutions of the optimal control problem (17) with initial condition (18) and objective functional (19) with fixed final time Tf (20). Then there exists adjoint function
satisfying the following canonical equations
(23)
with transiversality conditions
(24)
Moreover, the corresponding optimal controls and
are given by
(25)
Proof: To obtain the form of the co-state equations we compute the derivative of the Hamiltonian function (), given in (22), with respect to
and
respectively. Then the adjoint or co-state equations obtained are given by:
(26)
with transiversality conditions
(27)
To obtain the control values, we compute the partial derivative of the Hamiltonian, given by:
(28)
Moreover, the corresponding optimal controls with the boundary condition of each control and
are given by
(29)
From the previous analysis, to get the optimal point, we have to solve the system
with the Hamiltonian
where
5. Numerical results
In this section we have presented the numerical result we have obtained using the parameters value collected in Table 3 below. We have collected data from a variety of sources, and have compiled the values in the table for the convenience of the constructed model numerical simulations and to verify the analytical results.
5.1. Numerical simulations and discussions of the deterministic model (3)
In this section, a numerical simulation of the entire HIV/AIDS and COVID-19 co-infection model given in Eq (3) is performed. We used ode45 fourth order Runge-Kutta scheme to examine the effect of various parameters on the spread and control of COVID-19 mono-infection, HIV/AIDS mono-infection, and HIV/AIDS and COVID-19 co-infection. The parameter values presented in Table 3 are used for numerical simulation. Moreover, we have investigated the stability of the endemic equilibrium point of the co-infection model (3), the effects of parameter on reproduction numbers, and the impact of treatment primarily on dually-infected individuals in the community.
5.2. Simulation of co-infection model (3) whenever 
The above Fig 3 was plotted using ode45 Runge-Kutta fourth order method to observe the numerical simulation of the full co-infection model (3) by using parameter values from Table 3. We can deduce from the figure that after a year, the solutions of the COVID-19 and HIV/AIDS co-infection dynamical system (3) are approaching the endemic equilibrium point of the given dynamical system whenever the co-infection effective reproduction number .
5.3. Numerical simulation to show the effect of k3 on 
The effect of the HIV protection rate on the HIV/AIDS effective reproduction number is depicted in Fig 4. The graph shows that as the value of protection rate k3 increases, the effective reproduction number
decreases and for k3 > 0.771 indicates that
is reduced to less than one. As a result, the public health and policymakers must focus on increasing the values of the HIV/AIDS protection rate k3 in order to control HIV/AIDS spread which may causes for existence of co-infection in the community.
5.4. Simulation to show the effect of κ on 
A numerical simulation in order to show the effect of COVID-19 treatment on the COVID-19 effective reproduction number is given by Fig 5. The graph shows that as the value of the treatment rate raises, the COVID-19 basic reproduction number decreases and for the value of κ > 0.776 implies that
.
5.5. Numerical simulation to show the effect of k2 on 
Fig 6 depicted the effect of the COVID-19 protection rate k2 on the COVID-19 effective reproduction number . As we can observe from the graph as the value of k2 increases, the COVID-19 effective reproduction number decreases, and k2 > 0.654 implies that
. As a result, all the stakeholders must focus on increasing the values of COVID-19 quarantine rate k2 in order to prevent and control COVID-19 spread in the community. Biologically, this means that COVID-19 infection decreases as the quarantine rate k2 rises.
5.6. Numerical simulation to show effect of β2 on 
Fig 7 shows the influence of the COVID-19 transmission rate β2 on the COVID-19 effective reproduction number . The graph shows that as the value of β2 rises, so does the COVID-19 effective reproduction number and the value of β2 < 0.225 means that
. As a result, public health authorities must focus on reducing the value of COVID-19 transmission rate β2 in order to avoid and regulate COVID-19 spread in the community.
5.7. Simulation to show effect of β1 on 
Fig 8 depicts a numerical simulation on the influence of HIV transmission rate β1 on the HIV/AIDS effective reproduction number . The graph shows that as the value of β1 grows, so does the HIV/AIDS effective reproduction number and whenever β1 < 0.193 significantly
reduces to less than unity. Therefore it is recommendable to give an attention on minimizing the value of the HIV transmission rate β1 to prevent and control HIV/AIDs expansion in the community. Biologically, this indicates that the HIV/AIDS infection lowers as the transmission rate β1 drops.
5.8. Simulation to show effect of k4 on 
Fig 9 looked at how the COVID-19 immunization (vaccination) rate k4 affected the COVID-19 effective reproduction number . The graph shows that when the value of k4 grows, the COVID-19 effective reproduction number decreases, and values of k4 > 0.9 suggest that
As a result, public health authorities must focus on increasing the COVID-19 immunization rate k4 in order to prevent and control COVID-19 spread in the community. Biologically, this indicates that the COVID-19 infection reduces as the immunization rate k4 rises.
5.9. Numerical simulation to show effect of κ on COVID-19 infectious (Ci)
Fig 10 examined the effect of COVID-19 treatment rate on the number of COVID-19 mono-infectious population. The graph shows that when the value of κ increases, the number of COVID-19 mono-infectious people decrease. As a result, public officials should focus on increasing the value of the treatment rate at which COVID-19 infected individuals recovered from COVID-19 illness increase.
5.10. Simulation to show effect of θ1 on the co-infectious (Mu)
Fig 11 looked at how θ1 affected the number of COVID-19 and HIV/AIDS co-infected individuals. The graph shows that when the value of COVID-19 treatment rate θ1 rises, the number of COVID-19 and HIV/AIDS co-infected individuals’ decreases. As a result, public officials should focus on maximizing the value of COVID-19 treatment rate θ1 in COVID-19 infected persons.
5.11. Simulation to show effect of θ2 on the co-infectious (Ma)
Fig 12 show that the impact of θ2 on the number of COVID-19 and HIV/AIDS co-infected people. The graph shows that when the value of the COVID-19 treatment rate θ2 rises, the number of COVID-19 and HIV/AIDS co-infected individuals decrease. As a result, public officials must focus on maximizing the value of COVID-19 treatment rate θ2 in COVID-19 infected persons.
5.12. Numerical simulations of optimal control strategies
To verify the analytical results, the optimal control model system (17) is simulated using the parameter values given in Table 3 with positive weight constants w1 = w2 = w3 = w4 = 18. The optimal control system is composed of two dynamical systems, the state dynamical system (17) and the adjoint dynamical system (27), each with its own initial and final-time conditions, with the control value state in Eq (26). The fourth forward-backward Runge-Kutta iterative method is used to solve this optimality system. The state Eq (17) is solved with the initial values of state variables using the fourth-order forward Runge-Kutta method. We used backward fourth order Runge-Kutta to solve the adjoint equations once we had the solution of the state functions and the value of optimal controls. To determine the impact of control measures on the reduction of the HIV/AIDS and COVID-19 co-infection we have the following three cases of optimal control strategies:
- Case 1: Controlling HIV infection Ha with the combinations of strategies: strategy 1: use
, and
, strategy 2: use
, and
and strategy 3: use
and
.
- Case 2: Controlling COVID-19 infection Ci with the combinations of strategies: strategy 1: use
, and
, strategy 2: use
, and
and strategy 3: use
, and
.
- Case 3: Controlling the total HIV/AIDS and COVID-19 co-infection Mu + Ma with the combinations of strategy 1: use the strategy
,
,
, and
strategy 2: use the strategy
,
,
, and
strategy 3: use the strategy
,
,
, and
strategy 4: use the strategy
,
5.13. HIV infection (Ha) simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the HIV/AIDS infection (Ha) when there is no control strategy in place and when there is only HIV/AIDS treatment control measure. Fig 13 shows that the HIV/AIDS treatment control measure efforts are implemented then the number of individuals infected with HIV decreases throughout time to zero.
5.14. HIV infection simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the HIV/AIDS infection (Ha) when there is no control strategy in place and when there is only HIV/AIDS protection control measure. Fig 14 shows that the HIV/AIDS protection control measure efforts are implemented then the number of individuals infected with HIV decreases throughout time to zero.
5.15. HIV infection simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the HIV/AIDS infection (Ha) when there is no control strategy in place and when there are HIV/AIDS protection and treatment control measures. Fig 15 shows that the HIV/AIDS protection and treatment control measures efforts are implemented then the number of individuals infected with HIV/AIDS decreases rapidly to zero after seven years.
5.16. COVID-19 infection simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the COVID-19 infection (Ci) when there is no control strategy in place and when there is COVID-19 treatment control measure. Fig 16 shows that the COVID-19 treatment control measure effort is implemented then the number of individuals infected with COVID-19 decreases to zero through time.
5.17. COVID-19 infection simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the COVID-19 infection (Ci) when there is no control strategy in place and when there is COVID-19 protection control measure. Fig 17 shows that the COVID-19 protection control measure effort is implemented then the number of individuals infected with COVID-19 decreases to zero after five years.
5.18. COVID-19 infection simulation with strategy 1 (
, and 
)
In this subsection simulation is done for the COVID-19 infection (Ci) when there is no control strategy in place and when there are COVID-19 protection and treatment control measures. Fig 18 shows that the COVID-19 protection and treatment control measures efforts are implemented then the number of individuals infected with COVID-19 decreases quickly to zero.
5.19. Co-infection simulation with strategy 1 (
, 
, 
, and 
)
In this subsection simulation is done for the cumulated HIV/AIDS and COVID-19 co-infection when there is no control strategy in place and when there are controls involving COVID-19 protection, treatments for both HIV and COVID-19 single infections without HIV protection measure. Fig 19 shows the result that all the prevention and control strategies except HIV protection efforts are implemented, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero after year seven.
5.20. Co-infection simulation with strategy 2 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are controls involving HIV protection, treatment strategies for both HIV and COVID-19 single infections without COVID-19 protection measure. Fig 20 shows the result that all the prevention and control strategies except COVID-19 protection efforts are implemented, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero.
5.21. Co-infection simulation with strategy 3 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are controls involving HIV protection, COVID-19 protection, and HIV treatment without COVID-19 treatment measure. Fig 21 shows the result that all the prevention and control strategies except HIV treatment strategy efforts are implemented, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero after 7 years.
5.22. Co-infection simulation with 4 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are controls involving HIV protection, COVID-19 protection, and COVID-19 treatment without HIV treatment measures. Fig 22 shows the result that all the prevention and control strategies except HIV treatment strategy efforts are implemented, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero after 8 years.
5.23. Co-infection simulation with strategy 5 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are controls involving treatment strategies for COVID-19 and HIV single infection without HIV and COVID-19 protection measures. Fig 23 shows the result that treatment strategies efforts are implemented without protection strategies, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero in the long run.
5.24. Co-infection simulation with strategy 6 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are control strategies involving protection strategies for COVID-19 and HIV single infection without HIV and COVID-19 treatment measures. Fig 24 shows the result that protection strategies efforts are implemented without treatment strategies, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero after 8 years later.
5.25. Co-infection simulation with strategy 7 (
, 
, 
, and 
)
In this subsection simulation is done when there is no control strategy in place and when there are all the control strategies involving protection and treatment for both COVID-19 and HIV single infections. Fig 25 shows the result that all the protection and treatment strategies efforts are implemented, the number of individuals co-infected with HIV and COVID-19 decreases drastically to zero after 3.
6. Conclusions
In this paper, we formulated and investigated a continuous time dynamical model for the transmission of HIV/AIDS and COVID-19 co-infection with protection and treatment strategies. The mode incorporate four non-infectious groups the susceptible group, the HIV protection group, the COVID-19 protection group, and the COVID-19 vaccinated group and this made the model highly nonlinear and challenging for the qualitative analysis of the co-infection model. The model has been mathematically analyzed both for the sub-models associating the cases that each disease type is isolated and in the case when there is co-infection. In addition an optimal control problem model that minimizes the cost of the infection as well as minimizes the control efforts to control the diseases transmission in the community is formulated and analyzed. The model includes the intervention strategies, protective as well as treatment and numerical simulations of both the deterministic model and optimal control problem models are presented. In the analysis it has been indicated that the effect of protection as well as treating the infected ones with the available treatment mechanisms affects significantly the optimal control strategy and its outcome. From the optimal control problem simulation results it can be concluded that applying both protective and treatment control mechanisms at the population level yields both economic as well as epidemiologic gains. Therefore, we recommended to the stake holders to give more attention and the overall optimal effort to implement both the protective as well as treatment control strategies to minimize the single infections as well as the co-infection diseases transmission in the community.
This study did not considered the stochastic method, fractional order method, impacts of the environment, structure of human age, the spatial structure, and real population primary epidemiological data. Based on these limitations potential researcher can consider to extend this study.
Acknowledgments
We would like to give credit to Mr.Stotaw Ehete and Adugna Safeyi for their personal Wi-Fi contribution.
References
- 1. Ahmed Idris, Doungmo Goufo Emile F, Yusuf Abdullahi, Kumam Poom, Chaipanya Parin, et al. "An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator." Alexandria Engineering Journal 60, no. 3 (2021): 2979–2995.
- 2.
Martcheva Maia. An introduction to mathematical epidemiology. Vol. 61. New York: Springer, 2015.
- 3. Teklu Shewafera Wondimagegnhu, and Terefe Birhanu Baye. "COVID-19 and syphilis Co-Dynamics Analysis using Mathematical Modelling Approach." Frontiers in Applied Mathematics and Statistics 8: 140.
- 4. Aggarwal Rajiv. "Stability analysis of a delayed HIV-TB co-infection model in resource limitation settings." Chaos, Solitons & Fractals 140 (2020): 110138.
- 5. Aslam Muhammad, Murtaza Rashid, Abdeljawad Thabet, Khan Aziz, Khan Hasib, and Gulzar Haseena. "A fractional order HIV/AIDS epidemic model with Mittag-Leffler kernel." Advances in Difference Equations 2021, no. 1 (2021): 1–15.
- 6. Awoke Temesgen Debas, and Kassa Semu Mitiku. "Optimal control strategy for TB-HIV/AIDS Co-infection model in the presence of behaviour modification." Processes 6, no. 5 (2018): 48.
- 7. Babaei Afshin, Jafari Hossein, and Liya Atena. "Mathematical models of HIV/AIDS and drug addiction in prisons." The European Physical Journal Plus 135, no. 5 (2020): 1–12.
- 8. Huo Hai-Feng, and Chen Rui. "Stability of an HIV/AIDS treatment model with different stages." Discrete Dynamics in Nature and Society 2015 (2015).
- 9. Nwankwo A., and Okuonghae D. "Mathematical analysis of the transmission dynamics of HIV syphilis co-infection in the presence of treatment for syphilis." Bulletin of mathematical biology 80, no. 3 (2018): 437–492. pmid:29282597
- 10. Teklu Shewafera Wondimagegnhu, and Mekonnen Temesgen Tibebu. "HIV/AIDS-pneumonia co-infection model with treatment at each infection stage: mathematical analysis and numerical simulation." Journal of Applied Mathematics 2021 (2021).
- 11. Nthiiri Joyce K., Lavi G. O., and Mayonge A. "Mathematical model of pneumonia and HIV/AIDS coinfection in the presence of protection." Int J Math Anal 9, no. 42 (2015): 2069–2085.
- 12. Omondi E. O., Mbogo R. W., and Luboobi L. S. "Mathematical analysis of sex-structured population model of HIV infection in Kenya." Letters in Biomathematics 5, no. 1 (2018): 174–194.
- 13. Seidu Baba, Makinde O. D., and Bornaa Christopher S. "Mathematical analysis of an industrial HIV/AIDS model that incorporates carefree attitude towards sex." Acta Biotheoretica (2021): 1–20. pmid:33502640
- 14. Teklu Shewafera Wondimagegnhu, and Rao Koya Purnachandra. "HIV/AIDS-Pneumonia Codynamics Model Analysis with Vaccination and Treatment." Computational and Mathematical Methods in Medicine 2022 (2022). pmid:35069778
- 15. Zeb Anwar, Alzahrani Ebraheem, Erturk Vedat Suat, and Zaman Gul. "Mathematical model for coronavirus disease 2019 (COVID-19) containing isolation class." BioMed research international 2020. pmid:32685469
- 16. Bae Seongman, Joon Seo Lim Ji Yeun Kim, Jung Jiwon, and Kim Sung-Han. "Transmission characteristics of SARS-CoV-2 that hinder effective control." Immune network 21, no. 1 (2021). pmid:33728102
- 17. Baek Yae Jee, Lee Taeyong, Cho Yunsuk, Jong Hoon Hyun Moo Hyun Kim, Sohn Yujin, Kim Jung Ho et al. "A mathematical model of COVID-19 transmission in a tertiary hospital and assessment of the effects of different intervention strategies." PloS one 15, no. 10 (2020): e0241169. pmid:33104736
- 18. Barbera Lauren K., Kamis Kevin F., Rowan Sarah E., Davis Amelia J., Shehata Soraya, Carlson Jesse J., Johnson Steven C., and Erlandson Kristine M. "HIV and COVID-19: review of clinical course and outcomes." HIV Research & Clinical Practice 22, no. 4 (2021): 102–118. pmid:34514963
- 19. Chen Tian-Mu, Rui Jia, Wang Qiu-Peng, Zhao Ze-Yu, Cui Jing-An, and Yin Ling. "A mathematical model for simulating the phase-based transmissibility of a novel coronavirus." Infectious diseases of poverty 9, no. 1 (2020): 1–8
- 20. Cirrincione Luigi, Plescia Fulvio, Ledda Caterina, Rapisarda Venerando, Martorana Daniela, Raluca Emilia Moldovan Kelly Theodoridou, and Cannizzaro Emanuele. "COVID-19 pandemic: Prevention and protection measures to be adopted at the workplace." Sustainability 12, no. 9 (2020).
- 21. Inayaturohmat Fatuh, Anggriani Nursanti, and Supriatna Asep Kuswandi. "A mathematical model of tuberculosis and COVID-19 coinfection with the effect of isolation and treatment." Frontiers in Applied Mathematics and Statistics (2022).
- 22. Mekonnen Temesgen Tibebu. "Mathematical model analysis and numerical simulation for codynamics of meningitis and pneumonia infection with intervention." Scientific Reports 12, no. 1 (2022): 1–22.
- 23. Mugisha Joseph YT, Ssebuliba Joseph, Nakakawa Juliet N., Kikawa Cliff R., and Ssematimba Amos. "Mathematical modeling of COVID-19 transmission dynamics in Uganda: Implications of complacency and early easing of lockdown." PloS one 16, no. 2 (2021): e0247456. pmid:33617579
- 24. Musa Salihu S., Baba Isa A., Yusuf Abdullahi, Sulaiman Tukur A., Aliyu Aliyu I., Zhao Shi, and He Daihai. "Transmission dynamics of SARS-CoV-2: A modeling analysis with high-and-moderate risk populations." Results in physics 26 (2021): 104290. pmid:34026471
- 25. Omame Andrew, Sene Ndolane, Nometa Ikenna, Nwakanma Cosmas I., Nwafor Emmanuel U., Iheonu Nneka O., and Okuonghae Daniel. "Analysis of COVID‐19 and comorbidity co‐infection model with optimal control." Optimal Control Applications and Methods 42, no. 6 (2021): 1568–1590. pmid:34226774
- 26. Riyapan Pakwan, Shuaib Sherif Eneye, and Intarasit Arthit. "A Mathematical model of COVID-19 Pandemic: a case study of Bangkok, Thailand." Computational and Mathematical Methods in Medicine 2021 (2021). pmid:33815565
- 27. Sun Deshun, Long Xiaojun, and Liu Jingxiang. "Modeling the COVID-19 Epidemic With Multi-Population and Control Strategies in the United States." Frontiers in Public Health 9 (2021). pmid:35047470
- 28. Teklu Shewafera Wondimagegnhu. "Mathematical analysis of the transmission dynamics of COVID-19 infection in the presence of intervention strategies." Journal of Biological Dynamics 16, no. 1 (2022): 640–664 pmid:35972455
- 29. Wangari Isaac Mwangi, Sewe Stanley, Kimathi George, Wainaina Mary, Kitetu Virginia, and Kaluki Winnie."Mathematical modelling of COVID-19 transmission in Kenya: a model with reinfection transmission mechanism " Computational and Mathematical Methods in Medicine 2021 (2021).
- 30. Tolossa Tadesse, Tsegaye Reta, Shiferaw Siraj, Wakuma Bizuneh, Ayala Diriba, Bekele Belay, and Shibiru Tesfaye. "Survival from a triple co-infection of COVID-19, HIV, and tuberculosis: a case report." International Medical Case Reports Journal 14 (2021): 611. pmid:34512043
- 31. Yang Hyun Mo, Lombardi Luis Pedro Junior, Morato Castro Fábio Fernandes, and Yang Ariana Campos. "Mathematical modeling of the transmission of SARS-CoV-2, Evaluating the impact of isolation in São Paulo State (Brazil) and lockdown in Spain associated with protective measures on the epidemic of CoViD-19." Plos One 16, no. 6 (2021): e0252271
- 32. Daniel D. O. "Mathematical model for the transmission of Covid-19 with nonlinear forces of infection and the need for prevention measure in Nigeria." J. Infect. Dis. Epidem 6 (2021): 158.
- 33. Danwang Celestin, Jean Jacques Noubiap Annie Robert, and Yombi Jean Cyr. "Outcomes of patients with HIV and COVID-19 co-infection: a systematic review and meta-analysis." AIDS research and therapy 19, no. 1 (2022): 1–12.
- 34. Gesesew Hailay Abrha, Mwanri Lillian, Stephens Jacqueline H., Woldemichael Kifle, and Ward Paul. "COVID/HIV co-infection: A syndemic perspective on what to ask and how to answer." Frontiers in public health 9 (2021): 193. pmid:33791266
- 35. Ssentongo Paddy, Heilbrunn Emily S., Ssentongo Anna E., Advani Shailesh, Chinchilli Vernon M., Nunez Jonathan J., and Du Ping. "Epidemiology and outcomes of COVID-19 in HIV-infected individuals: a systematic review and meta-analysis." Scientific reports 11, no. 1 (2021): 1–12.
- 36. Tamuzi Jacques L., Ayele Birhanu T., Shumba Constance S., Adetokunboh Olatunji O., Jeannine Uwimana-Nicol Zelalem T. Haile, Inugu Joseph, and Nyasulu Peter S. "Implications of COVID-19 in high burden countries for HIV/TB: A systematic review of evidence." BMC infectious diseases 20, no. 1 (2020): 1–18. pmid:33036570
- 37. Tchoumi S. Y., Diagne M. L., Rwezaura H., and Tchuenche J. M. "Malaria and COVID-19 co-dynamics: A mathematical model and optimal control." Applied mathematical modelling 99 (2021). pmid:34230748
- 38. Ambrosioni Juan, José Luis Blanco Juliana M. Reyes-Urueña, Davies Mary-Ann, Sued Omar, Maria Angeles Marcos Esteban Martínez et al. "Overview of SARS-CoV-2 infection in adults living with HIV." The lancet HIV 8, no. 5 (2021): e294–e305. pmid:33915101
- 39. Goudiaby M. S., Gning L. D., Diagne M. L., Dia Ben M., Rwezaura H., and Tchuenche J. M. "Optimal control analysis of a COVID-19 and tuberculosis co-dynamics model." Informatics in Medicine Unlocked 28 (2022): 100849. pmid:35071729
- 40. Hezam Ibrahim M., Foul Abdelaziz, and Alrasheedi Adel."A dynamic optimal control model for COVID-19 and cholera co-infection inYemen." Advances in Difference Equations 2021, no. 1 (2021).
- 41. Mekonen Kassahun Getnet, Obsu Legesse Lemecha, and Habtemichael Tatek Getachew. "Optimal control analysis for the coinfection of COVID-19 and TB." Arab Journal of Basic and Applied Sciences 29, no. 1 (2022): 175–192.
- 42. Mirzaei Hossein, McFarland Willi, Karamouzian Mohammad, and Sharifi Hamid. "COVID-19 among people living with HIV: a systematic review." AIDS and Behavior 25, no. 1 (2021): 85–92. pmid:32734438
- 43. Ringa N., Diagne M. L., Rwezaura H., Omame A., Tchoumi S. Y., and Tchuenche J. M. "HIV and COVID-19 co-infection: A mathematical model and optimal control." Informatics in Medicine Unlocked (2022): 100978. pmid:35663416
- 44. Zhang Jiu-Cong, Yu Xiao-Hui, Ding Xiao-Han, Ma Hao-Yu, Cai Xiao-Qing, Kang Sheng-Chao, and Xiang Da-Wei. "New HIV diagnoses in patients with COVID-19: two case reports and a brief literature review." BMC infectious diseases 20, no. 1 (2020): 1–10 pmid:33076830
- 45. Kotola Belela Samuel, and Teklu Shewafera Wondimagegnhu. "A Mathematical Modeling Analysis of Racism and Corruption Codynamics with Numerical Simulation as Infectious Diseases." Computational and Mathematical Methods in Medicine 2022 (2022). pmid:35991135
- 46. Kotola Belela Samuel, Gebru Dawit Melese, and Alemneh Haileyesus Tessema. "Appraisal and Simulation on Codynamics of Pneumonia and Meningitis with Vaccination Intervention: From a Mathematical Model Perspective." Computational and Mathematical Methods in Medicine 2022 (2022). pmid:36479316
- 47. Teklu Shewafera Wondimagegnhu, and Terefe Birhanu Baye. "Mathematical modeling investigation of violence and racism coexistence as a contagious disease dynamics in a community." Computational and mathematical methods in medicine 2022 (2022). pmid:35928967
- 48. Takele Rediat. "Stochastic modelling for predicting COVID-19 prevalence in East Africa Countries." Infectious Disease Modelling 5 (2020): 598–607. pmid:32838091
- 49. Aggarwal Rajiv, and Raj Yashi A. "A fractional order HIV-TB co-infection model in the presence of exogenous reinfection and recurrent TB." Nonlinear Dynamics 104 (2021): 4701–4725. pmid:34075277
- 50. Omame A., Abbas M., & Onyenegecha C. P. (2021), A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative. Chaos, Solitons & Fractals, 153, 111486. pmid:34658543
- 51. Keno Temesgen Duressa, Makinde Oluwole Daniel, and Obsu Legesse Lemecha. "Optimal Control and Cost Effectiveness Analysis of SIRS Malaria Disease Model with Temperature Variability Factor." Journal of Mathematical & Fundamental Sciences 53, no. 1 (2021).
- 52. Keno Temesgen Duressa, Dano Lemessa Bedjisa, and Makinde Oluwole Daniel. "Modeling and Optimal Control Analysis for Malaria Transmission with Role of Climate Variability." Computational and Mathematical Methods 2022 (2022).
- 53. Asamoah Joshua Kiddy K., Okyere Eric, Abidemi Afeez, Moore Stephen E., Sun Gui-Quan, Jin Zhen, Acheampong Edward, and Gordon Joseph Frank. "Optimal control and comprehensive cost-effectiveness analysis for COVID-19." Results in Physics 33 (2022): 105177. pmid:35070649
- 54. Okosun K.O., Makinde O.D. and Takaidza I., 2013. Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives. Applied mathematical modelling, 37(6), pp.3802–3820.
- 55. Keno Temesgen Duressa, Obsu Legesse Lemecha, and Makinde Oluwole Daniel. "Modeling and optimal control analysis of malaria epidemic in the presence of temperature variability." Asian-European Journal of Mathematics 15, no. 01 (2022): 2250005.
- 56. Rabiu Musa, Willie Robert, and Parumasur Nabendra. "Optimal control strategies and sensitivity analysis of an hiv/aids-resistant model with behavior change." Acta biotheoretica 69 (2021): 543–589. pmid:34331152
- 57. Teklu Shewafera Wondimagegnhu, and Terefe Birhanu Baye. "Mathematical modeling analysis on the dynamics of university students animosity towards mathematics with optimal control theory." Scientific Reports 12, no. 1 (2022): 1–19
- 58. Abba B. Gumel, Lubuma Jean M.‐S., Sharomi Oluwaseun, and Terefe Yibeltal Adane. "Mathematics of a sex‐structured model for syphilis transmission dynamics." Mathematical Methods in the Applied Sciences 41, no. 18 (2018): 8488–8513.
- 59. Bakare E. A., and Nwozo C. R. "Bifurcation and sensitivity analysis of malaria–schistosomiasis co-infection model." International Journal of Applied and Computational Mathematics 3, no. 1 (2017).
- 60. Castillo-Chavez Carlos, and Song Baojun. "Dynamical models of tuberculosis and their applications." Mathematical Biosciences & Engineering 1, no. 2 (2004): 361. pmid:20369977
- 61.
Castillo-Chavez C., Blower S., van den Driessche P., Kirschner D., & Yakubu A. A. (Eds.). (2002). Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (Vol. 126). Springer Science & Business Media.