A MATHEMATICAL STUDY OF TUBERCULOSIS INFECTIONS USING A DETERMINISTIC MODEL IN COMPARISON WITH CONTINUOUS MARKOV CHAIN MODEL

Mathematical model of the transmission of tuberculosis infection was widely studied to capture the transient behavior of the disease transmission. In this study we model the dynamic of the disease by considering an epidemiological model called SEIIR model to capture the deterministic behavior of the disease. We also applied a continuous-time Markov chain model to take into consideration the randomness of the system. In the deterministic model, a disease-free equilibrium point and a basic reproduction number of the model are found which are mainly influenced by the contact rate of susceptible individuals with infective individuals. Other parameters such as progression rate of the latent individuals to be infectious individuals and the treatment rate of latent individuals also influence not only the deterministic model but also the stochastic sample paths. For a certain critical value of the treatment rate of latent class, deterministic and stochastic solutions show different behavior at the final time of observation. A high degree of randomness is observed in the latent and infected class (hospitalize or not-hospitalized). While in the susceptible class, effect of randomness is almost not observed. This suggests the robustness of deterministic model of susceptible class to the stochastic perturbations.


INTRODUCTION
Tuberculosis is a contagious disease caused by infection with the bacteria Mycobacterium tuberculosis (Mtb). Some of the factors that trigger infection are an unhealthy lifestyle and low human immunity since the bacteria can spread from one person to another through tiny droplets released into the air through coughing and sneezing of people with acute infection. When the human body contains bacteria, the immune system can usually prevent it from getting sick. People with this condition are called latent TB where the bacteria in their bodies are inactive and cause no symptoms. However, without treatment, latent TB can turn into active TB. If not handled immediately, active tuberculosis can be dangerous. Therefore, treatment is needed for people with latent TB and to help control the spread of TB [1], [2]. Furthermore, in order to clear the infection and prevent the development of antibiotic resistance, people with active TB have to take several types of drugs for months. Without drug treatment, Infectious tuberculosis has the potential to become a serious disease because it attacks the lungs and also affects other parts of human body, including kidneys, spine or brain. According to the WHO data, the deadliest infectious killer in the world is still caused by this TB disease. More than 4,000 people lose their lives every day due to the infection of TB. Nearly 30,000 people fall ill from this preventable and curable disease. Due to the population growth, tuberculosis remains a major global health problem in the world [1], [3]. Some diagnostics and novel therapies have been developed to bring great potential to reduce TB burden and mortality. However, limitations on the resources of TB endemic settings remain exist. A theoretical approach is needed to estimate the impact of various interventions on the outcome of interest. An intensive research will also provide an area for developing diagnostic 3 A MATHEMATICAL STUDY OF TUBERCULOSIS INFECTIONS tests and treatment regimens for TB. One of theoretical approaches that can present a useful insight is mathematical modelling. Mathematical epidemiology modelling provides useful insights by explaining the types of interventions that might maximize impact at the population level and highlighting gaps in the current knowledge that are most important for making such judgments [4][5][6][7][8][9]. Epidemiological models focus on reviewing the impact of TB control interventions by designing the transmission models to assess or understand the population-level (epidemiological) [9]. Various mathematical models have been developed to model the transmission of TB and to highlight the major contribution of TB transmission models in general such as the slow and fast dynamic of TB transmission [10][11][12], reinfection and drug-resistant strains [13-18], prevention of TB [19][20][21][22][23][24]. Most of them used compartmental model which described the transmission of infectious disease using the flow rate between compartments based on the characteristic of the infectious disease [25][26][27]. Nevertheless, other types of models were also developed to model the specific transmission dynamic of TB such as [28][29][30]. The use of mathematical approach and simulation gained more attention due to a convenient summary in predicting future outbreak as the infection progresses. It also provided better understanding regarding the transmission of the infection disease and decisions for the control of the underlying disease. In this study, we also develop a compartmental model to improve understanding of the behaviour of TB transmission based on the epidemiological understanding. We formulate the model deterministically using systems of ordinary differential equations and stochastically via continuous-time Markov chains. The deterministic model provides a framework for formulating the stochastic model with taking into consideration the stochastic perturbations. We believe that this study will provide a better understanding regarding the dynamic of TB infection to design an appropriate treatment strategy to reduce the infection probability of the pathogen.

SEIIR Deterministic Model of Tuberculosis Epidemic
In order to capture the key relevant complexities in the study of transmission dynamics of Tuberculosis (TB), SEIIR epidemiology model is used to model the dynamic of Susceptible individuals ( ), Exposed individuals ( ), Infected individuals but not hospitalized ( ), Infected individuals and hospitalized ( ℎ ), and Recovered individuals ( ). Our model follows the line of Zhang et al. [11] with some improvements. The connection and interaction between the five classes are depicted in Figure 1. The uninfected individuals move into the latent class E after getting the infection from the active TB with contact rate 1 . The latent individuals are then progressed to active TB with a fractional which stands for the proportion of which active TB is hospitalized or not. There is no directly recovery for the non-hospitalized active TB. Self-treatment for active TB produces self-recovered that is not totally eliminate the virus such that it will be moved into the latent class again with selfrecovery rate . After getting some treatments, the latent individuals can move into the recovery class with successfully treatment rate for latent individual 1 . While the hospitalized active TB 5 A MATHEMATICAL STUDY OF TUBERCULOSIS INFECTIONS may recover and move to the R class with successfully treatment rate 2 . We assume that, after recovering individuals can experience a relapse. Recurrence of tuberculosis infection can occur due to a new infection from the interaction of active TB with contact rate 2 . We assume that 1 > 2 meaning the possibility of recovered individuals for recurrence of tuberculosis infection is lower than the susceptible individuals. Natural mortality is the removal state for all compartments with constant rate μ. While mortality due to infection is employed for active TB compartments with mortality rate 1 for non-hospitalized individuals and 2 for hospitalized individuals. Using the above assumptions leads to the following non-linear system of equations for the transmission dynamic of TB: We set total population, = + + + ℎ + . By adding the equations of the system (1), we have Λ − ( + + + ℎ + ) − ( 2 + 1 ℎ ) ≤ Λ − . Hence, Therefore, the considered region for the system (1) is with positively invariant Γ meaning that the vector field points into the interior of Γ on the part of the boundary when ( + + + ℎ + ) = Λ such that

SEIIR Stochastic Model of Tuberculosis Epidemic
Let ( ), ( ), ( ), ℎ ( ), and ( ) are random variables which refer to the susceptible class, latent or exposed class, infectious not-hospitalized class, infectious hospitalized class, and recovered class, respectively. Let ∆ refers to the small-time interval such that at the time interval ( , + ∆ ) , there exists at most one event occurs. Based on the assumptions applied in the deterministic model, the events that may occur at the time interval ∆ are described as follows: 1. Event for one susceptible gets infection at the time interval ( , + ∆ ) has the transition with transition → − 1 and → + 1.

Event for one latent is moving into the infection stage and not-hospitalized at the time interval
( , + ∆ ) has the transition probability with transition → − 1 and → + 1.
4. Event for one latent becomes infectious and hospitalized at the time interval ( , + ∆ ) has the transition probability with transition → − 1 and ℎ → ℎ + 1.

Event for one infected not-hospitalized becomes latent again at the time interval ( , + ∆ )
has the transition probability with transition → − 1 and → + 1. 7. Event for one infected hospitalized and not-hospitalized is naturally die or due to the disease at the time interval ( , + ∆ ), respectively, have the transition probability 8. Event for one latent is recover from infection at the time interval ( , + ∆ ) has the transition probability 1 ∆ + (∆ ), with transition → − 1 and → + 1.

Event for one infected hospitalized is recover from infection at the time interval ( , + ∆ )
has the transition probability with transition ℎ → ℎ − 1 and → + 1.
By taking the limit of (4) as ∆ → 0 leads to the forward Kolmogorov differential equation for the bivariate process, The moment generator technique can be applied to solve the differential equation (5). However, the technique will generate a partial differential equation with five independent variables which is difficult to solve analytically. Therefore, in the next section, numerical technique will be applied to approximate the solution of (5). For the free-disease case, = 0, = 0, and ℎ = 0, we get the differential equation for the expectation of R, i.e.

Disease-free equilibrium and local stability
Setting the left-hand side of equations in (1) to zero and solving for the equilibrium values we find a disease-free equilibrium, ̂= ( Λ ,0,0,0,0).
Therefore, the disease-free equilibrium ̂ is locally asymptotically stable when the condition (9) is fulfilled.

The basic reproduction number (R0)
The ].
Since (̂) is a non-singular matrix, we have Since the basic reproduction number is the spectral radius of −1 , we get where 1 = ( + + 1 ), 2 = ( + 2 + ), and 3 = ( + 1 + 2 ). It can be observed that the value of 0 is directly proportional to the infection contact rate 1 . It is not affected by the infection contact rate 2 which comes from the recovered class meaning that the endemic condition is controlled by the infection that comes from the susceptible class not from the recovered class who has TB recurred.

Numerical results
This section deals with numerical simulations for the deterministic and stochastic model. Several control scenarios are designed to study the dynamical behavior of all individual classes. Adjusting parameters are chosen to study the behavior of the system such as the rate of the progression to infectious ( ), the treatment rate of the latent individuals ( 1 ), the treatment rate of hospitalized individuals ( 2 ), and proportion of the infectious to hospitalized ( ). One parameter is variated and other parameters are fixed (the values of parameter are shown in Table 1 Table 1 with final time = 50 months. Figure 2 shows the percentage of changes of all classes when the rate of the progression of latent to the infectious class ( ) is variated. As is increased, the number of latent is also increased. There exists a certain , says * , in which the value of ∆ reaches the maximum and then decreases as the number of susceptible also decreases (see top picture in Figure 2). The range in which the system should be controlled from high infectious is when the value of is defined in interval ( * , * * ). In that interval, the value of generates high number of latent and infected even though the infected individuals are hospitalized (see the bottom picture in Figure 2). To prevent the high number of infection class, the progression rate to the latent class should be controlled below * . It implies that an early treatment is needed to prevent the high infection rate.
In the next simulation (see Figure 3), hospitalized is variated, ∈ (0,1]. 10 We next simulate the numerical solution of the stochastic model. We investigate the sample 11 paths of the stochastic model to study the random process in the tuberculosis infection. The random 12 process will change the state according to the random variable, and then move to the different state 13 as specified by the probabilities of the stochastic matrix. Figure 5 shows how the sample path of 14 each compartment changes in time. Using the parameters in Table 1, we found that randomness in  Table 1.