Analysis of SVEIL Model of Tuberculosis Disease Spread with Imperfect Vaccination

ABSTRACT

This study proposes a SVEIL model of tuberculosis disease spread with imperfect vaccination. Susceptible individuals can receive imperfect vaccination, but over the time the vaccine efficacy will decrease. Vaccinated individuals are in vulnerable class since they still have probability to get reinfected. The proposed model includes treatment for both high-risk latent and active TB patients. In fact, after getting appropriate treatment (get recovered) the individuals still have bacteria in their body and it is classified to low-risk laten class. Dynamical behaviour of the model is analyzed to understand the local stability equilibrium. The Routh-Hurwitz criterion is used to analyze the local stability equilibrium in disease free equilibrium (DFE) point and Center Manifold theorem is used to prove the local stability of the endemic equilibrium (EE) point. The local stability equilibrium state totally depends on the effective reproduction number ℜ . If ℜ < 1 then the DFE point is locally asymtotically stable, while if ℜ > 1 the EE point is locally asymptotically stable The parameter used in this paper is based on the previous researches related to TB and the initial subpopulations are assumed. Numerical simulations show that the disease transmission rate affect the effective reproduction number, therefore it influences the stability of equilibrium points.
Although the incidence rate of tuberculosis spread is globally decreasing during the Covid-19, but not fast enough to reach the 2020 milestone of a 20% reduction between 2015 and 2020. The cumulative reduction from 2015 to 2019 was 9% (from 142 to 130 new cases per 100,000 population), including a reduction of 2.3% between 2018 and 2019 (World Health Organization, 2021). In Indonesia, based on the Ministry of Health data, 568,987 tuberculosis cases were found. However, in 2020, it was only recorded 271,750 cases, while the estimation the number of the case in that year should be around 840,000 cases (Ika, 2021). Based on the tuberculosis information online system (SITB), in Indonesia the case finding in 2019 was 67%, while in 2020, the case finding was only 41.7%. It means that there was a decrease in the case finding/detection by 25.3% (Dinas Kesehatan Daerah Istimewa Yogyakarta, 2021). According to data in 2017, it is only 24% of the tuberculosis symptomatic individuals who came to the health service. Nowadays, World Health Organization (WHO) has targeted to end the tuberculosis disease in 2030 (Ika, 2021). In general, there are three alternative solutions for TB treatments: treating the latent individual so that they do not become actively infected, treating the actively TB infected individuals, and giving vaccination to reduce the spread of tuberculosis (Sulayman et al., 2021). The elimination strategy will be dependent on preventing disease reactivation through vaccination or preventive treatment of those who are latently infected (Harris et al., 2016).
Vaccination can be done to overcome this tuberculosis problem, even the WHO has declared currently that developing a new vaccine for tuberculosis has become a top priority for health workers (Brennan et al., 2012). Depending on the TB epidemiology of a nation, the BCG vaccination is advised as a component of the national children immunization programs. 153 nations reported receiving BCG vaccines in 2018, and 113 of those had at least 90% BCG coverage (Martin et al., 2020). WHO advises giving children in leprosy and tuberculosis (TB) endemic nations a single dose of BCG, and almost 130 million infants worldwide receive the vaccine each year (Aspatwar et al., 2022). BCG was first introduced in 1921 as a licensed tuberculosis vaccine and still used today for the prevention of tuberculosis infection (Bhargava et al., 2016). BCG can provide decades of protection against tuberculosis disease, and it has been a proof that vaccines are a possible medium to protect the body from tuberculosis (Fletcher et al., 2018). Despite the facts that BCG lowers the incidence of disseminated tuberculosis in children, research on how well it works to prevent pulmonary tuberculosis in adults has been inconsistent (Mangtani et al., 2014). Because of the limitation of BCG, various potential TB vaccine candidates have been created and divided into three following groups: priming vaccines as preventive pre-exposure vaccines, boosting vaccines as preventive post-exposure vaccines targeted at adolescents and adults with LTBI and prior BCG immunization, and therapeutic vaccines to persons at higher risk of developing recurrent disease (Kaufmann et al., 2017).
In dealing with tuberculosis, medical science has an important role. Several fields of applied science and epidemiology also contributed (Sari & Rachmawati, 2019), including mathematics which plays an important role in the prevention and decision-making against the spread of tuberculosis disease, precisely through the branch of mathematics: mathematical modelling. Through differential equations, mathematical modelling can be applied to present phenomenon of change, such as epidemiology field. Therefore, the addition of vaccination variable into a mathematical model is a right decision. Vaccination in mathematical model is often represented as a linear exchange between the compartment of susceptible individuals and cured individuals (Buonomo & Lacitignola, 2011). Some vaccinations are known to prevent possible infection but not to guarantee vaccinated individuals to always be safe from getting infected and also transmitting the infection to others (Buonomo & Marca, 2019).
Mathematical models in the field of epidemiology, especially models of the spread of tuberculosis that applies vaccination have been widely used. Egonmwan and Okuonghae (2019) constructed a model of SVEIT (susceptible, vaccinated, exposed, infected, treated). In the same year, Mengistu and Witbooi (2019) also constructed the SVEIL (susceptible, vaccinated, exposed, infected, low-risk latent) model included vaccinations for newborns. Instead of being cured, individuals who have been treated are assumed to be in the low-risk latent class (L) because tuberculosis bacterium cannot be completely eliminated from the patient's body. In this model vaccination is assumed to be perfect vaccination, therefore there is no disease transmission from vaccinated class to exposed class. In addition, treatment is also applied in the exposed class (E) and infected class (I). Then, Sulayman et al. (2021) were inspired by Kar & Mondal (2012) and constructed the SVEIRE (susceptible, vaccinated, exposed, infected, recovered, exposed) model that contains imperfect vaccination. Sulayman et al. (2021) considered that the vaccinated class is like the susceptible class since it allows a decrease in vaccine efficacy over the time between these two classes. Moreover, it is a fact that treated individuals can relapse again when the patient's immunity is down (Kar & Mondal, 2012).
Nowadays, susceptible individuals are likely to be able to receive imperfect vaccination to reduce their susceptibility to tuberculosis. But over the time, the efficacy of the vaccine will also decrease. As a result, vaccinated individuals can be classified into vulnerable classes since they still have probability to get reinfected. Thus, according to Sulayman et al. (2021), it is important to understand the effects of the use of imperfect vaccination on the modeling of tuberculosis which is currently not widely studied by many scientists. Motivated by these previous studies, we proposed an SVEIL tuberculosis model. In this research, we assume that vaccinated individuals can still be infected and go to exposed class as an impact of imperfect vaccination. As mentioned above, since the TB bacteria can't be totally eliminated we proposed SVEIL, instead of SVEIRE and SVEIT model. We also consider the fast and slow progression of infection rate from susceptible class. This study also applies treatments to both latent class (E) and infected class (I), so that latent and actively infected individuals could later become low-risk latent (L) after getting appropriate treatment.
Our paper is divided in some sections. In section B, the model is formulated and the nonnegativity and the boundedness of the solution are analysed. Section C describes the equilibrium points and its conditions, and the effective reproduction numbers. The local stability is analysed and numerical simulation are also presented in this section. At the end, our conclusions and suggestions are presented in Section D. The steps of this research are depicted by Figure 1.

B. METHOD
To study the spread of tuberculosis disease with imperfect vaccination, we construct a mathematical model which is a modification of some existing models (Egonmwan & Okuonghae, 2019;Mengistu & Witbooi, 2019;Sulayman et al., 2021). We divide the total population ( ) at time into five subpopulations, namely susceptible ( ( )), vaccinated ( ( )), exposed ( ( )), actively-infected ( ( ) ), and low-risk latent ( ( )) subpopulations. Hence, we have ( ) = ( ) + ( ) + ( ) + ( ) + ( ) . In this case, the recruitment rate into the population is assumed to be . All subpopulations have the same rate of natural death , while the death caused by tuberculosis disease only happens in actively-infected subpopulation and it is denoted by . This SVEIL model describes the flow of disease spread from susceptible individuals who get vaccinated so they go to vaccinated class, and by the time the efficacy of the vaccine decreases so they might go to susceptible class again. Since the given vaccine is an imperfect vaccine, so the vaccinated individuals can still be infected and go to the exposed class with a rate of 1 . Since 1 is a transmission of tuberculosis due to imperfect vaccination, it is assumed that 1 < and thus we can write 1 = , where is a rate of of risk reduction due to vaccination (0 < < 1) and is the rate of tuberculosis transmission in general. Meanwhile, due to interaction with infected individuals in the susceptible class, the vulnerable individual will become latent (exposed) at a rate (1 − ) , or directly can go to infected class at a rate of .
The exposed class individuals can develop and become actively infected due to certain factors. In this work, we assumed that there are two treatments carried out: treatment in the exposed class as an initial treatment and consultation of new symptomatically exposed patients, and the treatment in the actively-infected class in the form of treatment for patient with active tuberculosis. Exposed individuals who have undergone the treatment process properly, will be cured and moved to low-risk latent class with a rate of γ. On the other hand, those who do not obey the recommendation on handling, they will become worse and go to actively-infected class with a rate of . Meanwhile, the treatment in the actively-infected class at a rate of allows a successful treatment with a rate (1 − ) so that they become cured (low-risk latent) and the rests with a rate of will just get better and go back to latent class. Based on assumptions above, we have the following compartment diagram that describes the interactions between classes, as shown in Figure 2. The system of ODE that describes the dynamics of tuberculosis is formulated as follows: with non-negative initial conditions given by (0) = 0 ≥ 0, (0) = 0 ≥ 0, (0) = 0 ≥ 0,

Proof.
By adding all the class equations of the model (1), then we get The inequality (2) where * will be described in the last part of this subsection. We now determine the effective reproduction number of the model (1). First, we notice that system (1)  .
Notice that * in equation (3) is the positive solution of the following quadratic equation

Local Stability Analysis of Equilibrium Points
This subsection discusses the local stability of equilibrium points, i.e. by studying the Jacobian matrix of the system (1) at each equilibrium point. Lemma 3. For model (1), the disease-free equilibrium point is locally asymptotically stable if ℜ < 1.
It is straighforward to show that Since < 0 and > 0, the well-known Center Manifold theorem, see Theorem 4.1 in (Castillo-Chavez & Song, 2004), says that a forward bifurcation occurs at ℜ = 1. This implies that if ℜ > 1 then the disease-free equilibrium point is unstable, while the endemic equilibrium point is locally asymptotically stable.

Numerical Simulation
In this subsection, we present several numerical simulations to illustrate and understand the spread of tuberculosis disease. We generate numerical simulations using the fourth-order Runge-Kutta method with step size ℎ = 0.001. The parameters for these simulations are shown in Table 1.

D. CONCLUSION AND SUGGESTIONS
This work presents a proposed basic tuberculosis SVEIL model with imperfect vaccination. As an impact of imperfect vaccination, we assume that vaccinated individuals can still be infected and go to exposed class. We also consider the fast and slow progression of infection rate from susceptible class. Our proposed model includes the treatment in both latent and actively-infected subpopulation.
The positivity and the bounded of each solution in this model has been proven. This model has two equilibrium points: a disease-free equilibrium point and an endemic equilibrium point.
The disease-free equilibrium point always exist whenever the effective reproduction number (ℜ ) is less than unity. Otherwise, a unique endemic equilibrium point exists whenever the effective reproduction number is greater than unity. Using linearization, Jacobian matrix, and the Routh-Hurwitz criterion, we showed that the disease-free equilibrium point is locally asymptotically stable when ℜ < 1 and 2 + + + (1 − ) > ( 2 + 3 ). Using the Center Manifold theorem, we found that this model undergoes a forward bifurcation when ℜ = 1 and the endemic equilibrium point is locally asymptotically stable when ℜ > 1. The numerical simulations illustrate that the effective reproduction number is directly proportional to the rate of disease transmission and reversely proportional to the rate of vaccination. It confirms the result that the effectiveness of the imperfect vaccine implemented in the model can effectively reduce the tuberculosis disease.