Transmission dynamics of tuberculosis with multiple re-infections

Highlights

• TB transmission with re-infections and fast progression.

• Occurrence of backward bifurcation and its impact.

• Qualitative nature of backward bifurcation in different re-infection level.

• Global dynamics of the endemic equilibrium for R₀ > 1.

Abstract

We propose and analyze an epidemic model describing the transmission dynamics of tuberculosis (TB) with the possibilities of re-infections and fast progression of the disease. The qualitative behavior of the system is studied, covering several distinct aspects of disease transmission. The epidemiological threshold, known as the basic reproduction number, R₀, is determined using the next-generation matrix approach. It is observed that the present epidemic system may exhibit a backward bifurcation for R₀ < 1. Therefore, we may conclude that reducing R₀ to less than unity is not sufficient for eradication of tuberculosis. However, reducing R₀ to less than $R_0^{*},$ the sub-threshold obtained in the absence of recurrent TB, it is possible to eradicate the disease. We notice that a sufficient proportion of newly infected individuals developing a direct progression to the active stage can overcome the possibility of backward bifurcation. We also insight the qualitative nature of backward bifurcation with variation in re-infection level. It is found that increasing the level of re-infections makes the disease eradication more challenging. The theoretical investigations are being supplemented by numerical simulations whenever necessary.

Introduction

Tuberculosis is a chronic airborne bacterial disease caused by infection with a bacterium called Mycobacterium tuberculosis (Mtb). It is one of the major global health concerns with almost a quarter of the world’s population as its reservoir. Moreover, $10-11$ million new active cases of the disease appear each year [37]. It is the most common infection in many developing countries. The South East Asian Region (SEAR) including India, Bangladesh, Nepal etc. are highly affected by TB infection. In this region, around 4.74 million new TB cases were reported and almost 784, 000 died in 2015. Despite having modern effective clinical therapies for the last two decades, the death toll caused by TB is still extraordinarily high. Hence, to reduce TB incidence globally, more effort is needed.

TB is an airborne disease and transmitted via the respiratory route. When an active TB individual coughs or sneezes, Mtb droplets are released in the air, these droplets contain Tubercle bacilli and stay alive for nearly two hours on the air. Inhaling these infectious droplets a susceptible person may be infected, depending on the duration of contact with contagious individuals. The bacteria primarily attacks lungs (pulmonary TB), but severe stages of TB (extra-pulmonary TB) can affect other organs like the central nervous system, bones-joint etc. There are various possible outcomes after infection with Mtb. In general, after infection, the innate mechanism seizes the bacilli, and the person enters in the period of latency, an asymptotic stage where the person does not suffer from any clinical symptoms and also not infectious. Fundamentally, latent TB can be considered as an equilibrium state between host and Mtb bacilli. The duration of the dormant stage can lead from months to decades, depending upon immunity power of the infected individual. The main difference between TB and other infectious diseases is that the disease progression from latent stage to active pulmonary TB is significantly time-consuming. Moreover, a very little proportion (approximately $5-10\%$) of latently infected individuals develop active TB, whereas remaining stays in the noninfectious state for a lifetime. This pattern of developing active TB from the latent stage is known as ’endogenous reactivation’ [16]. It mainly classifies the situations when an old infection which was in an asymptotic state becomes symptomatic. Besides the possibility of ’slow progression’ to the active stage, there is evidence of ’fast progression’ in which, an individual starts manifesting symptomatic active TB within a finite time frame (1 to 3 years). There are some theoretical studies on several aspects of the transmission dynamics of TB [1], [4], [9], [13], [19], [20]. In [1], the authors proposed an eight compartmental model to study the emergence and propagation of drug-resistant. Bowng and Tewa [2] developed an SEI type model to describe the transmission procedure of TB with general contact rate. They also provide a suitable Lyapunov function to study the global dynamics of their proposed model. In [3], one strain and two strain models of TB transmission are investigated. Okuonghae [23] reported a mathematical model incorporating genetic heterogeneity in TB epidemiology and suggests to develop new treatment drug which can reduce disease transmission rates rather than disease progression rates.

The primary purpose of studying infectious disease modelling is to design better health measures to control and ultimately to eradicate the disease. In the last few decades, both mathematicians and biologists have developed many mathematical models to analyze the transmission procedure of several infectious diseases [11], [15], [18], [21], [28], [31], [32], [33]. To determine appropriate public health measures, a threshold quantity called the basic reproduction number R₀, plays a significant role [29]. Generally, reducing R₀ to less than unity is sufficient to control the disease. However, some studies [4], [12], [14], [15], [22] also show the occurrence of a subcritical bifurcation, also known as backward bifurcation. In this scenario, the disease may persist even if the basic reproduction number R₀ < 1. This kind of adverse dynamical behavior makes the disease outbreaks less predictable, and hence it is a very significant phenomenon in infectious disease modelling.

In developing countries, re-infection is a major threat, even though it may not be significant in developed countries. In this context, exogenous re-infection [5] and recurrent TB are two essential aspects to be considered. The progression towards the active stage of TB from the latent stage may be accelerated due to exogenous re-infection. Further, Recurrent TB [4], [6] is another process through which a recovered person may follow a new episode of TB after the previous infection has been successfully cured. This often happens, as recuperation from TB infection not necessarily ensure permanent immunity from the disease. It has been observed that the person who had TB are at higher risk of developing TB when reinfected. Moreover, chances of re-infection after successful therapy is four times higher than a new infection [8]. Recurrent TB is a severe issue in TB epidemiology as it causes almost $10-30\%$ of the active cases. Thus to design a mathematical model, these re-infections must be incorporated. In these context, two most influential works are described in [12], [14]. Considering a TB model with exogenous re-infection, [12] suggest that reducing R₀ to less than one may not be adequate to get rid of the disease. On the other hand, considering a data-based model of TB, [14] showed that backward bifurcation should not be a serious concern for a TB eradication process. Thus from the above literature survey, it is found that even though there exists some literature describing re-infections in TB transmission dynamics and existence of backward bifurcation, however, impacts of backward bifurcation are not much clear.

Though TB progression depends on individual immunity level, still there are very few models considering the possibilities of fast progress. In the presence of re-infection, the consequences of rapid progression still are unknown. Existing literature studied only the local dynamics of TB model in the presence of reinfection. It gives no idea about the convergence of a trajectory starting from any arbitrary initial point. Therefore, it is important to analyze the fate of an epidemic process irrespective of the number of individual infectious present in the population. This particular observation motivates us to study the global dynamics of the proposed system. Finally, in the present study, investigations are focused on the substantial effects of re-infection and fast progression of the disease on the transmission dynamics of the disease. We aim to answer: (i) How the re-infection affects the backward bifurcation? (ii) What will be qualitative difference in backward bifurcation if the recurrent TB can be terminated? (iii) What will be the sufficient condition for disease eradication? (iv) How the fast progression of the disease affects the dynamics? (v) Is it possible to establish some parametric conditions for the global stability of the exiting endemic equilibrium point?

Rest of the paper is organized in the following way: In Section 2, a compartmental TB transmission model has been proposed. The qualitative dynamical behaviors of the model in terms of positivity and boundedness of solutions, the existence of equilibria and their stability has been discussed in Section 3. Next, we investigate our model in the absence of recurrent TB in Section 4. The next section is devoted to verifying our analytical results with the help of numerical simulations. Finally, in Section 6 we have discussed the epidemiological significance of our findings with some concluding remarks.