A mathematical model for the co-dynamics of COVID-19 and tuberculosis

Abstract

In this study, we formulated and analyzed a deterministic mathematical model for the co-infection of COVID-19 and tuberculosis, to study the co-dynamics and impact of each disease in a given population. Using each disease’s corresponding reproduction number, the existence and stability of the disease-free equilibrium were established. When the respective threshold quantities ${\mathcal{R}}_C$, and ${\mathcal{R}}_T$ are below unity, the COVID-19 and TB-free equilibrium are said to be locally asymptotically stable. The impact of vaccine (i.e., efficacy and vaccinated proportion) and the condition required for COVID-19 eradication was examined. Furthermore, the presence of the endemic equilibria of the sub-models is analyzed and the criteria for the phenomenon of backward bifurcation of the COVID-19 sub-model are presented. To better understand how each disease condition impacts the dynamics behavior of the other, we investigate the invasion criterion of each disease by computing the threshold quantity known as the invasion reproduction number. We perform a numerical simulation to investigate the impact of threshold quantities $({\mathcal{R}}_C,{\mathcal{R}}_T)$ with respect to their invasion reproduction number, co-infection transmission rate $\left({\beta }_{ct}\right)$, and each disease transmission rate $({\beta }_c,{\beta }_t)$ on disease dynamics. The outcomes established the necessity for the coexistence or elimination of both diseases from the communities. Overall, our findings imply that while COVID-19 incidence decreases with co-infection prevalence, the burden of tuberculosis on the human population increases.

1. Introduction

The infectious condition known as coronavirus disease (COVID-19) is the severe acute respiratory syndrome coronavirus 2 (SARS CoV 2). The virus can be transmitted from the mouth or nose in tiny liquid particles when an infected person talks, coughs or exhales. The novel COVID-19 has become widely known since it was first discovered in December 2019. It was classified as a pandemic by the World Health Organization (WHO) on March 12, 2020 [45]. Currently, only one medication has been approved to treat COVID-19, antiviral medications and monoclonal antibodies are the two categories into which the COVID-19 treatment can be divided. To prevent the virus from proliferating inside the body and causing serious sickness or death, antiviral medications target particular components of the virus, while monoclonal antibodies aid in the immune system’s ability to detect and combat the virus [8]. A COVID-19 vaccine aims to give acquired immunity against the disease. The use of vaccines to prevent infection is the most effective method for mitigating the COVID-19 pandemic. There are several COVID-19 vaccinations available worldwide. A current list of vaccine candidates that are being considered is kept up to date by the WHO [29]. According to official statistics from national public health organizations, 12.34 billion doses of the COVID-19 vaccination have been administered globally as of July 30th, 2022 [19].

Tuberculosis (TB) is a contagious bacterial disease that typically affects the lungs, but it can affect any organ in the body. It can form when bacteria spread through droplets in the air. This disease can be life-threatening but it is often preventable and treatable [53]. Its infection is caused by the bacterium called Mycobacterium tuberculosis (M.tb). Despite significant advances in prevention and treatment, tuberculosis remains a leading cause of death worldwide [22]. According to the WHO, two million individuals died from tuberculosis in 2020, including 214 000 people with HIV. TB is the 13th leading cause of death worldwide, and the second leading infectious killer after COVID-19. Tuberculosis spreads through the air from person to person. When people infected with TB cough, sneeze, or spit, the TB germs are released into the air, and only a few of these germs is been inhaled for a person to become infected. Chronic cough, bloody sputum, night sweats, fever, and weight loss are the most common symptoms of tuberculosis. Individuals with latent TB have no symptoms and cannot spread the infection, however, individuals at this stage may develop active TB and spread the disease to others [51]. The best way to stop the spread of TB is through prevention. It entails early detection and treatment of active TB to prevent transmission, vaccination of susceptible persons, and prevention of active disease in those who have been exposed to or are known to have latent infection. Chemoprophylaxis, a type of anti-TB pharmacological therapy, can lower the likelihood of a person with latent TB.

Co-infection, such as multi-parasite infections, is the simultaneous infection of a host by several pathogen species. It can also occur when two or more virus particles infect a single cell at the same time. This might happen progressively by first infecting the cell, then super-infecting it. According to a recent study, 0.37 to 4.47% of COVID-19 patients have TB cases. Experts and researchers have suggested that COVID-19’s effects are enhanced by tuberculosis. The lungs are affected by both diseases, which is a grave worry in the COVID-19 pandemic. According to the findings in [10], [68], TB infection increases a person’s vulnerability to COVID-19, as well as their likelihood of dying and having a slower recovery. There is widespread evidence that people with chronic respiratory conditions like TB run the risk of developing significant illnesses from COVID-19 and dying as a result. Patients with COVID-19 or TB may exhale microorganisms, and the treatment and prognosis of the co-infection are little understood [57]. Some studies have reported the impact of COVID-19 on TB dynamics and vice-versa. [4] developed a mathematical model on the impact of COVID-19 on TB control. The study examines an overview of the potential effects of COVID-19 on TB programs and the disease burden, as well as potential countermeasures.  [70] considers TB and COVID-19 interaction: A review of biological, clinical and public health effects. The study suggests that, based on the immunological mechanism at work, it has been discovered that COVID-19 and TB share a dysregulation of immune responses, which raises the possibility that co-infection poses a dual danger by aggravating COVID-19 severity and increasing the progression of TB disease. Different disease, same challenges, social determinants of TB and COVID-19 is considered in [17], the impact of COVID-19 on TB and the global TB burden is studied in [40]. The study suggests that, to efficiently allocate resources for the TB response, this requires a significant improvement in the TB data available. [64] considers patients’ perceptions regarding multidrug-resistant tuberculosis and barriers to seeking care in a priority city in Brazil during COVID-19 pandemic. Although, significant attention is still being devoted to COVID-19’s possible effects on TB patients. There has been a decline in the diagnosis of TB disease and infection in many nations and contexts, according to a recent study on the global impact of COVID-19 on TB services [12], [43]. According to a modeling study from the Philippines, COVID-19 patients with TB had a 2.17 times higher risk of death than non-TB patients and a 25% lower chance of recovering [68]. A vital aspect of the investigation of the dynamics of an infectious disease is mathematical modeling. Several mathematical models have been developed to study the dynamics and control of diseases in a given population. Many of these studies have employed different methods to quantify and mitigate disease burden (see [33], [59], [60], [61], [62] and the references therein). The dynamics of COVID-19 and TB co-infection transmission have also been described by several mathematical models. [38] developed a COVID-19 and tuberculosis model based forecasting in Delhi, India. The study suggests that primary prevention strategies should be addressed, particularly for TB patients. The management of severe COVID-19 in TB patients requires that TB treatment facilities and hospitals be ready for an early diagnosis. [23] proposed an optimal control analysis of a COVID-19 and TB co-dynamics model the study. Five control strategies are considered which are, TB awareness campaign, prevention against COVID-19 (e.g., face mask, physical distancing), control against co-infection, tuberculosis and COVID-19 treatment. The study considers effects of each and combinations of the control strategies on the dynamics and control of COVID-19 and TB. [55] studied a fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative. According to the study in [5], it is important to provide adequate care and treat other diseases during pandemics in order to prevent deaths from co-infection and lack of access to timely care. To the best of our knowledge, mathematical models that consider the co-infection of TB and COVID-19 are uncommon, prompting us to conduct further research. In light of this, this study is the first to use a deterministic model to examine the impact of COVID-19 on the burden of TB and vice versa. This is the founding motivation of this study. As a result, we develop and rigorously analyze qualitatively a mathematical model for the co-dynamics of COVID-19 and tuberculosis that incorporates the key epidemiological and biological features of both diseases. The order of the manuscript’s remaining sections is as follows. Section 2 presents the developed COVID-19-TB co-infection model, while a detailed theoretical analysis of the model is given in Section 3. This includes the analysis of the co-infection model, COVID-19-only, and TB-only sub-models. In Section 4, numerical simulations and a discussion of the results are provided, and Section 5 reports the study’s findings.

2. Model formulation

The current COVID-19 pandemic is still a major public health emergency that is seriously disrupting both the healthcare system and society as a whole. Since the first outbreak was discovered in Wuhan, China, in late 2019, the prevalence of the fatal disease has increased, resulting in high mortality rates across the globe [69]. Despite extensive research into the dynamics and management of this illness, the healthcare system continues to face a number of obstacles in reducing the danger it poses to humanity. The potential for various viruses to mutate, which has made it more difficult to control the emergence of disease cases, is one of many difficulties. An illustration of this is the recently discovered omicron variant, which, despite having mild symptoms, is extremely contagious in both vaccinated and unvaccinated people [6], [63]. The existence of other infectious diseases with similar infection symptoms, which makes it more challenging to diagnose infected people, presents another challenge to the healthcare system in reducing the risk of COVID-19. Malaria, influenza, and tuberculosis are a few examples of these illnesses [18], [56], [70]. It is important to note that despite the focus on containing the current pandemic, COVID-19 is harming some endemic diseases that pose a threat to human life, including TB. According to [4], the prevalence of COVID-19 has hampered efforts to control TB by raising the risk of drug-resistant TB development, delaying TB diagnosis and treatment, and increasing the transmission of the infection within the home during the lockdown. Additionally, according to some studies, there is a higher mortality rate for people with COVID-19 and TB co-infection [25], [35], [44], [70]. Several mathematical models have been developed and analyzed to investigate the impact of different control or prevention measures on the dynamic of COVID in different regions [20], [34], [46], [49], [67]. To better understand the co-dynamics of the two diseases, we present a ten-compartmental deterministic model to investigate how COVID-19 affects the burden of tuberculosis and vice versa. We grouped the total human population at time $t$ represented by $N\left(t\right)$ into a population of susceptible individuals $S\left(t\right)$, vaccinated individuals against COVID-19 $V\left(t\right)$, individuals exposed to COVID-19 $E\left(t\right)$, COVID-19 asymptomatic and symptomatic infectious individuals ($I_a\left(t\right),I_s\left(t\right)$ respectively). The total human population is further grouped into latent TB-infected individuals $L\left(t\right)$, active TB-infected individuals $A\left(t\right)$, infected individuals with both latent TB and COVID-19 $I_{lc}\left(t\right)$, individuals infected with both active TB and COVID-19 $I_{ac}\left(t\right)$, and recovered individuals $R\left(t\right)$. To make the notations more simple, we will subsequently remove the time $t$ from the model variables so that the total population of humans is given as

$$N=S+V+E+I_a+I_s+L+A+I_{lc}+I_{ac}+R.$$ (1)

It should be noted that while the people in the $E$ compartment are newly infected with COVID-19, they are not yet infectious (i.e., they cannot transmit the disease), whereas those in the $I_a$ and $I_s$ compartments are infectious people who can spread the disease. Individuals who are infectious but asymptomatic do not exhibit disease symptoms while those who are symptomatic do. Additionally, those who are latently infected with TB do not exhibit any clinical symptoms and cannot transmit the infection, whereas those with active TB exhibit symptoms and can spread the disease [9], [28]. The deterministic system of nonlinear differential equations below provides the COVID-19-TB co-infection model.

$$\frac{dS}{dt}=\pi +\omega V+({\kappa }_c+{\kappa }_t+{\kappa }_{ct})R-(\nu +\mu +{\lambda }_c+{\lambda }_t+{\lambda }_{ct}+{\lambda }_z)S$$

$$\frac{dV}{dt}=\nu S-(1-\varepsilon )({\lambda }_c+{\lambda }_{ct})V-(\omega +\mu )V$$

$$\frac{dE}{dt}=({\lambda }_c+{\lambda }_{ct})S+(1-\varepsilon )({\lambda }_c+{\lambda }_{ct})V-({\sigma }_c+\mu )E$$

$$\frac{d{I}_a}{dt}={\sigma }_c(1-\psi )E+p{\gamma }_5{I}_{lc}-(\mu +{\delta }_c+{\gamma }_1)I_a-\theta ({\lambda }_t+{\lambda }_z)I_a$$

$$\frac{d{I}_s}{dt}={\sigma }_c\psi E+m{\gamma }_6{I}_{ac}-(\mu +{\delta }_c+{\gamma }_2)I_s-\theta ({\lambda }_t+{\lambda }_z)I_s$$ (2)

$$\frac{dL}{dt}=({\lambda }_t+{\lambda }_z)S+q{\gamma }_5{I}_{lc}-({\sigma }_t+\mu +{\gamma }_3)L-\theta ({\lambda }_c+{\lambda }_z)L$$

$$\frac{dA}{dt}={\sigma }_{t}L+n{\gamma }_6{I}_{ac}-(\mu +{\delta }_t+{\gamma }_4)A-\theta ({\lambda }_c+{\lambda }_{ct})A$$

$$\frac{d{I}_{lc}}{dt}=\theta ({\lambda }_t+{\lambda }_z)(I_a+I_s)+\theta ({\lambda }_c+{\lambda }_z)L-(\mu +{\delta }_{ct}+{\sigma }_{ct}+{\gamma }_5)I_{lc}$$

$$\frac{d{I}_{ac}}{dt}={\sigma }_{ct}{I}_{lc}+\theta ({\lambda }_c+{\lambda }_{ct})A-(\mu +{\delta }_{ct}+{\gamma }_6)I_{ac}$$

$$\frac{dR}{dt}={\gamma }_1{I}_a+{\gamma }_2{I}_s+{\gamma }_{3}L+{\gamma }_{4}A+(1-(p+q)){\gamma }_5{I}_{lc}+(1-(m+n)){\gamma }_6{I}_{ac}-(\mu +{\kappa }_c+{\kappa }_t+{\kappa }_{ct})R.$$

with the initial conditions

$$\left\{\begin{array}{c}S\left(0\right)>0,V\left(0\right)\ge 0,E\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_s\left(0\right)\ge 0,L\left(0\right)\ge 0,A\left(0\right)\ge 0,I_{lc}\left(0\right)\ge 0,I_{ac}\left(0\right)\ge 0,R\left(0\right)\ge 0.\hfill \end{array}\right)$$ (3)

In Table 1, Table 2, the interpretations of the model variables and parameters are given, respectively, and Fig. 1 shows the schematic diagram. We note that the parameter values used are obtained from existing literature. The forces of infection ${\lambda }_c$, ${\lambda }_t$, ${\lambda }_{ct}$, and ${\lambda }_z$ in the co-infection model (2) are defined as follows:

$${\lambda }_c=\frac{{\beta }_c(\eta I_a+I_s)}{N},{\lambda }_t=\frac{{\beta }_{t}A}{N}{\lambda }_{ct}=\frac{{\beta }_{ct}(I_{lc}+I_{ac})}{N},{\lambda }_z=\frac{{\beta }_{ct}{I}_{ac}}{N}.$$ (4)

According to the co-infection model (2), the susceptible individuals become infected with the COVID-19 virus after effective contact with COVID-19 asymptomatic $I_a$ and symptomatic infectious $I_s$ human, and also co-infected individuals ($I_{lc}$ and $I_{ac}$), at the rates ${\lambda }_c$ and ${\lambda }_{ct}$ respectively. The infection modification parameter $\eta$ justifies the presumptive rise in the relative infectiousness of the symptomatic infectious individuals over the asymptomatic infectious individuals. Additionally, a susceptible person will become infected with latent TB after coming in contact with an active TB infectious individual $A$, and individuals infected with both diseases ($I_{lc}$ and $I_{ac}$), at the rate ${\lambda }_t$ and ${\lambda }_{ct}$ respectively. The parameters ${\beta }_c$, ${\beta }_t$, and ${\beta }_{ct}$ represent the effective contact rate resulting in COVID-19 and tuberculosis transmission. The susceptible humans are vaccinated against COVID-19 at the rate $\nu$. However, because the disease vaccine is imperfect (i.e. do not provide the hosts total immunity against disease) [32], [37], the population of people who have received COVID-19 vaccinations is decreased by the force of infection at a slower rate $(1-\varepsilon )({\lambda }_c+{\lambda }_{ct})$ with a protective vaccine efficacy satisfying $0<\varepsilon <1$.

Table 1.

Description of the COVID-19-TB model variables.

Variable	Description
$S$	Population of susceptible individuals
$V$	Population of individuals vaccinated against COVID-19
$E$	Population of individuals exposed to COVID-19 only
$I_a$	Population of COVID-19 asymptomatic infectious individuals
$I_s$	Population of COVID-19 symptomatic infectious individuals
$L$	Population of latent TB infected individuals
$A$	Population of active TB infectious individuals
$I_{lc}$	Population of co-infected individuals with both latent TB and COVID-19
$I_{ac}$	Population of co-infected individuals with both active TB and COVID-19
$R$	Population of recovered individuals

Table 2.

The parameter’s value and description.

Parameter	Description	Value	Source
$\pi$	Recruitment rate of susceptible individuals	$1.2\times 10^4$	[26]
$\eta$	Infection modification parameter for the asymptomatic infection rate	0.450	[1]
$\nu$	Vaccination rate against COVID-19	0.0203	[32]
${\beta }_c$	Transmission probability rate of COVID-19	0.5249	[26]
${\beta }_t$	Transmission probability rate of TB	2.9598	[55]
${\beta }_{ct}$	Transmission probability rate of co-infection	$\left({\beta }_c{\beta }_t\right)$	Assumed
${\kappa }_c$	Immunity waning rate of recovered individuals with COVID-19	0.011	[15]
${\kappa }_t$	Immunity waning rate of recovered individuals with TB	0.0027	[54], [65]
${\kappa }_{ct}$	Immunity waning rate of recovered individuals with co-infection	max (${\kappa }_c$, ${\kappa }_t$)	Assumed
$\omega$	Vaccine waning rate of COVID-19	0.000297	[26]
$\varepsilon$	COVID-19 vaccine efficacy	0.700	[26]
${\sigma }_c$	COVID-19 progression rate from exposure to either $I_a$ or $I_s$	0.400	[26]
${\sigma }_t$	TB progression rate from latent TB to active TB	0.0039	[42]
${\sigma }_{ct}$	Co-infection progression rate from $I_{lc}$ to $I_{ac}$	1.1148	[42]
$\theta$	Modification parameter for the infectiousness of co-infected individuals	1.000	[55]
$\psi$	Fraction of COVID-19 exposed individuals becoming symptomatic	0.600	[7]
$\mu$	Natural death rate of humans	0.0003516	[26]
${\delta }_c$	Mortality rate due to COVID-19 infection	0.008	[13]
${\delta }_t$	Mortality rate due to TB infection	0.00032	[55]
${\delta }_{ct}$	Mortality rate due to co-infection	0.002	[42]
${\gamma }_1$	Recovery rate of COVID-19 asymptomatic infectious individuals	0.13978	[2]
${\gamma }_2$	Recovery rate of COVID-19 symptomatic infectious individuals	0.100	[2]
${\gamma }_3$	Recovery rate of latent TB infected individuals	0.090	[41], [66]
${\gamma }_4$	Recovery rate of active TB infectious individuals	0.3500	[14], [23]
${\gamma }_5$	Movement rate of individuals from co-infected latent TB and COVID-19 population	1.3523	[42]
${\gamma }_6$	Movement rate of individuals from co-infected active TB and COVID-19 population	0.0422	[42]

Fig. 1.

The schematic diagram of the co-infection of COVID-19-TB model (2). The dotted black line denotes the extension of the population of recovered humans, while the dotted orange line denotes the extension of the susceptible population. Also, for depiction convenience we define ${\delta }_1=\mu +{\delta }_c$, ${\delta }_2=\mu +{\delta }_t$, ${\delta }_3=\mu +{\delta }_{ct}$, ${\gamma }_5^{\ast }={\gamma }_5(1-(p+q))$, ${\gamma }_6^{\ast }={\gamma }_6(1-(m+n))$, and ${\kappa }^{\ast }={\kappa }_c+{\kappa }_t+{\kappa }_{ct}$, and the forces of infection ${\lambda }_1={\lambda }_c+{\lambda }_{ct}$, ${\lambda }_2=(1-\varepsilon ){\lambda }_1$, ${\lambda }_3={\lambda }_t+{\lambda }_z$ and ${\lambda }_4={\lambda }_c+{\lambda }_z$.

Below are some of the key presumptions that were used to develop the co-infection model (2).

• 1.Since both COVID-19 and TB influence the human immune system, it stands to reason that those who have one will be more vulnerable to contracting the other. The increased infectiousness of co-infected individuals brought on by each disease is taken into account by the modification parameter $\theta$ (satisfying $\theta \ge 1$) [4], [10].
• 2.Individuals who are co-infected with COVID-19 and TB can spread either of the two diseases, but not both infections at once [23].
• 3.We assume that people with COVID-19 are susceptible to infection with TB and vice versa based on several case reports on the co-infection of the two diseases [25], [35].
• 4.Co-infected people cannot recover simultaneously from both infections; they can only recover from one disease at a time [41], [42]. For instance, the recovery rate of co-infected latent TB and COVID-19 individuals is given as $(1-(p+q)){\gamma }_5$ such that when the person recovers from tuberculosis, they move to the infected COVID-19 population at the rate $p{\gamma }_5$.

3. Model analysis

We conducted the qualitative analysis in this section to learn more about the dynamics of COVID-19 and TB transmission in a population. These include demonstrating each model’s positivity and boundedness, proving the existence and stability of steady-state solutions, calculating the threshold quantities, and looking into the potential for a backward bifurcation phenomenon. To do this, we first examine the COVID-19-only and TB-only sub-models ((5), (19) respectively), and then we present a generalized result of the co-infection model (2).

3.1. COVID-19 only model

By equating the variables $L=A=I_{la}=I_{ac}=0$ in (2), the COVID-19-only sub-model is derived as below.

$$\frac{dS}{dt}=\pi +\omega V+{\kappa }_{c}R-(\nu +\mu +{\lambda }_c)S$$

$$\frac{dV}{dt}=\nu S-(1-\varepsilon ){\lambda }_{c}V-(\omega +\mu )V$$

$$\frac{dE}{dt}={\lambda }_{c}S+(1-\varepsilon ){\lambda }_{c}V-({\sigma }_c+\mu )E$$

$$\frac{d{I}_a}{dt}={\sigma }_c(1-\psi )E-(\mu +{\delta }_c+{\gamma }_1)I_a$$

$$\frac{d{I}_s}{dt}={\sigma }_c\psi E-(\mu +{\delta }_c+{\gamma }_2)I_s$$ (5)

$$\frac{dR}{dt}={\gamma }_1{I}_a+{\gamma }_2{I}_s-(\mu +{\kappa }_c)R.$$

with the initial conditions $S\left(0\right)>0,V\left(0\right)\ge 0,E\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_s\left(0\right)\ge 0,R\left(0\right)\ge 0$. The force of infection is given as ${\lambda }_c=\frac{{\beta }_c}{N_c}(\eta I_a+I_s)$, where $N_c=S+V+E+I_a+I_s+R$.

3.1.1. Positivity and boundedness of solutions

We demonstrate that for the feasible region ${\Omega }_C$ to be bounded and the model to be epidemiologically significant, the state variables of the COVID-19 only model (5) are non-negative for all time $t>0$. The following result is given.

Theorem 1

The preliminary data for model (5) fulfilled $S\left(0\right)>0,V\left(0\right)\ge 0,E\left(0\right)\ge 0,I_a\left(0\right)\ge 0,I_s\left(0\right)\ge 0$ , and $R\left(0\right)\ge 0$ , for which the model solutions with non-negative data stay non-negative for the entire time $t>0$ .

Proof

Let $t_f=sup\{t>0:S\left(t\right)>0,V\left(t\right)>0,E\left(t\right)>0,I_a\left(t\right)>0,I_s\left(t\right)>0,R\left(t\right)>0\in [0,t]\}$, so that $t_f>0$. It is evident from the first equation of the sub-model (5) that

$$\frac{dS}{dt}=\pi +{\kappa }_{c}R+\omega V-(\nu +\mu +{\lambda }_c)S\ge \pi -{\lambda }_{c}S-(\nu +\mu )S.$$ (6)

We use the integrating factor method to resolve the equation shown in (6). This is therefore written as

$$\frac{d}{dt}\left(S\left(t\right)exp\left[(\mu +\nu )t+{\int }_0^t{\lambda }_c\left(\zeta \right)d\zeta \right]\right)\ge \pi exp\left[(\mu +\nu )t+{\int }_0^t{\lambda }_c\left(\zeta \right)d\zeta \right],$$

So that,

$$S\left(t_f\right)exp\left[(\mu +\nu )t_f+{\int }_0^{t_f}{\lambda }_c\left(\zeta \right)d\zeta \right]-S\left(0\right)\ge {\int }_0^{t_f}\pi \left(exp\left[(\mu +\nu )x+{\int }_0^x{\lambda }_c\left(\zeta \right)d\zeta \right]\right)dx,$$

Hence,

$$S\left(t_f\right)\ge S\left(0\right)exp\left[-(\mu +\nu )t_f-{\int }_0^{t_f}{\lambda }_c\left(\zeta \right)d\zeta \right]+exp\left[-(\mu +\nu )t_f-{\int }_0^{t_f}{\lambda }_c\left(\zeta \right)d\zeta \right]\times {\int }_0^{t_f}\pi \left(exp\left[(\mu +\nu )x+{\int }_0^x{\lambda }_c\left(\zeta \right)d\zeta \right]\right)dx>0.$$

It is clear from the inequality above that $S\left(t_f\right)\ge 0$ is non-negative. The rest of the variables $V\left(t\right)\ge 0,E\left(t\right)\ge 0,I_a\left(t\right)\ge 0,I_s\left(t\right)\ge 0$, and $R\left(t\right)\ge 0$ can also be demonstrated to be non-negative for $t>0$. This means that for all positive initial conditions, all solutions of the COVID-19-only model (5) remain positive. □

Additionally, for the COVID-19 only model (5) to be mathematically and epidemiologically meaningful, system (5) must be considered in a feasible region ${\Omega }_C\subset {\mathcal{R}}_{+}^6$ so that

$${\Omega }_C=\left\{\left(S,V,E,I_a,I_s,R\right)\in {\mathcal{R}}_{+}^6:S+V+E+I_a+I_s+R\le \frac{\pi }{\mu }\right\}.$$

All of the solutions to the model (5) are drawn to the positively invariant feasible region ${\Omega }_C$. This implies that all solutions that begin with ${\Omega }_c$ remain there. As a result, it can be said that the COVID-19-only sub-model is both mathematically and epidemiologically well-posed.

3.1.2. Existence and stability of the COVID-19 free-equilibrium (CFE)

We obtained the COVID-19-free equilibrium (CFE), by equating the infection variables to zero. Hence, the CFE represented by ${\mathcal{E}}_{C0}$ is derived as

$${\mathcal{E}}_{C0}=(S^{\ast },V^{\ast },E^{\ast },I_a^{\ast },I_s^{\ast },R^{\ast })=\left(\frac{\pi (\mu +\omega )}{\mu (\mu +\omega +\nu )},\frac{\pi \nu }{\mu (\mu +\omega +\nu )},0,0,0,0\right).$$ (7)

Using the next-generation matrix operator as described in [47], [52], the reproduction number threshold quantity is obtained in order to investigate the stability of the system. To do this, the Jacobian matrices for the new infection and the remaining transfer terms are respectively given as

$$F=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill \frac{{\beta }_c\eta (S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill & \hfill \frac{{\beta }_c(S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{ and }V=\left[\begin{array}{ccc}\hfill k_3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -k_4{\sigma }_c\hfill & \hfill k_5\hfill & \hfill 0\hfill \\ \hfill -\psi {\sigma }_c\hfill & \hfill 0\hfill & \hfill k_6\hfill \end{array}\right].$$

Hence, the reproduction number of the COVID-19 sub-model (5), which is determined by the highest eigenvalue of $F{V}^{-1}$ given by ${\mathcal{R}}_C=\rho \left(F{V}^{-1}\right)$, is derived as

$${\mathcal{R}}_C=\frac{{\beta }_c{\sigma }_c(\mu +\omega +\nu k_1)(\psi k_5+\eta k_4{k}_6)}{k_3{k}_5{k}_6(\mu +\omega +\nu )}.$$ (8)

The threshold quantity ${\mathcal{R}}_C$ provided in (8) is referred to as the control reproduction number, also known as the effective reproduction number. It calculates the typical amount of new infections that one infected person can generate throughout infectiousness in an entirely susceptible community where control interventions are implemented [27]. As a result, the threshold quantity ${\mathcal{R}}_C$ in (8) measures the anticipated amount of new COVID-19 cases that a single COVID-infected person can replicate in a wholly susceptible community. Furthermore, in the case where there are no control interventions (i.e. $\nu =\varepsilon =\omega =0$), then the threshold quantity known as the basic reproduction number is derived as

$${\mathcal{R}}_{△}=\frac{{\beta }_c{\sigma }_c(\psi k_5+\eta k_4{k}_6)}{k_3{k}_5{k}_6}.$$ (9)

The basic reproduction number ${\mathcal{R}}_{△}$ measures the average number of secondary infections that one infected individual can replicate in a population that is completely susceptible and without any control interventions (see, for example, [3], [21], [24], [48], [58]). The local stability of the COVID-19-free equilibrium ${\mathcal{R}}_C$ is established using Theorem 2 of [16] and the effective reproduction number ${\mathcal{R}}_C$ (henceforth referred to as COVID-19 reproduction number). By applying Theorem 2 of [16], the effective reproduction number ${\mathcal{R}}_C$ (henceforth referred to as reproduction number) is used to demonstrate the local stability of the COVID-19-free equilibrium ${\mathcal{E}}_{C0}$. The Theorem below shows the result.

Theorem 2

The COVID-19-free equilibrium ${\mathcal{E}}_{C0}$ , of the model (5) is locally asymptotically stable (LAS) in the feasible region ${\Omega }_C$ if ${\mathcal{R}}_C<1$ and unstable otherwise.

Proof

To demonstrate the aforementioned theorem, the Jacobian matrix of system (5) at the COVID-19 free-equilibrium ${\mathcal{E}}_{C0}$ was obtained as follows.

$$\mathcal{J}\left({\mathcal{E}}_{C0}\right)=\left[\begin{array}{cccccc}\hfill -k_0\hfill & \hfill \omega \hfill & \hfill 0\hfill & \hfill -\frac{{\beta }_c\eta S^{\ast }}{N_c^{\ast }}\hfill & \hfill \frac{{\beta }_c{S}^{\ast }}{N_c^{\ast }}\hfill & \hfill {\kappa }_c\hfill \\ \hfill \nu \hfill & \hfill -k_2\hfill & \hfill 0\hfill & \hfill -\frac{{\beta }_c\eta k_1{V}^{\ast }}{N_c^{\ast }}\hfill & \hfill -\frac{{\beta }_c{k}_1{V}^{\ast }}{N_c^{\ast }}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -k_3\hfill & \hfill \frac{{\beta }_c\eta (S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill & \hfill \frac{{\beta }_c(S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill k_4{\sigma }_c\hfill & \hfill -k_5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \psi {\sigma }_c\hfill & \hfill 0\hfill & \hfill -k_6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\gamma }_1\hfill & \hfill {\gamma }_2\hfill & \hfill -k_7\hfill \end{array}\right].$$ (10)

where $k_0=\mu +\nu$, $k_1=1-\varepsilon$, $k_2=\mu +\omega$, $k_3=\mu +{\sigma }_c$, $k_4=1-\psi$, $k_5=\mu +{\delta }_c+{\gamma }_1$, $k_6=\mu +{\delta }_c+{\gamma }_2$, and $k_7=\mu +{\kappa }_c$, while $S^{\ast }$ and $V^{\ast }$ are steady states given in (7). It is necessary to demonstrate that the eigenvalues of the Jacobian matrix $\mathcal{J}\left({\mathcal{E}}_{C0}\right)$ are all negative in order to establish the stability of the COVID-19-free equilibrium. The first eigenvalue of the matrix (10) is obtained as $-k_7$, and the remaining six eigenvalues are obtained from the reduced sub-matrix ${\mathcal{N}}_1$ which is given below as

$${\mathcal{N}}_1=\left[\begin{array}{ccccc}\hfill -k_0\hfill & \hfill \omega \hfill & \hfill 0\hfill & \hfill -\frac{{\beta }_c\eta S^{\ast }}{N_c^{\ast }}\hfill & \hfill \frac{{\beta }_c{S}^{\ast }}{N_c^{\ast }}\hfill \\ \hfill \nu \hfill & \hfill -k_2\hfill & \hfill 0\hfill & \hfill -\frac{{\beta }_c\eta k_1{V}^{\ast }}{N_c^{\ast }}\hfill & \hfill -\frac{{\beta }_c{k}_1{V}^{\ast }}{N_c^{\ast }}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -k_3\hfill & \hfill \frac{{\beta }_c\eta (S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill & \hfill \frac{{\beta }_c(S^{\ast }+k_1{V}^{\ast })}{N_c^{\ast }}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill k_4{\sigma }_c\hfill & \hfill -k_5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \psi {\sigma }_c\hfill & \hfill 0\hfill & \hfill -k_6\hfill \end{array}\right].$$ (11)

If $\text{Tr}\left({\mathcal{N}}_1\right)<0$ and $\text{Det}\left({\mathcal{N}}_1\right)>0$, then all of the eigenvalues of the sub-matrix ${\mathcal{N}}_1$ will be real and negative, according to the Routh–Hurwitz criterion. By using the sub-matrix (11), we obtain the following

$$Tr\left({\mathcal{N}}_1\right)=-(k_0+k_2+k_3+k_5+k_6)<0\text{ and }Det\left({\mathcal{N}}_1\right)=k_3{k}_5{k}_6(k_0{k}_2-\nu \omega )(1-{\mathcal{R}}_C)>0.$$

Based on the above findings, if ${\mathcal{R}}_C<1$, then all of the eigenvalues of the sub-matrix (10) are negative real parts. As a result, the COVID-19 free-equilibrium ${\mathcal{E}}_{C0}$ is locally asymptotically stable but otherwise unstable. □

Based on the epidemiological interpretation of Theorem 2, COVID-19 disease can be restricted in the communities when ${\mathcal{R}}_C<1$ and the initial sizes of the system (5) sub-population are in the ${\mathcal{E}}_{C0}$ region of attraction.

3.1.3. Evaluating the impact of vaccine on COVID-19 burden using the threshold quantities $({\mathcal{R}}_C,{\mathcal{R}}_{△})$

According to Theorem 2, the control of COVID-19 is dependent on the value of ${\mathcal{R}}_C$, implying that disease elimination is possible if ${\mathcal{R}}_C<1$. However, this is not always assured, especially in a system with two coexisting equilibria. Many studies have linked the usage of imperfect vaccines to the occurrence of backward bifurcation (see [30], [50] and the references therein). As a result, given that we formulate the COVID-19 model with an imperfect vaccine, it will be beneficial to examine the effect of such a vaccine, as well as the vaccine coverage, on the disease prevalence in the communities. To do this, the proportion of vaccinated individuals is defined as $\vartheta =\frac{V^{\ast }}{N_c^{\ast }}$, and the effective reproduction number is expressed as a function of $\vartheta$, so that

$${\mathcal{R}}_C={\mathcal{R}}_{△}\left(\frac{\mu +\omega +(1-\varepsilon )\vartheta }{\mu +\omega +\nu }\right)={\mathcal{R}}_{△}(1-\varepsilon \vartheta ).$$ (12)

We recall that ${\mathcal{R}}_{△}$ presented in Eq. (9) is the basic reproduction number in the absence of any intervention strategy, while ${\mathcal{R}}_C$ is the control reproduction number with an imperfect COVID-19 vaccine. It is worth noting that ${\mathcal{R}}_C\le {\mathcal{R}}_{△}$ is true, and the equality sign would only be satisfied if $\nu =\varepsilon =\omega =0$ (i.e. $\vartheta =0$). Despite using an imperfect vaccine, increasing the vaccinated proportion $\vartheta$ and vaccine efficacy $\varepsilon$ will decrease the prevalence of COVID-19 in the communities. To determine the required condition under which COVID-19 can be eliminated based on $\vartheta$, we calculated the least proportion of people that needs to be vaccinated as

$$\vartheta \ge \frac{1}{\vartheta }\left(1-\frac{1}{{\mathcal{R}}_{△}}\right)={\vartheta }_c.$$ (13)

so that ${\vartheta }_c$ is the critical minimum proportion of vaccinated individuals that are required to eliminate COVID-19 disease in the populace. The following theorem is based on the findings of the Refs. [27], [50].

Theorem 3

COVID-19 infection can be eradicated in the communities if $\vartheta >{\vartheta }_c$ .

We present a contour plot of the effective reproduction number ${\mathcal{R}}_C$ as a function of $\varepsilon$ and $\vartheta$ in Fig. 2. The figure depicts the effect of vaccine effectiveness $\varepsilon$ and the vaccine coverage $\vartheta$ on the disease burden through the effective reproduction number. It can be inferred that a greater proportion of immunized people must be acquired with increased vaccine efficacy in order to properly manage the dispersion of COVID-19 in the community, so that the condition ${\mathcal{R}}_C<1$ would be satisfied.

Fig. 2.

Contour plot of ${\mathcal{R}}_C$ with respect to vaccine coverage $(\vartheta =\frac{V^{\ast }}{N_c^{\ast }})$ and vaccine efficacy $\varepsilon$. Table 2 provides the parameter values that were used.

3.1.4. Existence of the COVID-19 endemic equilibria

In this section, we obtain the endemic equilibrium of the COVID-19-only model (5). This is the steady state solution of the system in the presence of infection, which is denoted by ${\mathcal{E}}_{C1}=(S^{\ast \ast },V^{\ast \ast },E^{\ast \ast },I_a^{\ast \ast },I_s^{\ast \ast },R^{\ast \ast })$. We define the force of infection ${\lambda }_c^{\ast \ast }$ as follows for convenience:

$${\lambda }_c^{\ast \ast }=\frac{{\beta }_c(\eta I_a^{\ast \ast }+I_s^{\ast \ast })}{N_c^{\ast \ast }},$$ (14)

such that the total population $N_c^{\ast \ast }=S^{\ast \ast }+V^{\ast \ast }+E^{\ast \ast }+I_a^{\ast \ast }+I_s^{\ast \ast }+R^{\ast \ast }$. The steady states solutions are given in the form of the force of infection ${\lambda }_c^{\ast \ast }$ as

$$S^{\ast \ast }=\frac{\pi k_3{k}_5{k}_6{k}_7(k_2+k_1{\lambda }_c^{\ast \ast })}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3},V^{\ast \ast }=\frac{\pi \nu k_3{k}_5{k}_6{k}_7}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3}$$

$$E^{\ast \ast }=\frac{\pi k_5{k}_6{k}_7{\lambda }_c^{\ast \ast }(k_2+k_1\nu +k_1{\lambda }_c^{\ast \ast })}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3},I_a^{\ast \ast }=\frac{\pi \sigma k_4{k}_6{k}_7{\lambda }_c^{\ast \ast }(k_2+k_1\nu +k_1{\lambda }_c^{\ast \ast })}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3}$$ (15)

$$I_s^{\ast \ast }=\frac{\pi {\sigma }_c\psi k_5{k}_7{\lambda }_c^{\ast \ast }(k_2+k_1\nu +k_1{\lambda }_c^{\ast \ast })}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3},R^{\ast \ast }=\frac{\pi {\sigma }_c{\lambda }_c^{\ast \ast }(\psi {\gamma }_2{k}_5+{\gamma }_1{k}_4{k}_6)(k_2+k_1\nu +k_1{\lambda }_c^{\ast \ast })}{a_1{\lambda }_c^{\ast \ast 2}+a_2{\lambda }_c^{\ast \ast }+a_3},$$

where the constants $a_1$, $a_2$, and $a_3$ are given as

$$a_1=k_1{k}_3{k}_5{k}_6{k}_7-k_1{\kappa }_c{\sigma }_c(\psi {\gamma }_2{k}_5+{\gamma }_1{k}_4{k}_6),$$

$$a_2=k_3{k}_5{k}_6{k}_7(k_2+k_0{k}_1)-{\kappa }_c{\sigma }_c(k_2+\nu k_1)({\gamma }_1{k}_4{k}_6+\psi {\gamma }_2{k}_5),$$

$$a_3=k_3{k}_5{k}_6{k}_7(k_0-\nu \omega ).$$

We substitute the above expressions (15) into the force of infection (14), and the resulting polynomial is given as

$$m_2{\lambda }_c^{\ast \ast 2}+m_1{\lambda }_c^{\ast \ast }+m_0=0,$$ (16)

where the coefficients of the polynomial equation are given as

$$m_2=k_1[\psi k_5{\sigma }_c({\gamma }_2+k_7)+k_4{k}_6{\sigma }_c({\gamma }_1+k_7)+k_5{k}_6{k}_7],$$

$$m_1=\nu {\sigma }_c{k}_1(\psi {\gamma }_2{k}_5+\psi k_5{k}_7+{\gamma }_1{k}_4{k}_6+k_4{k}_6{k}_7)+k_2{k}_5\psi {\sigma }_c({\gamma }_2+k_7)+k_2{k}_4{k}_6{\sigma }_c({\gamma }_1+k_7)+k_5{k}_6{k}_7(k_2+\nu k_1+k_1{k}_3)-{\beta }_c{\sigma }_c{k}_1{k}_7(\eta k_4{k}_6+\psi k_5),$$ (17)

$$m_0=k_3{k}_5{k}_6(\mu +\omega +\nu )(1-{\mathcal{R}}_C).$$

The possibility of multiple endemic equilibria is investigated using the quadratic equation (16) when the threshold quantity is less than unity. We observed that the constant term $m_0$ will be negative if ${\mathcal{R}}_C<1$, while the coefficient $m_2$ will always be positive for all positive parameter values. Consequently, the following result holds.

Theorem 4

The COVID-19 model (5) has

• (i) single endemic equilibrium if $m_0<0$ or ${\mathcal{R}}_C>1$ ;
• (ii) single endemic equilibrium if $m_1<0$ , and either $m_0=0$ or $m_1^2-4m_0{m}_2=0$ ;
• (iii) double endemic equilibria if $m_0>0$ , $m_1<0$ , and $m_1^2-4m_0{m}_2>0$ ;
• (iv) no endemic equilibrium otherwise.

The first case (i) of Theorem 4 implies that the COVID-19 model (5) has a unique endemic equilibrium whenever ${\mathcal{R}}_C>1$, while case (iii) suggests the possibility of backward bifurcation when the threshold quantity ${\mathcal{R}}_C$ is below one [47]. To check for the possibility of this phenomenon, we set the discriminant $m_1^2-4m_0{m}_2=0$ and solve for the critical value of ${\mathcal{R}}_C$ (which is denoted by ${\mathcal{R}}_C^{\ast }$), to obtain the backward bifurcation point of system (5) when the threshold quantity ${\mathcal{R}}_C<1$. Thus, we have

$${\mathcal{R}}_C^{\ast }=1-\frac{m_1^2}{4m_2(\mu +{\sigma }_c)(\mu +{\delta }_c+{\gamma }_1)(\mu +{\delta }_c+{\gamma }_2)(\mu +\omega +\nu )}.$$ (18)

for which the following result is claimed.

Theorem 5

The COVID-19 model (5) will undergo a backward bifurcation when the case (iii) of Theorem 4 holds and the condition ${\mathcal{R}}_C^{\ast }<{\mathcal{R}}_C<1$ is satisfied.

Backward bifurcation of a given system tends to happen when a steady endemic equilibrium and a disease-free state coexist under certain parameter values in which the threshold quantity is less than one. Epidemiologically speaking, Theorem 5 means that whilst the epidemiological criterion of having a threshold quantity below one is required for disease elimination, it is no longer adequate. As a result, this phenomenon would make COVID-19 elimination difficult since it would be required to satisfy the threshold condition (${\mathcal{R}}_C^{\ast }<{\mathcal{R}}_C<1$). Therefore, additional intervention strategies would be required to effectively control the prevalence of the disease in the communities.

It is worth mentioning that the existence of bifurcations is dependent on certain factors such as the parameter values used in the given system. With this being said, in Fig. 3 we simulate the dynamical behavior of the COVID-19 infected population under the influence of the effective transmission rate ${\beta }_c$ and vaccine efficacy $\varepsilon$. In Fig. 3(a), we illustrate the bifurcation behavior driven by the transmission rate of COVID-19 ${\beta }_c$ in the interval $0\le {\beta }_c\le 2$. We note that for the values of $0\le {\beta }_c<\widehat{{\beta }_c}$, for which $\widehat{{\beta }_c}\approx 0.4608$, the COVID-19 free equilibrium is stable (this can be affirmed by the result in Theorem 2). It should be noted that, at ${\beta }_c=\widehat{{\beta }_c}$, the effective reproduction number ${\mathcal{R}}_C=1$, and when ${\beta }_c$ surpasses the bifurcation point $\widehat{{\beta }_c}$ (i.e. ${\beta }_c>\widehat{{\beta }_c}$), then the system undergoes a fold bifurcation such that the stable COVID-19 free equilibrium is exchanged for the endemic equilibrium.

Fig. 3.

Bifurcation diagram for COVID-19 symptomatic infectious population $\left(I_s\right)$ in terms of (a) transmission probability rate of COVID-19 ${\beta }_c\in [0,2]$; (b) vaccine efficacy $\varepsilon \in [0,1]$. Table 2 gives the parameter values that were used.

We also depict the bifurcation behavior driven by the vaccine efficacy $\varepsilon$ in the interval $0\le \varepsilon \le 1$ in Fig. 3(b). For the values of $0\le \varepsilon <\hat{\varepsilon }$, for which $\hat{\varepsilon }\approx 0.739$, the COVID-19 endemic equilibrium is asymptotically stable. However, when $\varepsilon >\hat{\varepsilon }$, the stable endemic equilibrium is replaced with a disease-free equilibrium. Overall, we can infer that to keep the system of the model at the COVID-19 free equilibrium state, the transmission rate of the disease ${\beta }_c$ must be kept below $\widehat{{\beta }_c}$, and the vaccine efficacy must satisfy $\varepsilon >\hat{\varepsilon }$. In other words, keeping the transmission rate of COVID-19 infection low with an increase in vaccine efficacy would help reduce the burden of COVID-19 in the communities.

3.2. TB only model

To obtain the TB-only sub-model, we equate the variables $V=E=I_a=I_s=I_{lc}=I_{ac}=0$ in the co-infection model (2). Thus, the model used in studying the dynamics of tuberculosis in this work is given below

$$\frac{dS}{dt}=\pi +{\kappa }_{t}R-(\mu +{\lambda }_t)S$$

$$\frac{dL}{dt}={\lambda }_{t}S-({\sigma }_t+\mu +{\gamma }_3)L$$

$$\frac{dA}{dt}={\sigma }_{t}L-(\mu +{\delta }_t+{\gamma }_4)A$$ (19)

$$\frac{dR}{dt}={\gamma }_{3}L+{\gamma }_{4}A-(\mu +{\kappa }_t)R.$$

with the initial conditions $S\left(0\right)>0,L\left(0\right)\ge 0,A\left(0\right)\ge 0,R\left(0\right)\ge 0$. The force of infection is given as ${\lambda }_t=\frac{{\beta }_{t}A}{N_t}$, where $N_t=S+L+A+R$.

3.2.1. Positivity and boundedness of solutions

The state variables of the TB-only model (19) can be demonstrated to be non-negative for the entire time $t>0$ by using the same approach described in Section 3.1.1. Further to that, the feasible region ${\Omega }_T$ for the TB-only model, defined as

$${\Omega }_T=\left\{\left(S,L,A,R\right)\in {\mathcal{R}}_{+}^4:S+L+A+R\le \frac{\pi }{\mu }\right\}.$$

is positively invariant and attracts all the solutions of model (19). As a result, all solutions with a starting point in ${\Omega }_T$ remain for all time. Thus, the TB-only model can be examined in the feasible region ${\Omega }_T$, where it is said to be mathematically and epidemiologically well-posed.

3.2.2. Existence and stability of the TB free-equilibrium (TFE)

We obtained the TB-free equilibrium (TFE), by equating the infections variables $(L,A)$ to zero. Hence, the TFE represented by ${\mathcal{E}}_{T0}$ is derived as

$${\mathcal{E}}_{T0}=(S^{+},L^{+},A^{+},R^{+})=\left(\frac{\pi }{\mu },0,0,0\right).$$ (20)

By following the same procedure as in sub- Section 3.1.2, the basic reproduction number is calculated by using the next-generation matrix operator, so that the Jacobian matrices for the new infection and the remaining transfer terms are respectively given as

$$F=\left[\begin{array}{cc}\hfill 0\hfill & \hfill {\beta }_t\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{ and }V=\left[\begin{array}{cc}\hfill \mu +{\delta }_t+{\gamma }_3\hfill & \hfill 0\hfill \\ \hfill -{\sigma }_t\hfill & \hfill \mu +{\delta }_t+{\gamma }_4\hfill \end{array}\right].$$

Using the above Jacobian matrices, the basic reproduction number of the TB-only model is obtained below by computing the highest eigenvalue of $F{V}^{-}$ given by ${\mathcal{R}}_T=\rho \left(F{V}^{-}\right)$. Thus, the basic reproduction number is given as

$${\mathcal{R}}_T=\frac{{\beta }_t{\sigma }_t}{(\mu +{\delta }_t+{\gamma }_3)(\mu +{\delta }_t+{\gamma }_4)}.$$ (21)

We should note that the threshold quantity ${\mathcal{R}}_T$ represents the average number of new TB cases that a TB-infected individual can replicate over an infectiousness period in an entirely susceptible population. The result below can be deduced by using the same method as in Section 3.1.2.

Theorem 6

The TB-free equilibrium ${\mathcal{E}}_{T0}$ , of the sub-model (19) is LAS in the region ${\Omega }_T$ if ${\mathcal{R}}_T<1$ and unstable otherwise.

Proof

The Jacobian matrix of system (19) at the TB free-equilibrium ${\mathcal{E}}_{T0}$ is computed to prove the theorem above. We defined the Jacobian matrix $\mathcal{J}\left({\mathcal{E}}_{T0}\right)$ (which is henceforth denoted by ${\mathcal{N}}_2$) as

$${\mathcal{N}}_2=\left[\begin{array}{cccc}\hfill -\mu \hfill & \hfill 0\hfill & \hfill -{\beta }_t\hfill & \hfill {\kappa }_t\hfill \\ \hfill 0\hfill & \hfill -k_8\hfill & \hfill {\beta }_t\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\sigma }_t\hfill & \hfill -k_9\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\gamma }_3\hfill & \hfill {\gamma }_4\hfill & \hfill -k_{10}\hfill \end{array}\right],$$ (22)

where $k_8={\sigma }_t+\mu +{\gamma }_3$, $k_9=\mu +{\delta }_t+{\gamma }_4$, and $k_{10}=\mu +{\kappa }_t$. In accordance with the Routh–Hurwitz criterion, the entire eigenvalues of the Jacobian-matrix ${\mathcal{N}}_2$ will be real and negative if $\text{Tr}\left({\mathcal{N}}_2\right)$ and $\text{Det}\left({\mathcal{N}}_2\right)>0$. Thus, by using the Jacobian-matrix (22), we derived the following

$$Tr\left({\mathcal{N}}_2\right)=-(\mu +k_8+k_9+k_{10})<0\text{ and }Det\left({\mathcal{N}}_2\right)=k_8{k}_9(1-{\mathcal{R}}_T)>0.$$

From the above findings, all eigenvalues of the Jacobian-matrix (22) are negative real parts if ${\mathcal{R}}_T<1$. Therefore, the TB free-equilibrium ${\mathcal{E}}_{T0}$ is LAS if ${\mathcal{R}}_T<1$ and unstable otherwise. □

3.3. Existence of the endemic equilibria

In this section, we obtain the endemic equilibria of the TB-only model (19). The TB-endemic equilibria denoted by ${\mathcal{E}}_{T1}=(S^{++},L^{++},A^{++},R^{++})$ are obtained by first defining the force of infection ${\lambda }_t^{++}$ as

$${\lambda }_t^{++}=\frac{{\beta }_t{A}^{++}}{N_t^{++}},$$ (23)

where the total human population $N_t^{++}=S^{++}+L^{++}+A^{++}+R^{++}$. The steady states solutions are obtained in the form of the force of infection ${\lambda }_t^{++}$ as

$$S^{++}=\frac{\pi k_8{k}_9{k}_{10}}{b_1{\lambda }_t^{++}+b_2},L^{++}=\frac{\pi k_9{k}_{10}{\lambda }_t^{++}}{b_1{\lambda }_t^{++}+b_2},A^{++}=\frac{\pi {\sigma }_t{k}_{10}{\lambda }_t^{++}}{b_1{\lambda }_t^{++}+b_2},R^{++}=\frac{\pi {\lambda }_t^{++}({\gamma }_3{k}_9+{\gamma }_4{\sigma }_t)}{b_1{\lambda }_t^{++}+b_2},$$ (24)

where the constants $b_1$, and $b_2$ are given as $b_1=k_8{k}_9{k}_{10}-{\kappa }_t({\gamma }_3{k}_9+{\gamma }_4{\sigma }_t)$, and $b_2=\mu k_8{k}_9{k}_{10}$. By substituting the above steady states solutions (24) into the force of infection (23), we obtain

$${\lambda }_t^{++}=\frac{k_8{k}_9{k}_{10}({\mathcal{R}}_T-1)}{k_9({\gamma }_3+k_{10})+{\sigma }_t({\gamma }_4+k_{10})}.$$ (25)

Thus, the following result is claimed.

Theorem 7

The TB-only model (19) has a unique endemic equilibrium when ${\mathcal{R}}_T>1$ .

3.4. COVID-19-TB co-infection model

The result for the co-infection model (2) is presented in this section. The region for model (2) is given by ${\Omega }_{CT}={\Omega }_C\times {\Omega }_T$, where ${\Omega }_C$ and ${\Omega }_T$ are defined in the preceding subsections. Using the same method as in Section 3.1, it can be demonstrated that all solutions of the co-infection model (2) with non-negative initial conditions will remain non-negative indefinitely. Furthermore, the region ${\Omega }_{CT}$ is positively invariant and attracts all model solutions. Thus, in the region ${\Omega }_{CT}$, the COVID-19-TB co-infection model (2) is claimed to be epidemiologically and mathematically well-posed.

3.4.1. Existence and stability of COVID-19-TB free-equilibrium

The disease-free equilibrium of the COVID-19-TB is given as

$${\mathcal{E}}_{CT0}=(S^{-},V^{-},E^{-},I_a^{-},I_s^{-},L^{-},A^{-},I_{lc}^{-},I_{ac}^{-},R^{-})=\left(\frac{\pi (\mu +\omega )}{\mu (\mu +\omega +\nu )},\frac{\pi \nu }{\mu (\mu +\omega +\nu )},0,0,0,0,0,0,0,0\right).$$ (26)

Following the computation of the reproduction numbers for the COVID-19 and TB sub-models in the preceding sections, the reproduction number of the co-infection model system (2) is given by

$${\mathcal{R}}_{CT0}=\text{max}\left\{{\mathcal{R}}_C,{\mathcal{R}}_T\right\}.$$ (27)

where ${\mathcal{R}}_C$ and ${\mathcal{R}}_T$ are the associated reproduction number for COVID-19 and TB given in (8), (21) respectively. Thus, the following result follows.

Theorem 8

The equilibrium ${\mathcal{E}}_{CT0}$ , of the COVID-19-TB co-infection model (2) is LAS in the feasible region ${\Omega }_{CT}$ when the threshold quantity ${\mathcal{R}}_{CT0}<1$ and unstable otherwise.

3.5. Investigating the invasion criterion of each disease by the other

When modeling co-infection diseases, it is critical to investigate how each disease condition impacts the dynamic behavior of the other. In other words, it will be necessary to determine whether the prevalence of one disease affects the prevalence of the other. According to [36], a community that is already infected with a disease could be invaded by a different disease under certain conditions. For example, it is widely acknowledged that TB is one of the ancient endemic diseases that is still affecting human health, but the newly COVID-19 illness has caused a shift of attention in the healthcare system due to its global prevalence. Due to this, we investigate the criterion for which COVID-19 would invade TB and vice-versa in this section. To achieve this, we compute the invasion reproduction number to determine the invasion conditions needed for the disease trade-off. The invasion reproduction number, according to [31], is the average number of secondary infections that can be generated by introducing one disease-infected individual into a community where a different endemic disease exists. Thus, to study the criterion for which COVID-19 would invade a TB endemic area, we define the COVID-19 invasion reproduction number ${\mathcal{R}}_T^C$ as the average number of secondary infections generated by introducing a COVID-19 infected individual in a TB endemic community. The COVID-19 invasion reproduction number ${\mathcal{R}}_T^C$ around the TB endemic equilibrium ${\mathcal{E}}_1^0$ is defined as

$${\mathcal{E}}_1^0=\left(S^{++},V^{\ast },0,0,0,L^{++},A^{++},0,0,R^{++}\right).$$

By following the same procedure in [11], [31], the COVID-19 invasion reproduction number is calculated by using the next generation matrix operator method around ${\mathcal{E}}_1^0$, where TB is endemic (i.e. ${\mathcal{R}}_T>1$). The COVID-19 infected compartments considered are $(E,I_a,I_s,I_{lc},I_{ac})$. The matrices for the COVID-19 infection terms $\mathcal{F}$, and the remaining transfer terms $\mathcal{V}$ are defined such that

$$\mathcal{F}-\mathcal{V}=\left[\begin{array}{c}\hfill {\lambda }_{c}S+k_1{\lambda }_{c}V\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \theta {\lambda }_{c}L\hfill \\ \hfill \theta {\lambda }_{c}A\hfill \end{array}\right]-\left[\begin{array}{c}\hfill k_{3}E\hfill \\ \hfill -{\sigma }_c{k}_{4}E-p{\gamma }_5{I}_{lc}+(k_5+\theta {\lambda }_t)I_a\hfill \\ \hfill -{\sigma }_c\psi E-m{\gamma }_6{I}_{ac}+(k_6+\theta {\lambda }_t)I_s\hfill \\ \hfill -\theta {\lambda }_t(I_a+I_s)+k_9{I}_{lc}\hfill \\ \hfill -{\sigma }_{ct}{I}_{lc}+k_{12}{I}_{ac}\hfill \end{array}\right].$$ (28)

Using the same approach as in sub- Section 3.1.2, the spectral radius of the next generation matrix associated with (28), denoted by $F{V}^{-1}$ is obtained as

$${\mathcal{R}}_T^C=\frac{{\beta }_c\left\{{\lambda }_t^{++}\left[K_0(K_1+K_2)+k_3{\theta }^2{K}_3{L}^{++}+m{k}_3{\gamma }_6{\theta }^2{K}_2{A}^{++}\right]+K_0{K}_4+K_5\right\}}{k_3{N}_t^{++}\left\{[m{\gamma }_6{\sigma }_{ct}+k_{12}(p{\gamma }_5-k_9)]{\theta }^2{\lambda }_t^{++2}+[m{\gamma }_6{\sigma }_{ct}+p{\gamma }_5{k}_6{k}_{12}-k_9{k}_{12}(k_5+k_6)]\theta {\lambda }_t^{++}-K_6\right\}}.$$ (29)

where $k_1=1-\varepsilon$, $k_3=\mu +{\sigma }_c$, $k_4=1-\psi$, $k_5=\mu +{\delta }_c+{\gamma }_1$, $k_6=\mu +{\delta }_c+{\gamma }_2$, $k_9=\mu +{\delta }_{ct}+{\sigma }_{ct}+{\gamma }_5$, $k_{12}=\mu +{\delta }_{ct}+{\gamma }_6$, $K_1=\theta {\sigma }_c{k}_4(\eta k_9{k}_{12}+m{\gamma }_6{\sigma }_{ct}-\eta m{\gamma }_6{\sigma }_{ct})$, $K_2=\theta {\sigma }_c\psi k_{12}(\eta p{\gamma }_5+k_9-p{\gamma }_5)$, $K_3=\eta p{\gamma }_5{k}_{12}+m{\gamma }_6{\sigma }_{ct}$, $K_4={\sigma }_c{k}_9{k}_{12}(\eta k_4{k}_6+\psi k_5)$, $K_5=\theta k_3(m{\gamma }_6{k}_5{k}_9{A}^{++}+k_3{k}_6{L}^{++})$, $K_6=k_5{k}_6{k}_9{k}_{12}$, and $K_0=S^{++}+k_1{V}^{++}$. We note that the force of infection is defined as ${\lambda }_t^{++}=\frac{{\beta }_t{A}^{++}}{N^{++}}$, with $N_t^{++}=S^{++}+L^{++}+A^{++}+R^{++}$. By following Theorem 3.3 of [11], we deduce the result below.

Theorem 9

The TB-endemic equilibrium, ${\mathcal{E}}_1^0$ , which exists whenever ${\mathcal{R}}_T>1$ , is locally asymptotically stable if ${\mathcal{R}}_T^C<1$ , and unstable if ${\mathcal{R}}_T^C>1$ .

Theorem 9 means that COVID-19 will be effectively controlled in a community where TB is endemic if the threshold quantity ${\mathcal{R}}_T^C$ can be reduced and retained at a value less than unity. As a result, in the COVID-19-TB model (2), the condition ${\mathcal{R}}_T^C>1$ can be used as a criterion for which COVID-19 invades a TB endemic populace. Fig. 4 provides a numerical illustration of the aforementioned result.

Fig. 4.

Numerical simulation of the co-infection model (2), showing the impact of threshold quantities on the total infected human population at different initial conditions. The parameters used are at baseline values except otherwise stated. (a) ${\beta }_t=5.9196$ (so that ${\mathcal{R}}_C>{\mathcal{R}}_T>1$, for which ${\mathcal{R}}_C=1.14,{\mathcal{R}}_T=1.09$, ${\mathcal{R}}_C^T=0.47$, ${\mathcal{R}}_T^C=1.53$); (b) ${\beta }_t=11.8392$ (so that ${\mathcal{R}}_T>{\mathcal{R}}_C>1$, for which ${\mathcal{R}}_C=1.14,{\mathcal{R}}_T=1.45$, ${\mathcal{R}}_C^T=0.93$, ${\mathcal{R}}_T^C=1.20$); (c) ${\beta }_t=17.7588$ (so that ${\mathcal{R}}_T>{\mathcal{R}}_C>1$, for which ${\mathcal{R}}_C=1.14,{\mathcal{R}}_T=2.18$, ${\mathcal{R}}_C^T=1.40$, ${\mathcal{R}}_T^C=0.73$).

In a similar way, we examine the condition for which TB would invade a COVID-19 endemic population. The invasion reproduction number ${\mathcal{R}}_C^T$ is defined as the average number of secondary infections generated by introducing a TB-infected individual in a COVID-19 endemic population. The TB invasion reproduction number ${\mathcal{R}}_C^T$ around the COVID-19 endemic equilibrium ${\mathcal{E}}_2^0$ defined as

$${\mathcal{E}}_2^0=\left(S^{\ast \ast },V^{\ast \ast },E^{\ast \ast },I_a^{\ast \ast },I_s^{\ast \ast },0,0,0,0,R^{\ast \ast }\right).$$

By following the same procedure from above, the TB invasion reproduction number is obtained around ${\mathcal{E}}_2^0$ where COVID-19 is endemic (i.e. ${\mathcal{R}}_C>1$). The TB infected compartments considered are $(L,A,I_{lc},I_{ac})$. The matrices for the COVID-19 infection terms $\mathcal{F}$, and the remaining transfer terms $\mathcal{V}$ are defined such that

$$\mathcal{F}-\mathcal{V}=\left[\begin{array}{c}\hfill {\lambda }_{t}S\hfill \\ \hfill 0\hfill \\ \hfill \theta {\lambda }_t(I_a+I_s)\hfill \\ \hfill 0\hfill \end{array}\right]-\left[\begin{array}{c}\hfill -q{\gamma }_5{I}_{lc}+k_{8}L+\theta {\lambda }_{c}L\hfill \\ \hfill -{\sigma }_{t}L-n{\gamma }_6{I}_{ac}+(k_9+\theta {\lambda }_c)A\hfill \\ \hfill -\theta {\lambda }_{c}L+k_{11}{I}_{lc}\hfill \\ \hfill -{\sigma }_{ct}{I}_{lc}-\theta {\lambda }_{c}A+k_{12}{I}_{ac}\hfill \end{array}\right].$$ (30)

It can be shown with the same approach in sub- Section 3.1.2 that the spectral radius of the next generation matrix associated with (30), denoted by $F{V}^{-1}$ is obtained as

$${\mathcal{R}}_C^T=\frac{{\beta }_t\left\{{\lambda }_c^{\ast \ast }{S}^{\ast \ast }+n\theta {\gamma }_6{\sigma }_{ct}({\lambda }_c^{\ast \ast }+k_8)(I_a^{\ast \ast }+I_s^{\ast \ast })+{\sigma }_t{k}_{12}[k_{11}{S}^{\ast \ast }+q\theta {\gamma }_5(I_a^{\ast \ast }+I_s^{\ast \ast })]\right\}}{N_c^{\ast \ast }\left\{(k_{12}-n{\gamma }_6)({\theta }^2{k}_{11}-q{\theta }^2{\gamma }_5){\lambda }_c^{\ast \ast }+[\theta k_{11}{k}_{12}(k_8+k_9)-\theta (n{\gamma }_6{k}_8{k}_{11}+q{\gamma }_5{k}_9{k}_{12})]{\lambda }_c^{\ast \ast }+K_7\right\}}.$$ (31)

where $k_8={\sigma }_t+\mu +{\gamma }_3$, $k_9=\mu +{\delta }_t+{\gamma }_4$, $k_{11}=\mu +{\delta }_{ct}+{\sigma }_{ct}+{\gamma }_5$, $k_{12}=\mu +{\delta }_{ct}+{\gamma }_6$, and $K_7=k_8{k}_9{k}_{11}{k}_{12}$. We note that the force of infection is defined as ${\lambda }_c^{\ast \ast }=\frac{{\beta }_c(\eta I_a^{\ast \ast }+I_s^{\ast \ast })}{N^{\ast \ast }}$, with $N_c^{\ast \ast }=S^{\ast \ast }+V^{\ast }+E^{\ast \ast }+I_a^{\ast \ast }+I_s^{\ast \ast }+R^{\ast \ast }$. By following Theorem 3.3 of [11], we deduce the result below.

Theorem 10

The COVID-19-endemic equilibrium, ${\mathcal{E}}_2^0$ , which exists whenever ${\mathcal{R}}_C>1$ , is locally asymptotically stable if ${\mathcal{R}}_C^T<1$ , and unstable if ${\mathcal{R}}_C^T>1$ .

Theorem 10 infers that TB would be effectively controlled in a community where COVID-19 is endemic if the threshold quantity ${\mathcal{R}}_C^T$ can be reduced and retained at a value less than one. As a result, in the COVID-19-TB model (2), the condition ${\mathcal{R}}_C^T>1$ can be used as a criterion for which TB invades a COVID-19 endemic population. The above result is illustrated numerically in Fig. 4.

4. Numerical simulations and discussion

This section will focus on using numerical simulations to examine how the two diseases and their co-infection behave dynamically in various scenarios. It should be noted that throughout this section, the total population infected with COVID-19 was defined as the sum of infectious people with the virus who are asymptomatic and symptomatic $(I_a+I_s)$, while the total population infected with TB was defined as the sum of people who have latent and active TB $(L+A)$. Similar to this, the total co-infected population is defined as $(I_{lc}+I_{ac})$. Except where otherwise noted, Table 2 contains the parameter values that were used. The COVID-19 and TB reproduction numbers of threshold quantities’ estimated values in relation to the baseline parameter values are given as ${\mathcal{R}}_C=1.14$, and ${\mathcal{R}}_T=0.36$, respectively.

4.1. The impact of threshold quantities $({\mathcal{R}}_C,{\mathcal{R}}_T)$ on disease dynamics

In this subsection, we investigated the effect of the threshold quantities on the abundance of each disease under different initial conditions, using the co-infection model (2). One of the questions that arise when modeling co-infection diseases is the competitive outcome of the co-dynamic behavior of the diseases at distinct equilibrium states. According to epidemiological theory, the disease is predicted to be under control in a population when the reproduction value is less than one, whereas the persistence of such a disease is still possible in the absence of this condition. When two different diseases are present in a community and each has the potential to spread, coexistence in such a population is expected. In other words, it is anticipated that the two diseases will coexist when they are at endemic equilibrium and have reproduction numbers greater than one [39]. However, in some circumstances where super-infection dynamics are a possibility, the replacement of a disease may be reliant on the outcomes of competition between hosts. As a result, we will investigate the co-dynamical behavior of COVID-19 and TB at various threshold quantity values. According to Theorem 2, epidemiologically it is expected that a disease would be eradicated in a community when its reproduction number value is below one. So, it is logical that when two diseases are competing in a population, one with a threshold quantity less than unity will go extinct and allows the competing disease to invade if its threshold quantity is above unity. Based on this knowledge, we only investigate the co-dynamics of the diseases at their endemic equilibrium states (i.e. ${\mathcal{R}}_C>1$, and ${\mathcal{R}}_T>1$).

As depicted in Fig. 4, the two competing diseases (COVID-19 and TB) will co-exist in the population at their endemic equilibrium for the values of reproduction numbers ${\mathcal{R}}_i>{\mathcal{R}}_j$; for $i,j=C,T$ such that $C\ne T$. Furthermore, as shown in Fig. 4(a) and Fig. 4(c), we observe that even though both diseases co-exist at their endemic state, the disease with the threshold quantities advantage will dominate but not drive the other competing disease to extinction. In other words, the disease with the highest reproduction number and invasion reproduction number would predominate in the population. For example, in Fig. 4(a), COVID-19 dominates over TB disease due to the threshold quantities advantage for which ${\mathcal{R}}_C>{\mathcal{R}}_T>1$ and ${\mathcal{R}}_C^T<1<{\mathcal{R}}_T^C$, while TB infection dominates in the population due to its threshold quantities advantage for which ${\mathcal{R}}_T>{\mathcal{R}}_C>1$ and ${\mathcal{R}}_T^C<1<{\mathcal{R}}_C^T$. Fig. 4(b) shows that TB does not predominate in the population despite having an advantage in reproduction number over COVID-19 infection. This is due to the invasion reproduction value’s lower fitness (i.e. ${\mathcal{R}}_C^T=0.93<1$). As a result, we can conclude from this result that for a disease to invade its competitor in the community, it must have a higher reproduction number than the other one and an invasion reproduction number greater than unity.

4.2. The impact of the co-infection transmission rate $\left({\beta }_{ct}\right)$ on disease burden

Numerous cases of co-infection with COVID-19 and TB have been documented in studies, along with the impact this has on people’s health. As a result, we want to simulate how the co-infection transmission rate will affect the two diseases. We want to learn more about the co-dynamic behavior, competitive outcome, and prevalence of the two competing diseases in co-infected communities. In other words, we are interested in how the two diseases’ capacity to compete in the population is impacted by the frequency of co-infection. To ensure that both diseases can be anticipated to persist in the population, we set the reproduction numbers of the diseases to be equal and at endemic states, $({\mathcal{R}}_C={\mathcal{R}}_T=1.14>1)$. Fig. 5 depicts the effect of co-infection transmission rate ${\beta }_{ct}$ on the total infected populations and their final sizes. In Fig. 5(a), we observe that with the value of ${\beta }_{ct}=0.225$, the two diseases co-exist in the population, but the prevalence of COVID-19 dominates in the population. Similarly, when the co-infection transmission rate is reduced $({\beta }_{ct}=0.113)$, the prevalence of COVID-19 still dominates as shown in Fig. 5(e). Additionally, we noticed a decline in the number of people with TB infection.

Fig. 5.

Numerical simulation of the co-infection model (2), showing the effect of the transmission rate of co-infection ${\beta }_{ct}$ on the dynamics and final sizes of the total infected human population when ${\mathcal{R}}_C={\mathcal{R}}_T=1.14>1$. (a, b) Baseline parameter values with ${\beta }_t=9.2914$, ${\beta }_{ct}=0.225$; (c, d) Baseline parameter values with ${\beta }_t=9.2914$, ${\beta }_{ct}=0.450$; (e, f) Baseline parameter values with ${\beta }_t=9.2914$, ${\beta }_{ct}=0.113$.

The effects of a rise in the fitness of co-infection transmission rate at ${\beta }_{ct}=0.450$ are depicted in Fig. 5(c). The outcome indicates that an increase in ${\beta }_{ct}$ would increase the overall population of TB infected population. Additionally, we observe a decline in the total size of COVID-19-infected people, allowing TB to predominate in the population. Overall, the results of this simulation indicate that as the rate of co-infection transmission rises, the prevalence of TB infection will rise significantly in the communities. This suggests that the potential for COVID-19 and TB co-infection would increase the risk of TB in the public. This subsection’s outcome is consistent with the report provided in [4]. The authors discussed the effects of the ongoing pandemic on TB control, diagnosis, treatment, and prevention in human communities.

4.3. The impact of each disease transmission rate $({\beta }_c,{\beta }_t)$ on disease dynamics

We simulate the effects of each disease’s transmission rate on the entire infected population in order to effectively investigate how the two diseases affect the other. In Fig. 6, we simulate how the COVID-19 transmission rate ${\beta }_c$ will affect the total COVID-19 infected population, total TB-infected population, and the total co-infected population. As shown in Fig. 6(a), the number of people who are infected with COVID-19 rises over time as its transmission rate increases. This outcome is anticipated because the parameter ${\beta }_c$ directly affects the prevalence of the reproduction number ${\mathcal{R}}_C$ of the disease. Although the number of people with TB infection decreased over time, as shown in Fig. 6(b), an increase in COVID-19 transmission increases the burden of TB infection in the populace. A similar result is observed in Fig. 6(c). We can conclude that an increase in COVID-19 prevalence adds to the burden of the two diseases and their co-infection.

Fig. 6.

Numerical simulation of the co-infection model (2), showing the effect of the transmission rate of COVID-19 ${\beta }_c$ on (a) Total COVID-19 infected population; (b) Total TB infected population; (c) Total co-infected population. Parameters are at baseline values except for ${\beta }_c=0.3499$, ${\beta }_c=0.5249$, ${\beta }_c=0.7874$.

Furthermore, we look at how the TB infection rate ${\beta }_t$ affects the total COVID-19 infected population, the total TB-infected population, and the total co-infected population, in Fig. 7. As predicted, Fig. 7(b) shows that an increase in the TB transmission rate would increase the number of TB infections. Once more, this outcome is anticipated because the parameter $bet{a}_t$ directly affects the prevalence of the disease’s reproduction number ${\mathcal{R}}_T$. A similar outcome is shown in Fig. 7(c), where an increase in the value of ${\beta }_t$ leads to an increase in the number of co-infected people. It is worth mentioning that in Fig. 7(a), we see fascinating dynamical behavior of the COVID-19 infected population under different values of the TB infection rate ${\beta }_t$. It was found that the total COVID-19-infected population did not experience any appreciable effects from the TB infection rate in the first 50 days. However, as the TB infection rate rises after the first fifty days of the simulation, we notice a slight decline in the COVID-19-infected population. This finding alerts people to the possibility that a decline in COVID-19 infections in communities could occur if TB prevalence rises in the general population. This outcome is consistent with that of Fig. 5, where it was demonstrated that a rise in the co-infection transmission rate increased the burden of TB and allowed it to predominate in the population.

Fig. 7.

Numerical simulation of the co-infection model (2), showing the effect of the transmission rate of TB ${\beta }_t$ on (a) Total COVID-19 infected population; (b) Total TB infected population; (c) Total co-infected population. Parameters are at baseline values except for ${\beta }_t=5.9196$, ${\beta }_t=8.8794$, ${\beta }_t=11.8392$.

5. Conclusions

In this manuscript, we have investigated the co-dynamics of COVID-19 and TB in the communities via a co-infection deterministic mathematical model. We also used the formulated COVID-19-TB co-infection model to examine the impact of each disease on the other under different scenarios. We first examined each sub-model and thereafter presented a generalized result for the co-infection model. The threshold quantities called the reproduction numbers were calculated and further used to establish the stability of each disease-free equilibrium. The result shows that when the associated reproduction number ${\mathcal{R}}_C$ and ${\mathcal{R}}_T$ are below one, the COVID-19 free equilibrium (CFE) and TB-free equilibrium (TFE) would be locally asymptotically stable. By using the same threshold quantity ${\mathcal{R}}_C$, we examine the effect of vaccine effectiveness and its coverage in curtailing the prevalence of COVID-19 in the communities. Our result shows that COVID-19 infection can be removed in the communities if the proportion of COVID −19 vaccinated individuals is above the critical minimum proportion of vaccinated individuals that are required to eradicate the disease. By simulating a contour plot to examine the effect of the vaccine, we deduce that a greater proportion of vaccinated people must be attained with an elevated vaccine efficacy to effectively control the disease dispersion in the communities. To enable us to know the condition for which each disease can be invaded by the other, we computed the invasion reproduction number of each disease. Our result shows that COVID-19 can be eliminated in the TB-endemic region of its threshold quantity ${\mathcal{R}}_T^C$ can be reduced and maintained below one. Thus, we deduce that COVID-19 would invade a TB- endemic population when ${\mathcal{R}}_T^C>1$. Similarly, the result shows that TB can be eliminated in a region where COVID-19 is endemic if ${\mathcal{R}}_C^T$ is brought and kept below one. As a result, we presume that when ${\mathcal{R}}_C^T>1$, TB would invade a COVID-19 endemic population.

We perform several numbers of simulations to investigate the effect of threshold quantities, and transmission rates on the co-dynamics of each disease in the communities. Our result shows that the threshold quantities (both the reproduction number and the invasion number) are the determining factors for disease invasion in the communities. Furthermore, we deduce that both diseases would co-exist in the community when their reproduction number is above one, however, the infection with the highest invasion reproduction number would dominate but not drive the other into extinction. Our numerical simulation results also show that an increase in the transmission rate of co-infection will upsurge the prevalence of TB infection in the communities. Thus, to lower the burden of TB in the human population, the implementation of control and preventative strategies should be prioritized not just for TB infection but also for COVID-19 disease. This among many other studies has enabled us to investigate the burden of COVID-19 on the abundance of TB and vice-versa. Also, we have established the conditions necessary for the invasion of each disease by using the invasion reproduction number.