Mathematical modelling and analysis of COVID-19 and tuberculosis transmission dynamics

In this paper, a mathematical model for assessing the impact of COVID-19 on tuberculosis disease is proposed and analysed. There are pieces of evidence that patients with Tuberculosis (TB) have more chances of developing the SARS-CoV-2 infection. The mathematical model is qualitatively and quantitatively analysed by using the theory of stability analysis. The dynamic system shows endemic equilibrium point which is stable when R0<1 and unstable when R0>1. The global stability of the endemic point is analysed by constructing the Lyapunov function. The dynamic stability also exhibits bifurcation behaviour. The optimal control theory is used to find an optimal solution to the problem in the mathematical model. The sensitivity analysis is performed to clarify the effective parameters which affect the reproduction number the most. Numerical simulation is carried out to assess the effect of various biological parameters in the dynamic of both tuberculosis and COVID-19 classes. Our simulation results show that the COVID-19 and TB infections can be mitigated by controlling the transmission rate γ.


Introduction
Recently, COVID-19 is posing a big threat to mankind around the world. Almost the entire world has been reeling under the repeated waves of COVID-19. The very first case of COVID-19 was reported in Wuhan city of China, in late 2019. This deadly virus was silently spreading to many countries till March 2020. Since 5 April 2020, every country has been suffering because of this virus. The COVID-19 patient asymptotically carried this virus and spread it to other susceptible individuals. Asymptotically and exposed persons are the main carriers of this virus. Till date there have been approximately 23,97,216 established cases and 1,62,956 deaths due to this epidemic due to environmental factors such as open drainage of sewage in residential areas, discharge of household wastes in a residential area, open water storage tanks, etc. In 2017, almost ten million new tuberculosis populations were estimated and appeared globally [6]. However, approximately three million cases have been registered in India, and it is the top in the universe [7]. But, tuberculosis-related deaths went down from 0.00056 year −1 in 2000 to 0.00032 year −1 , although they still caused approximately one and a half million death cases in India in 2016. Moreover, the prevalence of latent mycobacterium tuberculosis infection is also at a very high stage in India [8].
Both COVID-19 and tuberculosis spread in human body through a common path via upper respiratory tract [9][10][11][12][13]. There are shreds of evidence that patients with tuberculosis and latent mycobacterium tuberculosis infection have a high risk of SARS-CoV-2 infection, and afterwards, their immune system gets weakened and they receive COVID-19 infection very early [14][15][16][17][18][19]. The relationship between COVID-19 and tuberculosis is particularly common in the public health system in India. Therefore, India is one of the top contributors due to endemic tuberculosis, with the highest cases in the world [6,20]. Also, isolation of contacts and cases for the control of COVID-19 gives problems in low socioeconomic tuberculosis households [21].
Considering these circumstances, the large burden of active TB cases in India along with community transmission, localized estimation of the infection, a hot-spot of COVID-19 and birth due to the COVID-19 epidemic among tuberculosis patients in India is warranted. This would enable the allocation of resources towards implementing and developing effective public health interventions in such defected populations. With this area of study, we try to estimate the number of tuberculosis people infected due to SARS-CoV-2 and vital illnesses during the COVID-19 epidemic in Delhi, India. Markov's chance system was applied to assume the amount of severe and mild/moderate illness [22].
Two frameworks are used to approximate the amount of infectious COVID-19 cases, (a) Taking public health strategies/interventions (b) Taking public health strategies like social distancing and isolation of contacts and cases. The total contributed to the case pool was approximately 31% for workplace contact, 50% for household contact, 21% for social contact. The estimated mitigation gains by the social distancing, intervention lockdown, case and contact isolation 96% for workplace contacts, was 25% for household contacts and 70% for social contacts [23,24].
We also considered that the impact of social distancing and lockdown interventions might be shown after seven days which is half of the maximum incubation period. The basic reproduction number  0 under person health interventions was measured by multiplying the fractions of COVID-19 infected patients. The total number of confirmed cases of TB infection in 2018 for the world was obtained from the WHO report 2019, which are approximately 2.64 million cases [6,13]. The total number of COVID-19 and TB co-infected patients with acute illness was considered as 54% as per a study conducted in China [21]. The various mathematical models on endemic disease are analysed by the authors [25,26].
Both infectious diseases Tuberculosis and COVID-19 mostly effect on lungs of a human. On this aforementioned consequence, we make a mathematical co-infection epidemic model which is useful to our society.
Our study aims to formulate compartmental model to evaluate the total number of COVID-19 infected patients with the help of the progression rate from exposed COVID-19 class to infectious class to control this epidemic. This proposed compartmental model assumes a closed population that might not be considered in earlier models [12]. But, due to the lockdown, people's movements are effectively controlled in the world. This study needs to control with precautions to assess the risk of vital illness COVID-19 in tuberculosis patients. This data is captured from China, a highly affected country in the world. The prevalence of tuberculosis in general individuals might not be the same as the prevalence of tuberculosis in COVID-19 between TB cases; the study is already applied the above assumption.
This paper develops a deterministic compartmental mathematical model to incorporate the COVID-19 and tuberculosis co-infection. We extend the work of author [27] in which the authors presented the mathematical model for tuberculosis transmission in the zoonotic areas with the existence of positive equilibrium. But in our work, we incorporate the effects of COVID-19 on TB cases. This co-infection is reported in India in which TB and COVID-19 affected the people simultaneously [11].
The paper is arranged as follows. Section 2 is devoted to the mathematical model formulation. The basic preliminaries of the proposed model are given in Section 3. The mathematical analysis of the model is done in Section 4. In Section 5, the optimal model is analysed. In Section 6, the bifurcation analysis is derived. Sensitivity analysis is carried out in Section 7. The numerical simulation is performed in Section 8. The conclusion is drawn in Section 9.

Formulation of mathematical model
In order to formulate the mathematical model for transmission dynamics of TB and COVID-19 virus in a homogeneous population. We divide the entire population into five classes, namely susceptible class ( ), exposed Tuberculosis class ( ), exposed COVID-19 class ( ), infected class ( ) and recovered class ( ), respectively. Therefore, the total host population is constant and we have Thus, the mathematical model in the form of a set of differential equations is given below.
The new population entering in susceptible class is at the constant rate . Let be the rate at which the population is dying naturally. The susceptible population are getting exposed to TB infection at the rate . A fraction (1 − ) of population is exposed COVID-19 infection. Also, is fraction of people who are in contact with infected population and become infected and got active TB disease. It is also assumed that the TB population is vaccinated against COVID-19 at the constant rate . is assumed to be the death rate due to COVID-19 and is the infection contact rate, respectively. After vaccination some people are recovered at the rate and is the TB-induced death rate. The exposed population is being migrated to TB exposed class at the rate (1 − ) ( ).
Further, the TB patients are being caught COVID-19 infection at rate (1 − )(1 − ) ( ). and are the progression rates of TB exposed to infectious class and COVID-19 compartmental, respectively. The relationship between these biological parameters is explicated stated in the model (see Table 1).

Assumption
( ) The total population size + + + + = is supposed to be constant throughout the study.
( ) All the rates are supposed to be constant.
( ) The migration factor has negligible effects on tuberculosis diseases.

Properties of the model
In this section, we discuss some preliminary properties of the proposed model, which are useful in finding the existence and uniqueness of the solution.

Uniqueness of Solution for the Model
The general first order ODE in the form One will be interested in asking the following equations: To answer these, let We use the following theorem to establish the existence and uniqueness of the solution for our model.
Our interest in the domain that is ℵ ∈ [1, ]. Thus, we look for a bounded solution > 0. Then, we prove the below existence theorem.

Existence of Solution Theorem 3.2. Let be the domain defined in equation such that (2) and
(3) hold. Then, there exist a solution of the system (1) which is bounded in the domain .

Proof. Let
We have established that these twenty-five partial derivatives are continuous as well as bounded hence, theorem (2), yields that there exists a unique solution of the system (1) in the domain .

Invariant region and attractively
The dynamical transmission of the system (1) will be analyzed in the below biological feasible region, Theorem 3.3. The biological feasible region ⊂ 5 + is positively invariant for the system (1) with respect to ICs in 5 + .
Proof. We have Now, let us discuss the solution of Eq. (5) and we get Therefore, for positive value of , ( ) converges to infinity. So, the solutions of the system (1) with ICs remain in . Thus, the biological feasible region is positively invariant and attracts all the solutions in 5 + .

Positively and boundedness
Theorem 3.4. The solution of the system (1) is positively bounded for all , (0), (0)) ∈ 5 + and also define for the positive value of time .
Proof. In order to demonstrate the positive solution, it is required to verify that on every hyperplane bounding the positive orhant, the vector field point 5 + . From the system (1), we have, Thus, the solution obtained from the above set problem will remains in 5 + , and thus we have the following bounded feasible region.

The analysis
In this section, the model analysis is achieved by measuring the points of equilibrium.

Existence of steady state of the system
The critical points are obtained by taking all the five equations of the system (1)
Thus, the above equation has one positive root when √ 2 2 + 4 1 3 > 2 . So, it demonstrates that there exists at least one positive critical point.

Computation of basic reproduction number ( 0 )
By, the next generation matrix method [28], we evaluate the  0 : * = Jacobian matrices after taking the partial derivatives of * and * at 0 Thus, the  0 yields the average quantity of infected populations generated by the infected in a fully susceptible population and for our system (1). This yields by the above expression of  0 [28].

Local stability of disease free equilibrium point
Local stability of a critical point means that if we put the system somewhere nearby the point then it will move itself to the same point after some time. Now, we will discuss the local stability by computing the Jacobian matrix as follow.

Global stability of disease free equilibrium points
The global stability of disease free equilibrium points id given in the form of theorem as below: Proof. See Appendix A.

Global stability of endemic equilibrium points
In this subsection, we present the global asymptotically stability of endemic equilibrium points * ( * , * , * , , ) which is given in the form of theorem as follows:  Proof. See Appendix B.

Optimal control
We have used the optimal control theory as a powerful tool to understand the ways to reduce the COVID-19 spread the worked by devising disease intervention strategies. We consider many important issues pertaining to TB and COVID-19 infection that have not yet been adequately addressed in the literature earlier. The proposed mathematical model enables the application of optimal control theory to assess containment scenarios while maintaining the response capability of health scenarios. Since the pandemic has shown that public health is more than a treatment issue as it also affects the entire society.

Existence of optimal control model
COVID-19 is spread by a host contact with the infected population, we find the best optimal control strategies while keeping the following essential constraints in mind.
(1) 1 is the number of exposed individuals with COVID-19 infection get vaccinated, so that no more infection can spread.
where, is the set of all compartmental variables, 1 , 2 , 3 , 4 , 5 are the non-negative weight constants for the state variable , , , , , respectively. Now, we will find every value of control variable 1 , 2 , 3 from the initial value of base 0 to final value and we have where, is the smooth function for the interval [0, 1]. Thus, the Lagrangian function is related to objective function and is given by Hence, the above work gives us, * 1 = ) .

Bifurcation analysis
Bifurcation analysis in dynamical systems means a small change in bifurcation parameters leads to a sudden qualitative change in the system behaviour. Generally, at a bifurcation junction, properties of equilibrium points and periodic orbits changes. The region of stable solutions can be determined by computing stability boundaries directly as functions of the relevant system parameters. For the sake of convenience, we will take = 1 , = 2 , = 3 , = 4 , = 5 . Since, by the vector notation = ( 1 , 2 , 3 , 4 , 5 ) , the dynamical system (1) can be expressed in the form as below: where, The Jacobian matrix of ( = 1, 2, 3, 4, 5) at DFE is evaluated as: Now, consider the case when the  0 is equal to unity, then we will take = * as the bifurcation parameter. Solving for this parameter from the  0 which is equal to unity yields = * = √ (1− ) ( + + ) . Therefore, by using the [29], we will be able to find whether or not the system (12) demonstrate the backward bifurcation at basic reproduction number at unity.

Computation of characteristic values of *
It is easily observed that the Jacobian with = * of the linearized system has a simple zero characteristic value and all other characteristic values have negative real parts. Thus, center manifold theory may be applied to analyse the dynamical system (12) near = * . At the point when basic reproduction number is equal to unity, this can be demonstrated that the matrix * has a right characteristic vector that are given by =

Sensitivity analysis
In epidemiology, sensitivity analysis may be used to determine the impact of an unmeasured dose on the casual finding of a sample. The use of sensitivity analysis in mathematical modelling of infectious disease is suggested in the coronavirus disease 2019 outbreak [31][32][33][34][35][36][37]. In model (1), we use three key biological parameters, namely as , (1 − ), which significantly affect the basic reproduction number  0 [38]. The sensitivity of  0 to the change of these biological parameters is represented by the partial derivative listed below: Since all the calculated partial derivatives are non-negative, we conclude that  0 increases with the increase of any parameter given above.
To know the effect of corresponding changes to these parameters on  0 , we will calculate elasticities since elasticity is nothing but the corresponding responses to a corresponding perturbation.
From the Eq. (14), we observe that minor changes in and (1 − ) have same result on  0 . We also observe that > = (1− ) if √ ( 0 ) < 2 and in this case minor change in have a great effect on  0 compared to same corresponding changes in other biological parameters. Biologically it means that infection rate has larger effects on disease dynamics. The impact of these biological parameters on  0 is shown in Fig. 17.

Numerical simulation
In this section, we simulate the proposed mathematical model (1) by simulating various biological parameters given in Table 2. We use the default set of parameters cited in the literature and compute those values which are not found in the literature. In order to assess the impacts of various rates of TB cases and COVID-19 on the different populations, we simulate , , , , and parameters to see how these parameters affect the transmission dynamics of both diseases. In order to demonstrate this co-infection numerically via Runge-Kutta Method of fourth order. This numerical method is given as below: where, Further, The Eq. (16) is solved by applying Runge-Kutta Method of fourth order based on algorithm given in Eq. (15). Hence, the system is given as follow: ( 1 + 2 2 + 2 3 + 4 ) , +3 = +2 + 1 6 ( 1 + 2 2 + 2 3 + 4 ) , +4 = +3 + 1 6 ( 1 + 2 2 + 2 3 + 4 ) , ( 1 + 2 2 + 2 3 + 4 ) , with step size , we have Fig. 1. Effect of on .
Hence, the Eq. (17) is a numerical solution of the proposed mathematical model. Figs. 1-16 demonstrate the simulations of different biological parameters on the different populations. Further, we will discuss the effect of these state variables on different biological parameters in the preceding figures.
In Figs. 1-3 we observe the effects of , and on . Now, Fig. 1 explicit that as we increase the value of infection contact rate, the graph of susceptible class decreases. Fig. 2 demonstrates that vaccination has a significant effect on the susceptible population as the susceptible population increases as we increase the vaccination programme among the masses. Fig. 3 shows the variation of the susceptible population with respect to the death rate induced by COVID-19. It is clear that the COVID-19 cause mortality rate has demonstrated effects in susceptible cases.
Figs. 4-7 depict the effect of various populations on classes. Fig. 4 indicates that the exposed population exponentially decrease up to = 10 and it remains constant afterwards, as we increase the value of . In Fig. 5 it is seen that decays exponentially up to = 30 and remains constant afterwards. Similarly, in Figs. 6 and 7, similar trends can be seen. In Figs. 6 and 7, we see the effects of and on transmission dynamics of the COVID-19 class. Fig. 8 exhibits that the COVID-19 class curve decreases as we decrease the infection rate value from 0.1 to 0.001. Also, similar effects can be seen in Fig. 10 Figs. 10-14 show the effects of various parameters , , , and on . In Fig. 10, we observe that as we rise the utility of progression rate from exposed tuberculosis class to infectious class rate, the graph R. Singh et al.          of infectious human class increases. In Fig. 11, we observe that as we rise the utility of progression rate from exposed COVID-19 population to infectious population, the graph of infectious human class increases. In Fig. 12, we observe that as we increase the value of infection contact rate, the graph of infectious human class decreases. In Fig. 13, we observe that as we increase the value of fast progress to the active disease rate, the graph of infectious human class decreases. From Fig. 14 we observe that as we rise the value of recovery rate from infectious class rate, the graph of infectious human class decreases. In Fig. 15, we observe that as we rise the value of recovery rate from infectious class rate, the graph of recovered human class increases. Fig. 16 depicts that as we increase the value of COVID-19 causing mortality rate, the graph of recovered human class decreases.
In order to show the stability of disease-free critical point, we take

Conclusion
In this paper, a novel non-linear mathematical epidemic model for transmission dynamics of COVID-19 and tuberculosis is analysed. The essential basic control measures need to be highlighted particularly in tuberculosis cases. The tuberculosis hospitals and medication centres are required to be in early management and identification of COVID-19 in tuberculosis patients. It is also found that if we quarantine the TB-infected individual, they could be avoided the COVID-19 infection. In sensitivity analysis, we have seen that the rate of infection has significant effects on the propagation of COVID-19 transmission. We see that a small change in infective rate has a larger effect on COVID-19 transmission. It means that COVID-19 infection is spreading at a faster rate as compared to TB infection. Further, from our simulation, we computed the value of  0 = 0.521 which indicates that COVID-19 can be controlled through proper immunization and vaccination programmes. Our study further suggests that the need of the hour is to control SAR-CoV-2 viruses as it weakens the immunity which acts as a launching pad for other diseases. The use of optimal control helps us to find a better solution to deal with such a COVID-19 pandemic.
R. Singh et al.

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent
Informed consent was obtained from all individual participants included in the study.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
Data is available from the authors upon reasonable request.