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An epidemiological model is presented that provides a comprehensive description of the transmission pathways involved for recurrent tuberculosis TBwhereby cured individuals can become reinfected.

### Backward bifurcation and hysteresis in models of recurrent tuberculosis

Our main goal is to determine conditions that lead to the appearance of a backward bifurcation. This occurs when an asymptotically stable infection free equilibrium concurrently exists with a stable non-trivial equilibria even though the basic reproduction number R 0 is less than unity. The model used here incorporates progressive primary infection, exogenous reinfection, endogenous reactivation and recurrent TB as transmission mechanisms that contribute to TB progression.

Unlike other studies of TB dynamics that make use of frequency dependent transmission rates, our analysis provides exact analysr bifurcation threshold conditions without resorting to commonly applied approximations and simplifying assumptions. Furthermore, recurrent TB can independently induce backward bifurcation phenomena if it exceeds a certain threshold. In Mycobacterium tuberculosis was identified by Robert Koch [ 1 ] as the aetiological agent responsible for tuberculosis TB.

Despite all attempts to control its spread by modern medical science, TB has become one of the most widespread and serious of all infectious diseases today [ 2 ]. The disease is transmitted from one person to another in tiny microscopic droplets when a person with pulmonary TB expels bacteria into the air by either through coughing, sneezing, singing, laughing or other related activities that involve airborne pathways.

Amongst all infectious diseases, TB is one of the leading causes of death worldwide, and second only to human anslyse virus HIV [ 34 ]. Approximately a quarter of the global population harbours the Sjia bacteria and another eight to nine million new cases of tuberculosis emerge every year [ 5 ]. Extraordinarily, TB is a treatable sjia and can be prevented and cured through the use of prophylaxis and therapeutics amalyse individuals with latent and clinically active TB respectively. Such treatment should in analse be an effective strategy sia controlling the spread of TB.

Moreover, erratic treatment has led to the evolution of multi-drug resistant tuberculosis, giving rise to the fear that TB may become an untreatable disease in the not too distant future.

## Backward bifurcation and hysteresis in models of recurrent tuberculosis

A TB episode may have an exogenous analye endogenous origin. Exogenous refers to a analyze episode that results from recent exposure to some external infectious source typically, contact with an infectious person.

Endogenous designates situations where the individual is already harbouring the causing agent, which is under some healthy control by the immune system, but which destabilizes and leads to disease. TB is unusual in that it can arise via both exogenous and endogenous infection. The pathogenesis of TB is characterized by the infection either remaining dormant, often d1 a long period that may last years, or progress directly to active TB where clinical symptoms immediately manifest.

The latter process is referred to as fast primary progression. Thus, the exposure to tubercle bacilli does not necessarily result in the manifestation of clinical forms analys TB.

The other component of the population of exposed individuals develop dormant TB and may remain latently infected, possibly for the rest of their lifetime. However, destabilization of the immune system by the pathogen within the latently infected host can trigger endogenous reactivation, in smai latent bacilli are reactivated and analyae clinical Mycobacterium tuberculosis.

We are particularly interested in recurrent TB, which is defined as the emergence of a second episode of TB after the first episode has been successfully cured smiaa 9 ]. This often arises following treatment, because an individual that recovers from a first episode of the disease does not necessarily gain permanent immunity to a second. Although recurrent TB is recognized as a serious problem it receives little attention [ 10 ].

Through the use of advanced molecular fingerprinting techniques, TB recurrence has been classified into two fundamental forms of infection: The role of reinfection z1 relapse to the overall burden of tuberculosis recurrence is not well understood and this has potential public health implications [ 12 ]. It is important to note that the system adopted by the World Health Organization in recording and reporting TB cases does not differentiate between true relapse i.

This greatly affects the collection and analysis of data making it even more difficult to assess the specific role of reinfection. In [ s ] it was found that persons who had TB have a greatly increased risk of developing TB when reinfected. Moreover, the study suggests that the incidence rate of TB attributable to reinfection after successful treatment is four times higher than that of new TB [ 13 ].

A key goal of this paper is to investigate the effect of recurrent TB on the formation of backward bifurcations through detailed mathematical modelling. To our knowledge, no other study has examined this possibility. Similar to other infectious diseases, the ability for TB to persist in a population or go extinct is to a large degree governed by the basic reproduction number, R 0which is defined as the number of secondary cases smai infected individual will generate when introduced into a completely susceptible population [ 14 — 17 ].

Characteristically, when R 0 is below unity, the introduction of a few infected individuals in a susceptible population will only lead to disease die-out, as the disease is unable to reproduce itself or transmit through the population effectively. Conversely, when R 0 is above unity an epidemic may trigger and long-term disease persistence is feasible. Classical epidemic models therefore often have two intrinsic equilibria: Aanlyse endemic is sjia an equilibrium in which the number of infectives is greater than zero.

The red dotted line analysd unstable equilibria while the black solid line represent stable equilibria. However, this viewpoint has been recently challenged. When there are multiple stable equilibria coexisting simultaneously, the final equilibrium a population will reach is dependent on the initial conditions of its sub-populations. Fig 2 provides a typical bifurcation diagram that shows the key features of a backward bifurcation.

Multiple coexisting equilibria can lead to interesting dynamics when, for example R 0 varies slowly as shown in Fig 2. However, as soon as R analywe reduces below the threshold R cthe infective population immediately jumps towards zero the DFEwhile the endemic equilibrium disappears. Backward bifurcations have major implications for infectious diseases such as TB, since control programs based on reducing R 0 below unity may be ineffective given the disease might be able to easily persist indefinitely under such conditions.

Lipsitch and Murray [ 18 ] considered the phenomenon of backward bifurcation caused by exogenous reinfection in TB epidemic models with skepticism and suggested that it is unlikely to occur for realistic parameter values. However, TB epidemiology in the last two decades has greatly changed and challenges Lipsitch and Murray [ 18 ] line of argument.

More recent research supports the notion that individuals experience multiple infections throughout the course of their life as a result of reinfection, especially in high-incidence TB areas [ 19 — 21 ], all conditions which could promote the existence of backward bifurcations.

More on this controversy will be elaborated in the text. Although, there are numerous TB models that have attempted to include the recurrent TB pathway for instance see [ 22 — 27 ]none have explored its role in the formation of backward bifurcation. Their results suggest that a backward bifurcation is expected to occur when a critical value of the disease incubation period is exceeded.

However, Yang et al. The models of Feng et al. However, based on the natural history of TB, fast primary progression of TB forms an important process through which symptomatic TB emerges [ 28 ] and needs to be included. Indeed, Feng et al. The models developed by Gomes et al.

However, neither group examined analyxe recurrent TB reinfection pathways could lead to a backward bifurcation.

Instead their main objective was to study the reinfection threshold. Hence, our main aim is to study reinfection amongst recovered individuals and deduce the epidemiological implication it has especially with regards to the possible formation of a backward bifurcation. For this purpose we develop a TB model that includes fast primary progression and possible reinfection pathways. The model is represented graphically in Fig 3 and is based on the following processes.

We assume that every individual in the population belongs to one of four broad classes: Upon contact with an infective, a small proportion q of infected anlyse individuals follow the fast primary progression route i. Infected individuals who recover from the disease then move to the recovered class. The model includes three different pathways for exogenous reinfection as shown in Fig This relates to studies which have found that recovered individuals are more likely to be susceptible to future TB infection than TB-naive individuals [ 13 ].

The full model equation is given in terms of the rates of change of each of the sub-populations SEIand R namely:. The model assumes a frequency dependent incidence rate. This form of incidence is considered to be more appropriate for infections in human populations [ 33 ].

That is individuals become infected when they come into contact with infected individuals regardless of the size of the human population [ 3334 ]. Indeed the analysis becomes mathematically difficult or even intractable without making such assumptions, since otherwise the total population of the model N t might never remain constant.

Both Feng et al. In the analysis here, we make no such assumptions or simplifications and consider the total population to be xnalyse in time. Our model assumes all immigrants are susceptible. While this excludes the realistic possibility of immigration of infectives either latent individuals analsye individuals with active TBit helps untangle the conditions that result in backward bifurcation.

Previous modelling studies have already demonstrated that the immigration of individuals with latent TB or active TB are pathways that can trigger backward bifurcations [ 3536 ]. Progression to active TB is not uniform, as some infected individuals are more likely to progress to active TB than others. A number of models incorporating long and variable rate of progression have been constructed and analysed for instance see [ 233738 anlyse. Before the Feng et al. We also chose a single latent compartment over two latent compartments fast and slow TB progression for the purpose of mathematical tractability.

The increased number of reinfection pathways needed with two compartments would add a considerable degree of model complexity, and become anlyse difficult to analyse.

Moreover, the original study of [ 23 ], where the role of reinfection in inducing backward bifurcation was first identified, consisted of a single latent compartment.

### CV HAL : Isabelle BLANC Professeur at MINES ParisTech, PSL Research University

The paper by [ 18 ] that criticised Feng et al. Since our goal is to investigate how recurrent TB reinfection of recovered individuals can impact the backward bifurcation phenomenon found by Feng et al.

Table 1 provides a detailed list and description of model parameters as well as typical parameter values used here, as obtained from the literature. Following the methods in Garba et al. The equilibria of the model are found by setting the rates of all variables in the left-hand side of Eq 1 to zero. The stability of the DFE is controlled by the basic reproduction number R 0 which represents the average number of new infections generated by an infected individual when introduced into an entirely susceptible population.

R 0 may be determined using the next generation operator method see [ 47 ] as shown in S1 Appendixwhere it is found that:. It is possible to decouple the expression for R 0 to account for slow TB progression and fast primary progression. This general result has been reviewed in [ 48 ], and hence we do not prove Theorem 1 here.

Hence, this already suggests that R analyxe alone is unable to completely quantify some key dynamical features of the TB epidemic, and is in fact the first sign that a backward bifurcation might be involved. Substituting these equilibrium quantities into the force of infection Eq 2 yields.

Now the roots of the cubic equation. For mathematical convenience let. The associated backward bifurcation for the model without the reinfection pathways A and B i. Shortly we will apply center manifold to examine stability and confirm coexistence of these three equilibria see S2 Appendix. The blue solid line represent the stable equilibria while the red dotted line represent unstable equilibria.