EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6149 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Mathematical Model for COVID-19 and Tuberculosis Co-infection with the Effect of Quarantine Measure Julito A. Puebla Jr1,∗, Jay Michael R. Macalalag2, Kaye Y. Pajaron2 1 Department of General Education, Caraga State University Cabadbaran Campus, 8605 Cabadbaran City, Philippines 2 Department of Mathematics, Caraga State University, 8600 Butuan City, Philippines Abstract. In this paper, a deterministic mathematical model for the transmission dynamics of COVID-19 and tuberculosis co-infection, using a system of nonlinear ordinary differential equa- tions, is proposed and analyzed. We begin our mathematical analysis of the COVID-19 and tuber- culosis sub-models by establishing the existence, uniqueness, non-negativity, and boundedness of the solutions. Subsequently, we determine the equilibrium points and the reproduction numbers of the sub-models as well as the co-infection model. We then proceed to prove that the disease-free equilibrium point of each sub-model and the co-infection model is locally and globally asymp- totically stable if its corresponding reproduction number is less than 1 and unstable otherwise. Moreover, if the reproduction number is greater than 1, then the corresponding sub-model is lo- cally asymptotically stable at the endemic equilibrium point. Numerical simulations are conducted to validate the theoretical findings. This study demonstrates that effective quarantine measures are crucial for controlling and potentially eradicating COVID-19 and tuberculosis co-infections. 2020 Mathematics Subject Classifications: 92D30, 34D23, 92C60, 34C60, 37N25 Key Words and Phrases: COVID-19, tuberculosis, coinfection, stability analysis, reproduction number 1. Introduction Infectious diseases, caused by pathogenic microorganisms, invade the body and disrupt normal functions. They can be transmitted through direct contact, contaminated surfaces, or vectors. These diseases range from mild to life-threatening, affecting various organs and systems. Throughout history, they have shaped societies, caused epidemics, and influenced human history. Coronavirus disease (COVID-19) and tuberculosis (TB) exemplify the immense suffering, loss of life, and social disruptions caused by infectious diseases that persist worldwide [45]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6149 Email addresses: julito.puebla@csucc.edu.ph (J. Puebla), jrmacalalag@carsu.edu.ph (JM. Macalalag), kayepajaron@gmail.com (K. Pajaron) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 2 of 28 The COVID-19 pandemic is a global health crisis caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [20]. It was first identified in December 2019 in Wuhan, Hubei Province, China. On March 11, 2020, the virus quickly spread to other parts of China and subsequently to numerous countries around the world, leading to the declaration of a pandemic by the World Health Organization (WHO). The virus spreads primarily through droplets when an infected person talks, coughs, or sneezes, and can also spread by touching a contaminated surface and then touching one’s mouth, nose, or eyes [8]. In January 2025, there have been 777,349,085 confirmed cases of COVID-19 infection, with 7,086,633 confirmed deaths in the whole country [38]. In the Philippines, in January 2025 there were 4,140,383 confirmed cases of COVID-19 with 66,864 deaths reported to the WHO [30]. This indicates that COVID-19 continues to pose significant global challenges in terms of public health and economic impact. On the other hand, tuberculosis (TB) is a contagious respiratory infectious disease caused by the Mycobacterium tuberculosis. It affects primarily the lungs, but can also target other parts of the body such as the kidneys, spine, and brain. Tuberculosis spreads through the air when an infected individual coughs, sneezes, or talks, releasing tiny droplets containing the bacteria [36]. In total, approximately 10.6 million people fell ill with tuber- culosis and 1.6 million people died of tuberculosis in 2021 [39]. Eight countries with the highest number of new tuberculosis infections in 2023 are India (26%), China (8.5%) In- donesia (8.4%), Philippines (6.0%), Pakistan (5.8%), Nigeria (4.6%), Bangladesh (3.6%), and South Africa (3.6%) [41]. This indicates that tuberculosis is one of the major global health concerns in these countries, particularly in the Philippines. COVID-19 and tuberculosis are both infectious diseases primarily transmitted through close contact. Co-infection occurs when an individual is simultaneously infected with COVID-19 and tuberculosis. Co-infected individuals experience similar symptoms such as cough, fever, and difficulty breathing. The study reported in [9] has confirmed that latent and active TB are the primary risk factors for increasing the transmission of COVID-19 within the community. Furthermore, research work in [35] reveals that people with active or latent tuberculosis are more vulnerable to COVID-19. According to the report, the authors observed a more rapid and severe progression of symptoms in patients with both diseases. Clinical evidence indicates that TB is associated with COVID-19, resulting in a two to three-fold increase in mortality and a 25% relative reduction in the likelihood of recovery for individuals co-infected with both diseases [25]. Several research studies have shown that COVID-19 and tuberculosis rapidly transmit through close contact ([18], [40], [47]). Both these diseases primarily transmits rapidly in settings where there is close and prolonged contact between infected individuals and susceptible individuals. Some specific contexts where COVID-19 and tuberculosis trans- mission can occur rapidly include: poor ventilation, crowded places, and superspreading events. The implementation of the non-pharmaceutical interventions, such as isolation or quarantine help effectively mitigate the COVID-19 outbreak ([5],[15], [46], [47]). These measures are taken to prevent the potential spread of the disease by isolating and moni- toring individuals during the incubation period or when they show symptoms. Quarantine typically involves staying in a designated location, such as a healthcare facility or one’s J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 3 of 28 own home, and avoiding contact with others until the risk of spreading the disease has passed. It is an important public health strategy to control infectious diseases and protect the community. Hence, the quarantine measures plays a crucial role in mitigating the spread of co-infection between COVID-19 and tuberculosis. Amidst the chaotic spread of COVID-19, there has been a concerning neglect of sev- eral other diseases, including tuberculosis, in terms of treatment and care. The significant surge in COVID-19 cases has led to a drastic decline in the identification and reporting of TB cases, consequently reversing the progress made towards the global TB targets. It is of utmost importance to prioritize the implementation of essential control measures, particularly in managing tuberculosis cases. This entails promptly identifying and man- aging COVID-19 in individuals with tuberculosis, giving them precedence in tuberculosis hospitals and medication centers. To prevent the escalating number of individuals co- infected with COVID-19 and tuberculosis, it is essential to implement quarantine mea- sures. Several recent models have explored the dynamics of COVID-19 and tuberculosis (TB) co-infection. Studies have focused on vaccination and optimal control strategies without explicitly modeling quarantine ([1], [11]). Other works incorporated vaccination and isolation interventions, but did not systematically address quarantine for co-infected individuals ([17], [33]). Additional studies analyzed disease dynamics and sensitivity but lacked a quarantine component ([24], [26]). In contrast, the present study introduces quarantine compartments for individuals with COVID-19, TB, and co-infection, allowing a detailed assessment of quarantine as a non-pharmaceutical intervention. This study de- velops a mathematical model that systematically examines the impact of quarantine mea- sures on the transmission dynamics of COVID-19 and tuberculosis co-infection, providing new insights into effective control strategies and the potential eradication of co-infections, thereby addressing a critical gap in the existing literature. 2. Model Formulation We developed a mathematical model to study the dynamics of COVID-19 and tuber- culosis co-infection, dividing the human population into 11 compartments based on the health status of individuals: susceptible individuals (S), exposed individuals with COVID- 19 only (EC), exposed individuals with tuberculosis only (ET ), individuals exposed with both COVID-19 and tuberculosis (ECT ), infected individuals with COVID-19 only (IC), infected individuals with tuberculosis only (IT ), infected individuals with both COVID-19 and tuberculosis (ICT ), quarantined individuals with COVID-19 only (QC), quarantined individuals with tuberculosis only (QT ), quarantined individuals with COVID-19 and tu- berculosis only (QCT ), and recovered individuals (R). The human populationN at time t is N(t) = S(t)+EC(t)+ET (t)+ECT (t)+IC(t)+IT (t)+ICT (t)+QC(t)+QT (t)+QCT (t)+R(t). The model has the following assumptions: individuals infected with COVID-19 only IC are susceptible to tuberculosis infection and vice versa ([14], [17]); co-infected individu- als ICT can transmit either COVID-19 or tuberculosis but not the mixed infections at the same time ([14], [17]); co-infected individuals can recover either from COVID-19 or tuberculosis but not from the mixed infection at the same time ([14], [17], [27]); the trans- J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 4 of 28 missibility rates for single and co-infections are the same [14]; and recovered individuals R acquire lifelong immunity to both COVID-19 and tuberculosis. Every individual in all compartments may die naturally at a rate µ. More specifically, individuals enter the susceptible compartment at a recruitment rate ω; upon exposure to individuals from compartments EC and IC , they transition to com- partment EC ; while contact with individuals from compartment IT may lead to movement to compartment ET . Within compartment EC , individuals may progress to compartment IC at a rate rc; some may transition to compartment ECT upon exposure to individuals from compartment IT ; or some may move to compartment QC at a rate δe. Now, individ- uals in compartment IC may die due to COVID-19 at a rate µ1; and some may move to compartment QC at rate δc. Individuals in compartment QC may recover from COVID-19 at a rate γc; and some may die due to COVID-19 at a rate µ4. Once in compartment ET , individuals may move to compartment ECT following contact with individuals from EC and IC ; or some may transfer to compartment IT at a rate rt. Individuals in compartment IT may move to compartment QT at a rate δt; and some may die due to tuberculosis at a rate µ3. Meanwhile, individuals in compartment QT may recover at a rate γt; and some may die due to tuberculosis at a rate µ5. Individuals in ECT may move to compartment ICT at rate r. On the other hand, individuals in QCT may move to compartment QC at a rate γct; some may move to compartment QT at a rate γtc; and some may die due to co-infection at a rate µ6. Based on the aforementioned assumptions, Figure 1 presents the mathematical model depicting the transmission dynamics of COVID-19 and tuberculosis co-infection. Figure 1: Transmission Dynamics of COVID-19 and TB Co-infection Model The transmission dynamics of the COVID-19 and tuberculosis co-infection model, shown in Figure 1, are described by the following system of ordinary differential equa- tions. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 5 of 28  dS dt = ω − βc(EC + IC)S − βtITS − µS dEC dt = βc(EC + IC)S − αtβtITEC − (rc + δe + µ)EC dIC dt = rcEC − (δc + µ+ µ1)IC dQC dt = δeEC + δcIC + γctQCT − (γc + µ+ µ4)QC dECT dt = αtβtITEC + αcβc(EC + IC)ET − (rct + µ)ECT dICT dt = rctECT − (δct + µ+ µ3)ICT dQCT dt = δctICT − (γtc + γct + µ+ µ6)QCT dET dt = βtITS − αcβc(EC + IC)ET − (rt + µ)ET dIT dt = rtET − (δt + µ+ µ2)IT , dQT dt = δtIT + γtcQCT − (γt + µ+ µ5)QT dR dt = γcQC + γtQT − µR. (1) Parameter Description Value Source ω Recruitment rate [2,4] day−1 Assumed βc Effective contact rate transmission of COVID-19 0.015 Assumed βt Effective contact rate transmission of tuberculosis 0.012 Assumed αc COVID-19 co-infection rate from TB exposure 1 day−1 [14] αt Tuberculosis co-infection rate from COVID-19 exposure 1 day−1 [14] rc Progression rate from asymptomatic to symptomatic COVID-19 0.33 day−1 [23] rt Progression rate from asymptomatic to symptomatic tuberculosis 0.7 day−1 Assumed rct Progression rate from asymptomatic to symptomatic co-infection 0.02 day−1 [23] δe Rate at which exposed individuals are isolated in COVID-19 quarantine [0,1) day−1 Assumed δc Rate at which infected individuals are isolated in COVID-19 quarantine [0,1) day−1 Assumed δt Rate at which infected individuals are isolated in TB quarantine [0,1) day−1 Assumed δct Rate at which co-infected individuals are isolated in co-infected quarantine 0.79 day−1 Assumed γct Recovery rate of COVID-19 from co-infection 0.68 day−1 Assumed γtc Recovery rate of TB from co-infection 0.35 day−1 Assumed γc Recovery rate from COVID-19 infection 0.3 day−1 [17] γt Recovery rate from tuberculosis infection 0.2 day−1 [17] µ Natural death rate 0.0477 day−1 [23] µ1 Death rate of COVID-19 infected individuals 0.23 day−1 [23] µ2 Death rate of tuberculosis infected individuals 0.01 day−1 [23] µ3 Death rate of co-infected individuals 0.03 day−1 [17] µ4 Death rate of COVID-19 infected individuals in quarantine 0.23 day−1 [23] µ5 Death rate of tuberculosis infected individuals in quarantine 0.004 day−1 [17] µ6 Death rate of co-infected individuals in quarantine 0.03 day−1 [17] Table 1: Description of the Model Parameters J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 6 of 28 System (1) is defined with the following initial conditions S(0) ≥ 0, EC(0) ≥ 0, IC(0) ≥ 0, QC(0) ≥ 0, ECT (0) ≥ 0, ICT (0) ≥ 0, QCT (0) ≥ 0, ET (0) ≥ 0, IT (0) ≥ 0, QT (0) ≥ 0, and R(0) ≥ 0. All parameters in system (1) are described in Table 1. 3. Model Analysis This section examines the analysis of three models: the COVID-19 sub-model, the tuberculosis sub-model, and the co-infection model. 3.1. COVID-19 Sub-model Without considering tuberculosis-infected individuals from the co-infection model, the COVID-19 sub-model is given by dS dt = ω − βc(EC + IC)S − µS dEC dt = βc(EC + IC)S − (rc + δe + µ)EC dIC dt = rcEC − (δc + µ+ µ1)IC dQC dt = δeEC + δcIC − (γc + µ+ µ4)QC dR dt = γcQC − µR, (2) where NC = S + EC + IC + QC + R. By adding all the equations in the system (2), we obtained ω − µNC − µ1IC − µ4QC ≤ ω − µNC . The given initial conditions of the COVID-19 sub-model system (2) guarantee that N(0) ≥ 0. Thus, the total human population is positive and bounded for finite time t > 0. By differential inequality [4], NC(t) ≤ NC(0)e −µt+ ω µ (1−e−µt). As t → +∞, we get 0 ≤ NC(t) ≤ ω µ . The feasible region of the COVID-19 sub-model (2) is given by ΩC = { (S,EC , IC , QC , R) ∈ R5 + : NC(t) ≤ ω µ } . The set ΩC is positive invariant and attracting [16], and all the solutions of the COVID-19 sub-model (2) starting in ΩC remain in ΩC for all t ≥ 0. Thus, the model (2) is mathe- matically and epidemiologically well-posed, and is sufficient to study its dynamics in ΩC . Note that the first four (4) equations in system (2) are independent of the last equation, which means that R has no effect on the system. Without loss of generality, the last equation can be omitted from the system. Now, setting all differential equations in system (2) to zero and solving for all compartments, we obtain two equilibrium points, the disease- free equilibrium (DFE) and the endemic equilibrium (EE). The disease-free equilibrium point is obtained when EC = 0 and IC = 0 and will be denoted by E0C = (S,EC , IC , QC) = (ω µ , 0, 0, 0 ) . (3) J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 7 of 28 When EC ̸= 0 and IC ̸= 0, the endemic equilibrium point exists and is given by E1C = (S,EC , IC , QC) where S = (rc + δe + µ)(δc + µ+ µ1) βc(δc + µ+ µ1 + rc) EC = ωβc(δc + µ+ µ1 + rc)− µ(rc + δe + µ)(δc + µ+ µ1) βc(rc + δe + µ)(δc + µ+ µ1 + rc) IC = rcωβc(δc + µ+ µ1 + rc)− rcµ(rc + δe + µ)(δc + µ+ µ1) βc(rc + δe + µ)(δc + µ+ µ1 + rc)(δc + µ+ µ1) QC = δeωβc(δc + µ+ µ1 + rc)− δeµ(rc + δe + µ)(δc + µ+ µ1) βc(rc + δe + µ)(δc + µ+ µ1 + rc)(γc + µ+ µ4) + δcrcωβc(δc + µ+ µ1 + rc)− δcrcµ(rc + δe + µ)(δc + µ+ µ1) βc(rc + δe + µ)(δc + µ+ µ1 + rc)(δc + µ+ µ1)(γc + µ+ µ4) . (4) The basic reproduction number (R0) is defined as the average number of secondary infections produced by a single COVID-19-infected individual in a completely susceptible population [44]. The Next Generation Matrix (NGM) is used to compute the basic repro- duction number. The system (2) can be written as ẋ = f(x) = F(x) − V(x) where F is the transmission matrix and the V is the transition matrix are given as follows F = βc(EC + IC)S 0 0  and V =  (rc + δe + µ)EC −rcEC + (δc + µ+ µ1)IC −δeEC − δcIC + (γc + µ+ µ4)QC  . Evaluating the Jacobian matrix of F and V at the disease-free equilibrium E0C gives F = βcω µ βcω µ 0 0 0 0 0 0 0  and V = rc + δe + µ 0 0 −rc δc + µ+ µ1 0 −δe −δc γc + µ+ µ4  , respectively. Let A1 = rc + δe + µ, A2 = δc + µ+ µ1, and A3 = γc + µ+ µ4. (5) Then V −1 =  1 A1 0 0 rc A1A2 1 A2 0 δcrc+δeA2 A1A2A3 δc A2A3 1 A3  . Consequently, FV −1 =  βcω µA1 + βcωrc µA1A2 βcω ωA2 0 0 0 0 0 0 0  . J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 8 of 28 Accordingly, the largest eigenvalue of FV −1 is βcω µA1 + βcωrc µA1A2 . Hence, the basic reproduction number of the COVID-19 sub-model (2) is given by R0C = βcω µ(rc + δe + µ) + βcωrc µ(rc + δe + µ)(δc + µ+ µ1) . This quantity is a crucial epidemiological metric that indicates the potential spread of an infectious disease within a population. It determines whether the disease will continue to spread (R0C > 1) or eventually decline (R0C < 1) in the community. Hence, this will guide public health strategies and intervention measures. 3.2. Stability Analysis of the Equilibrium Points for the COVID-19 Sub- model This subsection examines the stability of the equilibrium points for the COVID-19 sub-model (2). Theorem 1. If R0C < 1, then the COVID-19 sub-model (2) is locally asymptotically stable at the disease-free equilibrium EOC and unstable otherwise. Proof. The Jacobian matrix of system (2) evaluated at EOC is given by JEOC =  −µ −βcω µ −βcω µ 0 0 βcω µ −A1 βcω µ 0 0 rc −A2 0 0 δe δc −A3  , where Ai, A2, and A3 are given in (5). Then the characteristic polynomial of the Jacobian matrix JEOC is JEOC (λ) = (−µ− λ)(−A3 − λ) [ λ2 + ( A1 +A2 − βcω µ ) λ+A1A2 − βcω(A2 − rc) µ ] . Hence, the characteristic roots are λ1 = −µ and λ2 = −A3 and the other roots are determined by the remaining factor λ2 + ( A1 +A2 − βcω µ ) λ+A1A2 − βcω(A2 − rc) µ . Observe that when R0C < 1, A1 +A2 − βcω µ > 0. Moreover, A1A2 − βcω(A2 − rc) µ = A1A2(1−R0C ) > 0 when R0C < 1. By Routh-Hurwitz criterion, all eigenvalues of the second order polynomial are negative, which means that COVID-19 sub-model system (2) is locally asymptotically stable at E0C when R0C < 1 and unstable otherwise. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 9 of 28 Theorem 2. If R0C < 1, then the COVID-19 sub-model (2) is globally asymptotically stable at disease-free equilibrium E0C . Proof. Consider the Lyapunov function below L = (δc + µ+ µ1)EC + (βcω µ ) IC . The time derivative of L computed along the solution of (2) is negative definite. That is, · L = (δc + µ+ µ1)E ′ C + (βcω µ ) I ′ C = (δc + µ+ µ1) [ βc(EC + IC)S − (rc + δe + µ)EC ] + (βcω µ )[ rcEC − (δc + µ+ µ1)IC ] ≤ (δc + µ+ µ1) [ βc(EC + IC) ω µ − (rc + δe + µ)EC ] + (βcω µ )[ rcEC − (δc + µ+ µ1)IC ] = βcω(δc + µ+ µ1)EC µ − (δc + µ+ µ1)(rc + δe + µ)EC + βcωrcEC µ = βcω(δc + µ+ µ1 + rc)EC µ − (δc + µ+ µ1)(rc + δe + µ)EC = [ (δc + µ+ µ1)(rc + δe + µ)EC ] (R0C − 1) ≤ 0 if R0C < 1. Because all model parameters are non-negative, it follows that · L is negative semi-definite, that is, · L ≤ 0 if R0C < 1. If the solution x∗ = E0C , then · L = 0. Conversely, if · L = 0, then S = ω µ , EC = 0 IC = 0 and QC = 0. Hence, · L = 0 if and only if x∗ = E0C . So, · L is a Lyapunov function on Ωc and the largest compact invariant set in {(S,EC , IC , QC) ∈ Ωc : · L = 0} is {(ωµ , 0, 0, 0)}. Thus, by LaSalle’s invariance principle, every solution of system (2) with the initial conditions in Ωc approaches E0C = (ωµ , 0, 0, 0) at t → ∞ whenever R0C < 1. Therefore, the COVID-19 sub-model (2) is globally asymptotically stable at the disease-free equilibrium point E0C = (ωµ , 0, 0, 0) if R0C < 1. The next result shows the existence of a unique positive endemic equilibrium for the COVID-19 sub-model (2). Note that the positive endemic equilibrium for the COVID-19 sub-model (2) can be expressed in terms of R0C , that is, S∗ = ω µR0C , E∗ C = ω(R0C − 1) (rc + δe + µ)R0C , I∗C = rcω(R0C − 1) (rc + δe + µ)R0C (δc + µ+ µ1) , and Q∗ C = (R0C − 1)(δeω(δc + µ+ µ1) + δcrcω) (rc + δe + µ)R0C (δc + µ+ µ1)(γc + µ+ µ1) . Hence, we have the following result. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 10 of 28 Theorem 3. If R0C > 1, then the COVID-19 sub-model (2) has a unique positive endemic equilibrium. Theorem 4. If R0C > 1, then the COVID-19 sub-model (2) is locally asymptotically stable at the endemic equilibrium E1C and unstable otherwise. Proof. The Jacobian matrix of system (2) evaluated at the endemic equilibrium point E1C is given by JE1C =  −βc(E ∗ C + I∗C)− µ −βcS ∗ −βcS ∗ 0 βc(E ∗ C + I∗C) βcS ∗ −A1 βcS ∗ 0 0 rc −A2 0 0 δe δc −A3  , where A1, A2 and A3 are given in (5). Then the characteristic polynomial of JE1C is JE1C (λ) =(−A3 − λ) ( λ3 − (−A2 + βcS ∗ −A1 − βcE ∗ − βcI ∗ − µ)λ2 − (−βcA1E ∗ − βcA1I ∗ − βcA2E ∗ − βcA2I ∗ + βcA2S ∗ + βcµS ∗ + βcrcS ∗ −A1A2 − µA1 − µA2)λ+ βcA1A2I ∗ − βcµA2S ∗ − βcµrcS ∗ + µA1A2 ) Certainly, the eigenvalue λ = −A3 < 0. The other eigenvalues are determined by the cubic poly- nomial factor. Set the coefficients and constant term of the cubic polynomial as follows: a1 as the coefficient of λ2, a2 as the coefficient of λ and a3 as the constant term. By Routh Hurwitz criterion, the eigenvalues that are determined by cubic polynomial have negative real parts if the following conditions are satisfied: a1, a2, and a3 are all positive and a1a2 > a3. By Theorem (3), a1, a2, and a3 can be expressed as a1 = A2 + βcωrcA1 βcωA2 + βcωrc + βcω(R0C − 1) A1R0C + βcrcω(R0C − 1) A1R0C + µ a2 = βcω(R0C − 1) R0C + βcrcω(R0C − 1) A2R0C + βcωA2(R0C − 1) A1R0C + βcrcω(R0C − 1) A1R0C + βcωrcµA1 βcωA2 + βcωrc + µA2 a3 = βcωA2(R0C − 1) R0C + βcrcω(R0C − 1) R0C Moreover, a1a2 = βcωA2(R0C − 1) R0C + βcrcω(R0C − 1) R0C +∆ = a3 +∆, where ∆ =A2 (βcωA2(R0C − 1) A1R0C + βcrcω(R0C − 1) A1R0C + βcωrcµA1 βcωA2 + βcωrc + µA2 ) + a2 ( βcωrcA1 βcωA2 + βcωrc + βcω(R0C − 1) A1R0C + βcrcω(R0C − 1) A1R0C + µ ) Note that a1, a2 and a3 are all positive and a1a2 > a3 if R0C > 1. Hence, the COVID-19 sub- model (2) is locally asymptotically stable at the endemic equilibrium point E1C when R0C > 1 and unstable otherwise. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 11 of 28 3.3. Sensitivity Analysis of COVID-19 Sub-model In this subsection, sensitivity analysis of the parameters in the COVID-19 sub-model is performed. The sensitivity of a parameter p is defined as the behavior of the model to a small change in its parameter value [22], and is given by Sp = ∂R0C ∂p × p R0C , where R0C = βcω µ(rc + ϵC + µ) + βcωrc µ(rc + ϵC + µ)(ρC + µ+ σC) (6) The sensitivity indices for each parameter in the COVID-19 sub-model (2) with respect to R0C is computed using the equation (6), and the corresponding sensitivity indices are shown in Figure 2. Figure 2: Sensitivity indices of the COVID-19 sub-model parameters The sensitivity analysis of the COVID-19 sub-model numerical results highlighted that the COVID-19 transmission rate βc, and the recruitment rate ω have a high positive impact on the spread of the virus. The analysis suggests that the magnitude of the impacts of βc and ω on the spread of COVID-19 are the same because the number of secondary infections increase with respect to increasing these parameters [22]. In contrast, the parameters µ1, µ, δc, rc, and δe have negative impact, which means that increasing the value of such parameters will decrease the number of people infected with COVID-19. 3.4. Tuberculosis Sub-model Without considering the infected individuals with COVID-19 from the COVID-19 and tuberculosis co-infection model (1), the tuberculosis sub-model is given by J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 12 of 28  dS dt = ω − βtITS − µS dET dt = βtITS − (rt + µ)ET dIT dt = rtET − (δt + µ+ µ2)IT dQT dt = δtIT − (γt + µ+ µ5)QT dR dt = γtQT − µR (7) where NT = S + ET + IT + QT + R. By adding all the equations in system (7), we obtained ω − µNT − µ2IT − µ5QT ≤ ω − µNT . The given initial conditions of the tuber- culosis sub-model system (7) ensure that N(0) ≥ 0. Hence, the total human population is positive and bounded for all finite time t > 0. From the theory of differential inequality [4], NT (t) ≤ NT (0)e −µt + ω µ (1 − e−µt). As t → +∞, we get 0 ≤ NT (t) ≤ ω µ . The feasible region of the TB sub-model (7) is given by ΩT = { (S,ET , IT , QT , R) ∈ R5 + : NT (t) ≤ ω µ } . The set ΩC is positive invariant and attracting [16], and all solutions of the tuberculosis sub-model (7) starting in ΩT remain in ΩT for all t ≥ 0. Hence, the model (7) is mathe- matically and epidemiologically well-posed, and it is sufficient to study its dynamics in ΩT . Without loss of generality, the last equation can be omitted since it is independent of the system (7). Setting all the differential equations in system (7) to zero and solve for all compartments, we obtain two equilibrium points, the disease-free equilibrium (DFE) and the endemic equilibrium (EE). The disease-free equilibrium point is obtained when ET = 0 and IT = 0 and given by E0T = (S,ET , IT , QT ) = ( ω µ , 0, 0, 0 ) . When ET ̸= 0 and IT ̸= 0, the endemic equilibrium point exists and is given by E1T = (S,ET , IT , QT ) where S = (µ+ rt)(δt + µ+ µ2) βtrt , ET = ωβtrt − µ(µ+ rt)(δt + µ+ µ2) (βtrt)(rt + µ) , IT = ωβtrt − µ(µ+ rt)(δt + µ+ µ2) βt(rt + µ)(δt + µ+ µ2) , QT = δt(ωβtrt − µ(µ+ rt)(δt + µ+ µ2)) βt(rt + µ)(δt + µ+ µ2)(γt + µ+ µ5) . This completes the solution for the existence of the equilibrium points for the tuberculosis sub-model (7). The basic reproduction number of the tuberculosis sub-model R0T is obtained using the next generation matrix [12]. The F is the transmission matrix and the V is the transition matrix of the system (7) are given as follows J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 13 of 28 F = βtITS 0 0  and V =  (rt + µ)ET −rtET + (δt + µ+ µ2)IT −δtIT + (γt + µ+ µ5)QT  . Then the Jacobian matrix of F and V evaluated in the disease-free equilibrium E0T F = 0 βtω µ 0 0 0 0 0 0 0  and V = rt + µ 0 0 −rt δt + µ+ µ2 0 0 −δt γt + µ+ µ5  . Setting B1 = rt + µ, B2 = δt + µ+ µ2, B3 = γt + µ+ µ5 (8) Hence, V −1 =  1 B1 0 0 rt B1B2 1 B2 0 δtrt B1B2B3 δt B2B3 1 B3  . Consequently, FV −1 =  βtωrt µB1B2 βtω µB2 0 0 0 0 0 0 0  . Accordingly, the largest eigenvalue of FV −1(λ) is βtωrt µB1B2 . Thus, the basic reproduction number of the tuberculosis sub-model (7) is R0T = βtωrt µ(rt + µ)(δt + µ+ µ2) . It is important to note that the endemic equilibrium can be expressed in terms of REOT , that is, S∗ = ω µR0T , E∗ T = ω(R0T − 1) (rt + µ)R0T , I∗T = rtω(R0T − 1) (rt + µ)R0T (δt + µ+ µ2) , Q∗ T = δtrtω(R0T − 1) (rt + µ)R0T (δt + µ+ µ2)(γt + µ+ µ5) . Hence, the unique positive endemic equilibrium for the tuberculosis sub-model (7) only exists when REOT > 1. In particular, we have the following result. Theorem 5. If R0T > 1, then the tuberculosis sub-model (7) has a unique positive endemic equilibrium. 3.5. Stability Analysis of the Equilibrium Points for the Tuberculosis Sub-model Theorem 6. If R0T < 1, then the tuberculosis sub-model (7) is locally asymptotically stable at the disease-free equilibrium EOT and unstable otherwise. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 14 of 28 Proof. The Jacobian matrix of system (7) evaluated at disease-free equilibrium E0T is given by JE0T = ( ω µ , 0, 0, 0 ) , that is JE0T =  −µ 0 −βtω µ 0 0 −B1 βtω µ 0 0 rt −B2 0 0 0 δt −B3  , where B1, B2 and B3 are given in (8). The characteristic polynomial of the Jacobian matrix JE0T is JE0T (λ) = (−µ− λ)(−B3 − λ) ( λ2 + (B1 +B2)λ+B1B2 − βtωrt µ ) . Certainly, the eigenvalues λ = −µ < 0 and λ = −B3 < 0. The other eigenvalues are determined by the second order polynomial factor which can be expressed as λ2 + (B1 +B2)λ+B1B2 ( 1−R0T ) . If R0T < 1, then the coefficient B1 + B2 and the constant term B1B2 ( 1 − R0T ) are all positive. Hence, by Routh-Hurwitz criterion, the tuberculosis-only submodel (7) is locally asymptotically stable at the disease-free equilibrium point E0T when R0T < 1 and unstable otherwise. Theorem 7. If R0T < 1, then the tuberculosis sub-model (7) is globally asymptotically stable at the disease-free equilibrium E0T . Proof. Consider the Lyapunov function below L = (δt + µ+ µ2)ET + (βtω µ ) IT . The time derivative of L computed along the solution of (7) is negative semi-definite, that is, · L = (δt + µ+ µ2)E ′ T (t) + (βtω µ ) I ′T (t) = (δt + µ+ µ2) [ βtITS − (rt + µ)ET ] + (βtω µ )[ rtET − (δt + µ+ µ2)IT ] ≤ (δt + µ+ µ2) [ βtIT (ω µ ) − (rt + µ)ET ] + (βtω µ )[ rtET − (δt + µ+ µ2)IT ] = −(δt + µ+ µ2)(rt + µ)ET + βtωrtET µ = [ (δt + µ+ µ2)(rt + µ)ET ] (R0T − 1) ≤ 0 if R0T < 1. If the solution of the system is x∗ = E0T , then · L = 0. Conversely, if · L = 0, then ST = ω µ , IT = 0, ET = 0 and QT = 0. Hence, · L = 0 if and only if x∗ = E0T . So, L is a Lyapunov function on ΩT and the largest compact invariant set in {(S,ET , IT , QT ) ∈ ΩT : · L = 0} is {(ωµ , 0, 0, 0)}. Thus, by LaSalle’s invariance principle, every solution of system (7) with the initial conditions in ΩT approaches E0T = (ωµ , 0, 0, 0) at t → ∞ whenever R0T < 1. Therefore, the tuberculosis sub-model (7) is globally asymptotically stable at the disease-free equilibrium point E0T = (ωµ , 0, 0, 0) when R0T < 1. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 15 of 28 Theorem 8. If R0T > 1, then the tuberculosis sub-model (7) is locally asymptotically stable at the endemic equilibrium point E1T and unstable otherwise. Proof. The Jacobian matrix of system (7) evaluated at the endemic equilibrium point E1T is given by JE1T =  −(ωβtrt−µB1B2) B1B2 − µ 0 −B1B2 rt 0 ωβtrt−µB1B2 B1B2 −B1 B1B2 rt 0 0 rt −B2 0 0 0 δt −B3  , where B1, B2 and B3 are given in (8). Hence, the characteristic polynomial of JE1T is JE1T (λ) = (−B3 − λ) ( −λ3 + (−B2 1B2 −B1B 2 2 − βtωrt)λ 2 B1B2 + (−B1βtωrt −B2βtωrt)λ B1B2 + (B2 1B 2 2µ−B1B2βtωrt) B1B2 ) Certainly, the eigenvalue λ = −B3 < 0. The other eigenvalues are determined by the cubic polynomial factor. Set the coefficients and constant term of the cubic polynomial as follows: b1 as the coefficient of λ2, b2 as the coefficient of λ and b3 as the constant term. By Routh Hurwitz criterion, the eigenvalues of the cubic polynomial have negative real parts when the following conditions are satisfied: b1, b2, and b3 are all positive and b1b2 > b3. Now, b1, b2, and b3 can be express as b1 = − (−B2 1B2 −B1B 2 2 − βtωrt) B1B2 = B2 1B2 +B1B 2 2 + βtωrt B1B2 > 0 b2 = − (−B1βtωrt −B2βtωrt) B1B2 = B1βtωrt +B2βtωrt B1B2 > 0 b3 = βtωrt −B1B2µ = B1B2µ ( βtωrt B1B2µ − 1 ) = B1B2µ(R0T − 1) Hence, the tuberculosis sub-model (7) is locally asymptotically stable at the endemic equilibrium point E1T when R0T > 1 and unstable otherwise. 3.6. Sensitivity Analysis of Tuberculosis Sub-model In this subsection, we conducted a sensitivity analysis of the parameters in the tuberculosis sub-model. For a parameter p, the sensitivity of p is defined as the behavior of the model to a small change in its value [22], and is given by: Sp = ∂R0T ∂p × p R0T where R0T = βtωrt µ(rt + µ)(δt + µ+ µ2) (9) The sensitivity indices for each parameter in the tuberculosis sub-model (7) with respect to R0T is computed by using the equation (9), and the corresponding sensitivity indices are shown in Figure 3. The sensitivity analysis of the tuberculosis sub-model revealed that the tuberculosis transmis- sion rate βt, and the recruitment rate ω have a high positive impact on the spread of the virus. The analysis suggests that the magnitude of the impacts of βt and ω on the spread of COVID-19 are the same because the number of secondary infections increases when the value of these parameters also increases [22]. Also, the progression rate of tuberculosis from asymptomatic to symptomatic rt has a positive impact on the spread of COVID-19 virus. In contrast, the parameters δt, µ, and µ2 have a negative impact, which means that increasing the value of such parameters will decrease the number of people infected with tuberculosis. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 16 of 28 Figure 3: Sensitivity indices of the tuberculosis sub-model parameters 3.7. COVID-19 and Tuberculosis Co-infection Model In this subsection, we examine the transmission dynamics of the co-infection model for COVID- 19 and tuberculosis which is described by system (1). When the environment is free from both diseases, the system reaches the disease-free equilibrium E0CT and is given by E0CT = (S,EC , IC , QC , ECT , ICT , QCT , ET , IT , QT ) = (ω µ , 0, 0, 0, 0, 0, 0, 0, 0, 0 ) . The basic reproduction number of the co-infection model is governed by the COVID-19 and tuberculosis sub-models, with the dominant value determining the overall transmission dynamics. Utilizing the Next Generation Matrix (NGM) approach, we compute the basic reproduction number of the co-infection model, where the transmission matrix F and transition matrix V are given by F =  βc(EC + IC)S 0 0 0 0 0 βtITS 0 0  and V =  αtβtITEC + (rc + δe + µ)EC −rcEC + (δc + µ+ µ1)IC −δeEC − δcIC − γctQCT + (γc + µ+ µ4)QC −αtβtITEC − αcβc(EC + IC)ET + (rct + µ)ECT −rctECT + (δct + µ+ µ3)ICT −δctICT + (γtc + γct + µ+ µ6)QCT αcβc(EC + IC)ET + (rt + µ)ET −rtET + (δt + µ+ µ2)IT −δtIT − γtcQCT + (γt + µ+ µ5)QT  . Accordingly, the Jacobian matrices of F and V evaluated at the disease-free equilibrium point E0CT are J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 17 of 28 F =  βcω µ βcω µ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 βtω µ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  and V =  A1 0 0 0 0 0 0 0 0 −rc A2 0 0 0 0 0 0 0 −ϵ C −ρ C A3 0 0 −γct 0 0 0 0 0 0 C1 0 0 0 0 0 0 0 0 −rct C2 0 0 0 0 0 0 0 0 −δct C3 0 0 0 0 0 0 0 0 0 B1 0 0 0 0 0 0 0 0 −θ T B2 0 0 0 0 0 0 −γtc 0 −ρ T B3  , respectively, where C1 = rct + µ,C2 = δct + µ+ µ3, and C3 = γtc + γct + µ+ µ5. (10) Hence, FV −1 =  βcω µA1 + βcωrc µA1A2 βcω µA2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 βtωrt µB1B2 βtω µB2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  Consequently, the nonzero eigenvalues are βcω(δc + µ+ µ1 + rc) µ(rc + δe + µ)(δc + µ+ µ1) and βtωrt µ(rt + µ)(δt + µ+ µ2) . Thus, the basic reproduction number of the COVID-19 and tuberculosis co-infection model (1) is R0CT = max { βcω(δc + µ+ µ1 + rc) µ(rc + δe + µ)(δc + µ+ µ1) , βtωrt µ(rt + µ)(δt + µ+ µ2) } . 3.8. Stability Analysis for the Co-infection Model of COVID-19 and Tuberculosis Theorem 9. If R0CT < 1, then the co-infection system (1) is locally asymptotically stable at the disease-free equilibrium point E0CT and unstable otherwise. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 18 of 28 Proof. The Jacobian matrix of system (1) evaluated at DFE E0CT is JE0CT =  −µ −βcω µ −βcω µ 0 0 0 0 0 −βtω µ 0 0 βcω µ −A1 βcω µ 0 0 0 0 0 0 0 0 rc −A2 0 0 0 0 0 0 0 0 δe δc −A3 0 0 γct 0 0 0 0 0 0 0 −C1 0 0 0 0 0 0 0 0 0 rct −C2 0 0 0 0 0 0 0 0 0 δct −C3 0 0 0 0 0 0 0 0 0 0 −B1 βtω µ 0 0 0 0 0 0 0 0 rt −B2 0 0 0 0 0 0 0 γtc 0 δt −B3  Then the linear factors of the characteristic polynomial of JE0CT are (−µ− λ), (−B3 − λ), (−A3 − λ), (−C3 − λ), (−C2 − λ), (−C1 − λ) and the second order polynomial factors are λ2 + (B1 +B2)λ+B1B2 − βcωrt µ and λ2 + ( A1 +A2 − βcω µ ) λ+A1A2 − βcωA2 µ − βcωrc µ . Similar to the proofs of Theorem (1) and Theorem (6), we can see that the second order polynomials above have roots with negative real parts. Therefore, the co-infection system (1) of COVID-19 and tuberculosis is locally asymptotically stable at the disease-free equilibrium point E0CT when R0CT < 1 and unstable otherwise. To examine the global stability of the co-infection model (1) at the disease-free equilibrium E0CT , we employ the method developed by Castillo-Chavez [7]. Rewriting the co-infection model (1) as  dX dt = F (X,Z) dZ dt = G(X,Z), where X = S and Z = (EC , IC , QC , ECT , ICT , QCT , ET , IT , QT ) denote the noninfectious and infectious states, respectively. The disease-free equilibrium is denoted now as E0CT = (X0, 0) where X0 = ω µ . The conditions stated below are necessary to ensure the global asymptotic stability of E0CT for R0CT < 1: (i) For dX dt = F (X, 0), X0 is globally asymptotically stable (ii) G(X,Z) = AZ − Ĝ(X,Z), Ĝ(X,Z) ≥ 0, for (X,Z) ∈ ω where A = DZG(E0CT , 0) is a Metzler matrix (the non-diagonal entities of A are non-negative), and ω is the biological feasible region of the model. From the co-infection model (1), we have dX dt = F (X,Z) = ω − βc(EC + IC)− βtITS − µS. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 19 of 28 Hence, dX dt = F (X, 0) = ω − µS and X0 is globally asymptotically stable. Moreover, dZ dt = G(X,Z) =  βc(EC + IC)S − αtβtITEC − (rc + δe + µ)EC rcEC − (δc + µ+ µ1)IC δeEC + δcIC + γctQCT − (γc + µ+ µ4)QC αtβtITEC + αcβc(EC + IC)ET − (rct + µ)ECT rctECT − (δct + µ+ µ3)ICT δctICT − (γtc + γct + µ+ µ6)QCT βtITS − αcβc(EC + IC)ET − (rt + µ)ET rtET − (δt + µ+ µ2)IT δtIT + γtcQCT − (γt + µ+ µ5)QT  Consequently, A =  βcω µ −A1 βcω µ 0 0 0 0 0 0 0 rc −A2 0 0 0 0 0 0 0 δe δc −A3 0 0 γct 0 0 0 0 0 0 −C1 0 0 0 0 0 0 0 0 rct −C2 0 0 0 0 0 0 0 0 δct −C3 0 0 0 0 0 0 0 0 0 −B1 βtω µ 0 0 0 0 0 0 0 rt −B2 0 0 0 0 0 0 γtc 0 δt −B3  . where Ai, Bi, Ci, i = 1, 2, 3 are given in equations 5, 8, and 10, respectively. Then, AZ =  ( βcω µ −A1 ) EC + βcω µ IC rcEC −A2IC δeEC + δcIC −A3QC + γctQCT −C1ECT rctECT − C2ICT δctICT − C3QCT −B1ET + βtω µ IT rtET −B2IT γtcQCT + δtIT −B3QT  . In effect, Ĝ(X,Y ) =  Ĝ1(X,Y ) Ĝ2(X,Y ) Ĝ3(X,Y ) Ĝ4(X,Y ) Ĝ5(X,Y ) Ĝ6(X,Y ) Ĝ7(X,Y ) Ĝ8(X,Y ) Ĝ9(X,Y )  =  βcω µ (EC + IC)− βc(EC + IC)S + αtβtITEC 0 0 −αtβtEC − αcβc(EC + IC)ET 0 0 αcβc(EC + Ic)ET 0 0  . J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 20 of 28 Observe that Ĝ5(X,Y ) = −αtβtEC − αcβc(EC + IC)ET < 0, which means that the condition 2 is not satisfied. Therefore, the co-infection model may not be globally asymptotically stable at the disease-free equilibrium E0CT . 4. Simulation In the preceding sections, we examined the stability behaviors of the sub-models (2) and (7), as well as the co-infection model (1). In this section, we present numerical simulations to illustrate the theoretical findings. The parameter values utilized in these simulations are provided in Table 1. 4.1. Numerical Simulation of the COViD-19 Sub-model Simulation 1. The given parameter values are based on Table 1, ω = 3, δe = 0.74, and δc = 0.78. Subsequently, we obtain R0C = 0.9922 and the disease-free equilibrium is E0C = (63, 0, 0, 0). We take the following initial conditions of the compartments of the COVID-19 sub-model: a. (100,70,45,35), b. (80,135,105,47), c. (75,87,38,52), and d. (240,87,213,97). Figure 4 shows that for different initial conditions, the lines of the solutions converge to E0C = (63, 0, 0, 0). It implies that the COVID-19 sub-model is locally asymptotically stable at the disease-free equilibrium when R0C < 1. Figure 4: (Simulation 1) The COVID-19 sub-model is locally asymptotically stable at E0C when R0C < 1. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 21 of 28 Simulation 2. Consider the same parameter values as in Simulation 1, except for the decreased values of δe = 0.24 and δc = 0.31. As a result, R0C becomes 4.0117, and the disease-free equilibrium remains unchanged at E0C = (63, 0, 0, 0). By using the same initial conditions as in Simulation 1, we observe from Figure 5 that the lines of the solutions do not converge to E0C = (63, 0, 0, 0). This indicates that the COVID-19 sub-model is unstable at the disease-free equilibrium when R0C > 1. Moreover, since we have obtained R0C = 4.0117 and the endemic equilibrium point E1C = (23, 4, 3, 5) now exists, we can observe that the lines of the solutions converge to E1C = (23, 4, 3, 5). Therefore, the COVID-19 sub-model is locally asymptotically stable at the endemic point whenever R0C > 1. Figures 4 and 5 illustrate the effect of quarantine measures in the COVID-19 sub-model (2). The findings indicate that when the intensity of quarantine measures is low, exposed and infected individuals with COVID-19 continue to transmit the disease to susceptible individuals. As a result, the number of exposed and infected cases increases. As observed in Figure 4, the number of susceptible individuals remains higher than that of exposed and infected individuals when the quarantine measure is stringent, suggesting that quarantine measures effectively reduce the spread of infection. Furthermore, Figure 4 shows that the number of exposed and infected individuals approaches zero, indicating widespread compliance with quarantine protocols. Consequently, the disease will eventually be eradicated within the community over time. Figure 5: (Simulation 2) The COVID-19 sub-model is locally asymptotically stable at E1C when R0C > 1. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 22 of 28 4.2. Numerical Simulation for Tuberculosis Sub-model Simulation 3. The given parameter values are based on Table 1, ω = 4 and δt = 0.95. Subsequently, we obtain R0T = 0.9349 and the disease-free equilibrium is E0T = (84, 0, 0, 0). To support our result, we take the following initial conditions: (a). (50,90,27,53), (b). (150,34,67,93), (c). (60,200,87,52) and (d). (230,47,129,109). Figure 6 shows that for different initial conditions, the lines of the solutions converge to E0T = (84, 0, 0, 0). It means that the tuberculosis sub-model (7) is locally asymptotically stable at the disease-free equilibrium when R0T < 1. Simulation 4. Consider the same parameter values as in Simulation 3, except for the decreased value of δt = 0.21. As a result, R0T = 3.5192 and the disease-free equilibrium remains unchanged at E0T = (84, 0, 0, 0). We observe from Figure 7 that the lines of the solutions do not converge to E0T = (84, 0, 0, 0). This indicates that the tuberculosis sub-model is unstable at the disease-free equilibrium when R0T > 1. Furthermore, since we have obtained R0T = 3.5192 and the endemic equilibrium point E1T = (23, 4, 11, 9) now exists, we can observe that the lines of the solutions converge to E1T (23, 4, 11, 9). Therefore, the tuberculosis sub-model is locally asymptotically stable at the endemic equilibrium point whenever R0T > 1. Figures 6 and 7 illustrate the effect of quarantine measures in the tuberculosis sub-model (7). The results are similar to those of the COVID-19 sub-model. When the value of the quarantine measure is low, infected individuals continue transmitting the disease, leading to an increase in cases. Conversely, a high quarantine measure helps control the spread of tuberculosis, ultimately leading to its eradication over time. Figure 6: (Simulation 3) The TB sub-model is locally asymptotically stable at E0T when R0T < 1. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 23 of 28 Figure 7: (Simulation 4) The TB sub-model is locally asymptotically stable at E1T when R0T > 1. 4.3. Numerical Simulation for Co-infection Model Simulation 5. The given parameter values are based on Table 1, ω = 2 and δe = 0.57, δc = 0.61, and δt = 0.46. Subsequently, we obtain R0C = 0.9749 and R0T = 0.9099 and the disease- free equilibrium is (42, 0, 0, 0, 0, 0, 0, 0, 0). Hence, the basic reproduction number of the co-infection model (1) isR0C = 0.9749. We take the following initial conditions: a. (100,45,30,0,35,25,0,35,20,0), b.(100,55,64,10,45,35,10,70,40,5), c.(150,100,78,70,48,37,10,64,28,19), d.(300,10,20,0,10,9,0,15,15,0). Figure 8 shows that the lines of the solutions converge to (42, 0, 0, 0, 0, 0, 0, 0, 0, 0). This indicates that the co-infection model (1) is locally asymptotically stable at the disease-free equilibrium when R0CT < 1. Simulation 6. Consider the same parameter values as in Simulation 5, except for the decreased value of δe = 0.02, δc = 0.05 and δt = 0.06. As a result, R0C = 13.1021 and R0T = 4.0021 and the disease-free equilibrium remains unchanged at (42, 0, 0, 0, 0, 0, 0, 0, 0, 0). We observe from Figure 9 that the lines of the solutions do not converge to (42, 0, 0, 0, 0, 0, 0, 0, 0, 0). This indicates that the co-infection model is unstable at the disease-free equilibrium when R0CT > 1. J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 24 of 28 Figure 8: (Simulation 5) The Co-infection model is locally asymptotically stable at the disease-free equilibrium when R0CT < 1 Figure 9: (Simulation 6) The Co-infection model is locally asymptotically stable at the disease-free equilibrium when R0CT < 1 J. Puebla, JM. Macalalag, K. Pajaron / Eur. J. Pure Appl. Math, 18 (3) (2025), 6149 25 of 28 5. Conclusion In this paper, a deterministic mathematical model for the co-infection of COVID-19 and tuber- culosis, incorporating quarantine measures, is proposed and analyzed. It is demonstrated that the solutions of the COVID-19 and tuberculosis sub-models are unique, positive, and bounded. The equilibrium points and basic reproduction numbers of both the sub-models and the co-infection model were computed independently. Subsequently, it is shown that both sub-models exhibit lo- cally and globally asymptotically stable at the disease-free equilibrium when the basic reproduction number is less than 1, and unstable otherwise. Additionally, the endemic equilibrium is locally asymptotically stable when the basic reproduction number exceeds 1. Moreover, the co-infection model is locally and globally asymptotically stable at the disease-free equilibrium when the max- imum reproduction number of the sub-models is less than 1. To further support and illustrate these analytical findings, multiple numerical simulations were conducted. The results of these simulations indicate that quarantine measures effectively mitigate the spread of both diseases. Based on the model results, public health initiatives should prioritize the strengthening of quarantine measures for individuals infected with COVID-19, tuberculosis, or both. Quarantine interventions effectively reduce the basic reproduction number below unity, thereby demonstrat- ing their critical role as a fundamental public health strategy for controlling current co-infection outbreaks. Furthermore, integrating quarantine protocols could significantly enhance disease mit- igation and may ultimately lead to the eradication of COVID-19 and tuberculosis co-infections. This study assumes a homogeneous population structure without accounting for differences in age, immunity levels, or social behavior, which may affect disease transmission dynamics in real-world settings. Additionally, the model does not explicitly incorporate vaccination strategies or their effects, such as vaccine efficacy, coverage, or waning immunity, which are critical factors in controlling COVID-19 and tuberculosis co-infections. Future work could extend the current model to include population heterogeneity, vaccination and reinfection dynamics to provide a more comprehensive assessment of intervention strategies. Conflict of Interest The authors declare no conflicts interest. Acknowledgements The authors would like to thank everyone who supported and contributed to this study. Special thanks to our colleagues for their valuable insights, constructive feedback, and continued support throughout the research process. References [1] Appiah, R. F., Jin, Z., Yang, J., Asamoah, J. K. 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