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Analysis of a mathematical model in the spread of tuberculosis epidemic with vaccination and relapse effect

Tuberculosis (TB) is a contagious disease that still exists in a community. A mathematical model incorporating vaccinated treatment and the effect of relapse of the disease is studied. This research is modified with vaccinated to reduce the rate of transmission and relapse condition in a healing period. The model is analysed to verify the dynamical behaviour of the equilibriums. The nonendemic equilibrium (NE) state is determined by Castillo-Chaves theorem for the stability of global and the endemic equilibrium (EE) state using Lyapunov method. The existence of EE is determined by reproductive number (ℛ0) that derived by next generation matrix. When the ℛ0 less than one, NE state is stable. Then EE is stable if ℛ0 exceed unity. The simulation result is presented to describe the dynamic of TB spread in a long time.


Introduction
The most contagious disease that still exists in a community and belongs to the top ten causing death in the world is tuberculosis (TB). Robert Koch has identified that it is caused by Bacillus mycobacterium tuberculosis [1] [2]. The spread of TB can be via droplet infection or contaminated foods. The first organ to be infected are predominantly the lungs, but the infection can spread to any part of the body and affect everyone from infants to oldsters [1]. Based on the data reported by World Health Organization (WHO), there is an increase in TB cases in Indonesia since 2013 which makes it rank the third in the world after India and China. The cases increased by 70% from 331.703 in 2015 to 563.879 in 2018 [3]. From those cases, the control strategy is necessary to slow down the infection by vaccination and treatment [3] [4]. Bacille calmette guerin (BCG) is a vaccine that becomes a standard immunization for newborn baby [3] [5].
According to WHO, the treatment of TB infection is done by using a regimen. The duration of a regimen usually needs 6 months [6]. That treatment gives some possibilities, i.e. success, relapse or death. In describing transmission dynamics, SIR mathematical models of nonlinear system have been commonly used the schema to make it easier to understand about dynamics of the infection spread [5] [7][8][9][10][11]. Akinyemi et.al. described relapse condition where the recovered population becomes infected one [7], while other researchers described about the impact of giving vaccine and treatment as the strategies to reduce the rate of transmission [5][8] [9]. Widyaningsih et.al developed a TB transmission model of SVIR type with relapse and vaccination [10]. While, Intan et.al modeled TB transmission with SEIR type by taking into account the vaccination effect contained in susceptible group [11].
The article above explained about spreading of TB epidemic with SIR, SVIR, or SEIR models and the relapse condition was happened on SVIR type without exposed group. This article discusses about spreading of TB epidemic in SVEIR type model with vaccinated and exposed group which consists It is important to be discussed because the vaccination of TB disease is needed for host population caused it can reduce the rate transmission. And the other hand, the most population have a relapse condition in healing period. The vaccinated population is taken from susceptible group. The other part of susceptible population which infected move to the exposed group of TB disease.
The research method is descriptive with literature review about the problem about the mathematical model of TB epidemic i.e. the characteristic, cause, vaccination and the relapse effect of TB. We create the model and stability analysis of TB transmission. Then we obtain the reproductive number to determine the stability of NE and EE state. The analyze of NE state is global stability using Castillo-Chaves theorem, and the EE state using Lyapunov method because to determined EE state by bifurcation method not eligible for center manifold. The next section, we give the numerical simulation of this mathematical model.

Formulation of the model
The population is partitioned by five variables, i.e. Susceptible population (S), Vaccinated population (V), Exposed population (E), Infectious (I) and Recovered population (R). With the compartments where Λ is the recruitment of S, 1 is the level of contact with E, is fraction of the level of contact from V to E, is progression fraction to recovered stage, is vaccination coverage rate, is fraction of individuals who progress relapse in the infected stage, is progression relapse fraction to the exposed stage, is progression rate to I, is progression rate to R, is natural death rate every stage, and is infected population that natural death rate. The schema's SVEIR model is shown below,

The equilibrium state and number of basic reproduction
The mathematical model equations can be used to obtain the equilibrium state. The system is stable if the population is constant to sub population within a long time range. The NE state is.
Then, for EE state as follows Lemma [12] There exists a unique of EE state if under the condition of equations (2) as and * in the nonnegative orthant . Proof. In the eq.(5) that then we change 0 in eq.(7) become . Then, we get such that 0 is negative. We obtain . Thus, there exist a unique the EE state is proof. Next step is calculation the reproductive number . It is an important part to be calculated because it be used to determines NE and EE stability. The result of is

Stability analysis of EE state
The stability of EE state is determined by used global stability theorem as follows:

Theorem II
The EE state of system (1) is global stable if the reproductive number . Proof. In order to analyze, we use the Lyapunov function. So, we construct the Lyapunov function from the system (1)

The numerical simulation results
We illustrate the asymptotic state of the system (1). The graphic of the system (1) when are presented in the fig.2. Based on the graphic a, we can see that there is always susceptible population. The other population will disappear, while the vaccinated population needs a significantly longer time than the others. Thus, this graphic describes that the disease will disappear as in Theorem 1. The graphic  The other illustration describes about susceptible and infected population in the different conditions of (vaccination rate) and (progression fraction relapse). Fig. 3 is given different conditon of . These graphic illustrate about the change at the susceptible and infected population by different conditions. The fig.3a explained the susceptible population is constat and stability is occured. On the fig.3b can be seen that the infected population is slow down and disappeared. So, we can see the impact of vaccination from these illustrates that vaccination can reduce the rate of transmission.

Conclusion
The research paper explained about the model of TB epidemic with vaccination and relapse effect. Based on the discussion, theoretically is proven that the global stability of the model depends on reproductive number. The SVEIR model with vaccination and relapse cases can be written as the mathematic model (1). The model is analyzed globally stable. The NE state is global stable that was proven by using Castillo-Chaves theorem. Then, used Lyapunov method for the EE state that the model is global stable.
The simulation of the model can be concluded that the impact of vaccination is effective (see fig.3). The infected population have slowed down and disappeared. It means that the vaccination can reduce the rate of transmission. Then, for the relapse condition, relapse resulted susceptible and infected remain in the population as the fig.4. It can be concluded that the TB epidemic persists in population.