key: cord-1029010-5ijqhmie
authors: Fatima, Bibi; Zaman, Gul
title: Co-infection of Middle Eastern respiratory syndrome coronavirus and pulmonary tuberculosis
date: 2020-08-19
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110205
sha: ee1a44a96e51cc7998a18063662f7e85b5bb78d6
doc_id: 1029010
cord_uid: 5ijqhmie

Co-infection of Middle Eastern respiratory syndrome, coronavirus and tuberculosis, TB has a complex clinical entities that has estimated worldwide; mostly, in the Middle East. Clinical studies have shown that the propagation of disease is faster in (MERS-CoV) and TB co-infection compared to those of mono-infection. Clinical reports have shown that treatment of tuberculosis (TB) increase the risk of (MERS-CoV) reactivation. In this article, we propose an epidemic model to represents the Middle East respiratory syndrome coronavirus and tuberculosis (TB) co-infection. To do this, we first find the basic reproductive number and analyze the stability of the model. The stability conditions are obtained in term of the basic reproductive number. We also study the bifurcation analysis of the model, using the central manifold theory. Sensitivity of the basic reproductive number is performed to understand the most sensitive parameters. Finally, we show the feasibility of the analytical work, by numerical simulation.

Commonly known by its abbreviation as TB, the disease tuberculosis is a contagious bacterial infection disease which can affect any area in the body. However, it most commonly affects the lungs. The obvious reason behind this is that the bacteria causing TB, mycobacterium tuberculosis, is transmitted through the air. Though the disease has been discovered much earlier, yet it remained incurable till the recent times resulting in havoc on large scale, wiping a one third of the world population [1, 2] .

A new coronavirus was identified in Saudi Arabia in 2012, known as Middle Eastern respiratory syndrome coronavirus (MERS-CoV) [3] . MERS-CoV is usually associated in the Middle East with an animal source. Besides human, MERS-CoV has been found in camels in several countries. Possibly, contact with camels causes infection in human. Human to human infection spreads through coughing. MERS-CoV has spread from ill people to others through close contact, such as taking care for or living with an infected person. Since April 2012 till date, there have been a total of 536 cases with 145 deaths, a case fatality rate of 27% with the majority being reported in the Middle East (Saudi Arabia, Jordan, and Qatar) [4, 5] . The people who were affected with MERS-CoV developed severe acute respiratory illness along with the symptoms of fever, cough and shortness of breath.

Co-infection of MERS-CoV and tuberculosis, TB was first reported by Alfaraj et al. [6] in Saudi Arabia. They investigated two such cases in which MERS-CoV and pulmonary TB co existed. Case first was related to a child of 13 years who showed symptoms of fever, loss of weight, coughing and sweating during night for almost 2 months. Whereas, the second one was related to a female aging 30, who was coughing and had shortness of breath as well as weight loss of 2 kg during few weeks of the disease. The two patient had pulmonary tuberculosis TB and positive MERS-CoV. So the reported study suggest that a positive MERS-CoV attract the pulmonary tuberc ulosis, TB ( Table 1 ) , Fig. 1 .

Mathematical modeling and analysis are used to understand and forecast the dynamics of infectious diseases, see for instance, [8] [9] [10] [11] [12] [13] [14] [15] [16] . Several researchers developed various mathematical models by taking into account different biological feasible parameters to further elaborate the co-infection of the diseases in the community [17] [18] [19] . Particularly, Akinyi et al., had formulated a mathematical model elaborating the consequences of wrong diagnosis and cure of pneumonia as malaria [20] . The result of the study suggests that the diagnosis of pneumonia and malaria can accurately be done if the basic reproductive number is less then unity. Another study has been proposed by David et al., to investigate the co-dynamics of HIV/AIDS and pneumonia [21] . Similarly, Tilahun et al. [22] investigated the co-dynamics of pneumonia and typhoid fever diseases with optimal control and cost-effective analysis. In addition to this, various authors studied the co-infection of different infectious diseases [23] [24] [25] [26] . Moreover, a number of case studies have also perform to study the transmission and co-infection of MERS-CoV, Assiri et al. [4] , Azhar et al. [5] , Alfaraj et al. [6] , Abdulhaq et al. [7] . In the reported study, the authors used some statistical tools and laboratory confirm investigation rather than the theory of dynamical system.

In 2017, a case study has been done to investigate the coinfection of MERS-CoV with other respiratory diseases [6] , which suggest that a person infected with MERS-CoV have more chances to attract other respiratory diseases, especially, tuberculosis (TB) and pneumonia. However, to the best of our knowledge there is no study for the co-dynamics of MERS-CoV and tuberculosis, TB. Therefore, in the current study, we will formulate and analysis the co-infection model for MERS-CoV and TB to fill the gap.

In this paper, we study a susceptible infectious and recovered epidemic model of tuberculosis, TB and MERS-CoV that describe the co-infection of two infectious diseases spreading through a single population as motivated from the fact that a person who has MERS-CoV is having more chances to attract TB as compared to susceptible one. We review some of the literature associated with tuberculosis, TB and MERS-CoV, model and their theoretical impact. The host population consists of four epidemiological classes: susceptible to both diseases (TB, MERS CoV), infectious to TB and susceptible to MERS-CoV, infectious to MERS-CoV and susceptible to TB, infectious to both diseases and recovered from both the dis-eases. To find the transmission potential of diseases we develop a formula for the co-infection basic reproduction number by using next generation matrix method. We use Routh Hurwitz criteria for the local stability of the co-infection model. Our model revealed that the disease-free equilibrium of the co-infection model is locally asymptotically stable when the basic reproduction number is less than one, and therefore disease dies out after some period of time. While When the basic reproduction number is greater than one, epidemiology means that the disease will prevail and persist in the population. We also investigate the model for the global stability by using geometrical approaches. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. The backward bifurcation in a disease model has important qualitative implication; small change in certain parameters can produce large change in equilibrium behavior. We find backward bifurcation by using central manifold theory. Numerical simulation of the co-infection model was carried out and the results are displayed graphically and discussed.

The paper is organized as follows: In Section 2 , we propose a mathematical model containing non linear system of ODE, and define all parameters used in the model. In Section 3 , we find bounded ness and positivity of the solution. In Section 4 , we find equilibria and the basic reproductive no. In Section 5 , we analyze the local and global stability for both the equilibria. In Section 6 , we study the possibility of bifurcation and sensitivity analysis. We perform numerical simulation to verify the results in Section 7 . In Section 8 , we give conclusion and shed some light on possible future research direction.

The model propose MERS-CoV causing to camel population C ( t ) and also human population. We divide the human population into five classes, susceptible individuals S ( t ), those individuals which are infected with TB are represented by I t ( t ), while individuals that are infected with MERS-CoV are denoted by I m ( t ). Individuals who are co-infected by TB and MERS-CoV are I tm ( t ), TB and MERS-CoV co-infectious recovered is also examined and are denoted by R tm ( t ). The total population is classified into six different compartment. Thus the proposed model takes the following form:

with initial condition:

where π represents the birth or death rate of susceptible individuals. Individuals losing their transient defence from recovered subclass and moving to susceptible class at a rate ω. The TB infected individuals can get treatment or die due to TB with a rate k 1 . Similarly MERS-CoV infected individuals either can get treatment or dies due to MERS-CoV with a rate k 2 . The subclass infected with both TB and MERS-CoV could take cure with a rate σ and avail immunity for short time, one or both the disease. Each individuals can receive tuberculosis TB with a force of infection: γ t = (I t + I tm ) N and MERS infection with a force of infection: γ m = uC(t ) u + C(t ) . u is the rate of exertion of MERS causing from camel population. The inherent death causing rate in subclasses of the human population is μ. The magnitude of co-infection sub class is made more intense from infected sub-class through catching TB disease with γ t infections force as well as TB infected group through catching MERS fever with γ m force of infection. The MERS causing camel population ( C ) grows due to contact with camel or caring for an infected camel with a rate P and increase from the release of virus from the individuals who are infected and those who are co-infected individuals through a rate of τ 1 and τ 2 . Death rate resulting due to MERS is μ c . In order to prove mathematical and biological feasibility, we show the fundamental properties with the help of the following results.

Proof. Let N ( t ) denotes the number of total population, then

Differentiation N ( t ) with regards to time and putting the expression for

If there is no death from MERS and TB then the above Eq. (2) become

The solution of Eq. (3) and calculating it at some t → ∞ , we get

If there is no discharge of virus from camel then dC(t )

The solution of (4) yields

necessarily S or I t or I m or I tm or R tm or C is equal to zero at t 1 . So from Eq. (1) , let us take the first equation

Applying alternation of constant formula, we get solution of Eq. (5) ,

Furthermore, as the variables are positive in [0, t 1 ], S ( t 1 ) > 0. Similarly I t ( t 1 ) > 0, I m ( t 1 ) > 0, I tm ( t 1 ) > 0, and C ( t 1 ) > 0, that is contradiction to our supposition. Thus t 1 = ∞ . Consequently, all the solution are positive.

We discuss qualitative study of the model (1) . For this we find equilibria of the model (1) . There are two types of equilibria: one is the disease free equilibria and the other is endemic equilibria. The first one that is DFE of the suggested model (1) is representing by

Basic reproductive number is defined as the threshold amount which analyze if an epidemic appear or the infection dies out.

That is the predicted average number of recent infections produced by a single infection directly and indirectly, while being entered in a fully susceptible population. For the reproductive number, we use the method of Driessche and Watmough [27] [28] [29] . Sup-

In Eq. (7) , the matrices F and V contain the nonlinear and linear terms respectively which are expressed as

The Jacobian matrix of F and V at disease free equilibrium F 0 , is given by

R 0 is therefore the spectral radius of next generation matrix H = F V −1 . Hence the basic reproduction number R 0 for our proposed model (1) becomes 

To show the local asymptotic stability of the proposed model (1) at disease free equilibrium F 0 , we use the following result. Theorem 3. If R 0 < 1, then the model (1) at disease free equilibrium point F 0 is locally asymptotically stable, and for R 0 > 1 it is unstable.

Proof. The Jacobian matrix of the model (1) at disease free equilibrium point F 0 is

where

By row operation reducing the matrix (9) to echelon form, so the resulting matrix is given by

The above matrix shows that all the eigenvalues are negative if ( R 0 p , R 0 t ) < 1, so it ensures the local asymptotic stability of the disease free equilibrium.

For the global stability, we have the following results.

Theorem 4. For R 0 < 1, the model (1) at disease free equilibrium F 0 is globally asymptotically stable, and unstable otherwise.

Proof. Let κ 1 = (S(t) , R tm (t)) and κ 2 = (I t (t) , I m (t) , I tm (t)) represents the number of uninfected and infected person, [30, 31] . De- 

where c 11 = − (γ m + k 1 + μ) , c 22 = (k 2 + μ) ,c 32 = (1 − p) γ 1 , c 33 = (k 1 + k 2 + σ + μ) , so matrix B and H (κ 1 , κ 2 ) are given by

From Eq. (13) , it is clear that the matrix B is M-matrix that is the off diagonal element are non-negative. But G ( κ 1 , κ 2 ) < 0.

Thus condition.2, are not satisfied, and hence by Lemma.1, the disease free equilibrium point F 0 may not be globally asymptotically stable.

To described the impact of TB on MERS, we express R 0 t in term

. B. Fatima and G. Zaman / Chaos, Solitons and Fractals 140 (2020) 110205 5 The substitution of μ in R 0 t leads to the following equation

To find out the impact of both disease on each other, we have

which shows that due to the increase in MERS cases, TB is also increases. Similarly, if the number of cases in TB increase, MERS cases also increases.

Here, we explore to study bifurcation analysis by applying the method introduced in [32] . Such theorem consists of two essential quantities the coefficient that is a and b , of the usual form indicating the dynamics of the central manifold theory. The sign of the coefficient a and b determine the nature of bifurcation. Specially if "a < 0, b > 0 , their is forward bifurcation; if "a > 0 and b > 0 the bifurcation is backward. By applying the above technique, we present the following result. . Now we will make the following change of variables: S = w 1 , I t = w 2 , I m = w 3 , I tm = w 4 , R tm = w 5 , C = w 6 . Using the vector notation " w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) t ". The TB and MERS model can be written in the form dw dt = G (x ) with G = (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 ) t as shown below

The Jacobian matrix of model (1) at disease free equilibrium point F 0 is

where

First we calculate the right eigenvector J ( E 1 ), which are denoted as

The above matrix can be written in the following form

Now, we calculate the left eigen vectors of J 1 given by C =

which gives

Solving Eq. (18) gives,

. Hence the coefficients a and b are defined

Substituting the values, which becomes

where v 0 is defined in theorem (5) . Thus, the system (1) show forward or backward bifurcation on R 0 = 1 depending on the sign of v 0 .

In this section we present analysis of sensitivity of a few parameters used in the proposed model. This will help us to know the parameters that have a significant effect on reproductive number. In order to do this, we apply the technic given in [33, 34] . Sensitivity index of basic reproductive number R 0 , is given by

where h is parameter. Since R 0 = max { R 0 t , R 0 m } , we, separately, present the sensitivity analysis of R 0 t and R 0 m as: Fig. 2 and 3 show the sensitivity analysis of basic reproductive number R 0 . These indices allow us the importance of different factors involved in disease transmission and also to measure the relative change in the reproduction number with the change in a parameter. Using these indices, we find the parameters that highly affect the reproduction number. It is clear from the sensitivity in- 

In this section, we solved the proposed deterministic coinfection model by using Runge-Kutta method of order 4th, see for detail [35] . To further understand the dynamical behavior of the proposed model we used numerical simulation to verify our analytical findings. In order to do this, we assumed some value of parameters, and some are taken from publish data given in Table 3 , to simulate the co-infection model. The choice of numerical values of the parameter are taken in such a way that would be more biologically feasible. We also assume the time interval is 100 months with initial population for susceptible S ( t ), infected with TB I t ( t ), infected with MERS-CoV I m ( t ), co-infected with TB and MERS-CoV I tm ( t ), recovered with TB and MERS R tm ( t ) and camel population C ( t ). The application of Runge-Kutta method of order 4th on the proposed model leads to the following system:

m ,

tm ,

Step 1:

Step 2: for i = 1 , 2 . . . n − 1 .

,

,

Step 3:

Once we execute the above algorithm with the help of Matlab software, we generate the graphs presented in Figs. 4 and 5 , which represent the dynamics of susceptible ( S ( t )); infected with TB ( I t ); infected with MERS-CoV ( I m ( t )); co-infectious with TB and MERS-CoV ( I tm ); recovered with TB and MERS-CoV ( R tm ); and camel population ( C ( t )). Clearly the solution trajectories of the compartmental population reaches to its equilibrium position, which ensure the stability of the proposed model. Moreover, the biological interpretation of these results states that if the value of basic reproductive number is less then one, then the susceptible population decreases, while then become stable and shows that there will be always stable susceptible population, see Fig. 4 a. The dynamics of I t ( t ), I m ( t ), I tm ( t ) and R tm ( t ) reveals that the number of these populations will be decreases and reaches to zero as shown in Figs. 4 a-5 a, while the dynamics of camel population describes that the camel population decreases up to 60 days while then reaches to its equilibrium position, which ensure that there will be always camel population as shown in Fig. 5 . Which ensure the stability of the proposed model. One of the important factor is to find the relative impact of the basic reproductive number to epidemic parameters as shown in Figs. 4 and 5 .

In this paper, we developed a mathematical model to represents the Middle East respiratory syndrome and tuberculosis (TB) co-infection. We studied different mathematical analysis including positivity, boundedness and biological feasibility of the proposed model to investigate the well posedness of the model. The important quantity i.e., the basic reproductive ratio ( R 0 ) is the key to the transmission dynamics, by which major epidemic may be prevented and prospects for the eradication of an infection. We find the basic reproductive number by using next generation matrix method and analyze the stability of equilibria. The condition of stability are obtained in term of the basic reproductive number. For local stability we used linearization and Routh-Harwitz criteria, while for the global stability we used geometrical approach. We also study the bifurcation analysis of the model using the central manifold theory. Sensitivity of the basic reproductive number is performed to understand the most sensitive parameters. On the basis of stability analysis. Finally all the theoretical results are supported with the help simulations and for easy understanding for general readers.

Our analytical results show that the susceptible S(t), infected with TB I t ( t ), infected with MERS-CoV I m ( t ), co-infected with TB and MERS-CoV I tm ( t ) and recovered with TB and MERS CoV converge to equilibrium point. Which ensure the stability of the proposed model.

We also give clinical justification of patient having positive MERS-CoV and pulmonary tuberculosis TB. In future, we are planning to develop an optimal mechanism on the basis of local dynamics and sensitivity analysis. This control strategy will help that how to eradicate the infection from the community. Work on such issue is in progress and will be reported soon in the form of a new article.

None.

Bibi Fatima: Conceptualization, Data curation, Writing -original draft. Gul Zaman: Conceptualization, Data curation, Writingoriginal draft.

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