Introduction

In 1882 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 immunodeficiency virus (HIV) [3, 4]. Approximately a quarter of the global population harbours the TB bacteria and another eight to nine million new cases of tuberculosis emerge every year [5]. Extraordinarily, TB is a treatable disease and can be prevented and cured through the use of prophylaxis and therapeutics for individuals with latent and clinically active TB respectively. Such treatment should in theory be an effective strategy for controlling the spread of TB. However, it has failed in practice due to the inability to distribute sufficient drug treatment, usually in the form of antibiotics, to the world’s population combined with the difficulties of ensuring compliance to the required lengthy treatment program. 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. These problems, taken together, have led the World Health Organization (WHO) to formulate a post-2015 global “Stop TB Strategy” [2] to “end the global TB epidemic.”

A TB episode may have an exogenous or endogenous origin. Exogenous refers to a disease 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 for 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. The particular course of the disease depends on the host’s immune response towards the tubercle bacilli. Thus, the exposure to tubercle bacilli does not necessarily result in the manifestation of clinical forms of TB. Studies suggest that only 5-10% of individuals progress directly to the active stage after exposure to bacilli [6, 7]. 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 which latent bacilli are reactivated and cause clinical Mycobacterium tuberculosis. The lifetime risk of a latently infected individual to progress to the infectious stage is approximately 5-10% [8].

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 [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. Approximately 10-30% of all cases of TB are due to recurrent tuberculosis [10, 11], and multiple episodes are largely attributable to ineffective or poorly implemented tuberculosis control programs. 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: i) relapse of the original infecting strain, and ii) reinfection with a new strain of Mycobacterium tuberculosis. The role of reinfection and 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.e., reactivation of latent TB) and reinfection (exogenous acquisition of TB) and classifies as relapse any recurrence of TB [11]. This greatly affects the collection and analysis of data making it even more difficult to assess the specific role of reinfection.

In [13] 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, R₀, which is defined as the number of secondary cases an infected individual will generate when introduced into a completely susceptible population [14–17]. This fundamental epidemiological quantity determines whether an infection will be able, on average, to reproduce itself (R₀ > 1) in the population or not (R₀ < 1). Characteristically, when R₀ 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 R₀ is above unity an epidemic may trigger and long-term disease persistence is feasible. Classical epidemic models therefore often have two intrinsic equilibria: a disease free equilibrium (DFE) and a non-trivial endemic equilibrium. By endemic is meant an equilibrium in which the number of infectives is greater than zero. The stability of these equilibria switch at the (transcritical) bifurcation point when R₀ = 1. Fig 1 illustrates the more typical forward bifurcation by plotting the equilibrium number of infectives (I*) in a population as a function of R₀. This shows that a stable DFE in which I* = 0 exists when R₀ < 1. However, when crossing the threshold to a regime where R₀ > 1, there is a change of stability where the DFE becomes unstable while the previously unstable endemic equilibrium stabilizes.

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Fig 1. Qualitative illustration of forward bifurcation by plotting I* versus R₀.

The red dotted line represent unstable equilibria while the black solid line represent stable equilibria.

https://doi.org/10.1371/journal.pone.0194256.g001

Backward bifurcations

For decades, it has been widely accepted that the condition R₀ < 1 is an essential requirement for the elimination of a disease. However, this viewpoint has been recently challenged. Instead, the phenomenon of backward bifurcation offers a different interpretation since it shows that although R₀ < 1 and the DFE is stable, there might still be another stable endemic equilibrium coexisting simultaneously. Thus even though R₀ < 1, a population may still reside at an endemic equilibrium at which the disease persists indefinitely. 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. Note that three equilibria coexist when R₀ is in the range 0 < R_c < R₀ < 1, where R_c is a critical value, which in Fig 2 is R_c = 0.53. In this range, the “middle” equilibrium is unstable, while the other two outer equilibria (the DFE and the endemic equilibrium) are both stable. When R₀ < R_c, only the DFE equilibrium exists and is stable.

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Fig 2. Qualitative illustration of backward bifurcation at R₀ = 1.

The critical value of R₀, namely R₀ = R_c = 0.53. The red dotted line represent unstable equilibria while the black solid line represent stable equilibria.

https://doi.org/10.1371/journal.pone.0194256.g002

Multiple coexisting equilibria can lead to interesting dynamics when, for example R₀ varies slowly as shown in Fig 2. Suppose, for example, that the infective population is close to extinction I* = 0 (the DFE) and R₀ increases slowly. As soon as R₀ passes through the threshold point R₀ = 1 the number of infectives will suddenly jump from close to I* = 0 to the large endemic equilibrium (I* > 0), (see Fig 2). Similarly, when the infective population sits close to the endemic equilibrium and R₀ reduces slowly through the threshold point R₀ = 1, rather than switching to the DFE, the infectives remain close and converge to the stable endemic equilibrium. However, as soon as R₀ reduces below the threshold R_c, the infective population immediately jumps towards zero (the DFE), while the endemic equilibrium disappears.

Backward bifurcations have major implications for infectious diseases such as TB, since control programs based on reducing R₀ below unity may be ineffective given the disease might be able to easily persist indefinitely under such conditions. Certainly this non-intuitive possibility will have to be taken into account if the new WHO “Stop TB Strategy” is to succeed. 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.

Model of recurrent tuberculosis

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. Yang et al. [24] investigated the impact of multiple infections and long latency on the dynamics of recurrent tuberculosis. 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. [24] assume that the reinfection of recovered individuals is negligible, arguing that such a pathway increases non-linearity and makes the model mathematically intractable. The models of Feng et al. [23] and Kar et al. [22] were all based on the assumption that individuals pass through a long latency period before TB reactivated to clinically active TB. 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. [23] incorporated a particular recurrent TB pathway in their model; however, their simplifying assumptions hindered further exploration with regard to its role in causing a backward bifurcation. The models developed by Gomes et al. [25] and Herrera et al. [26] attempted to incorporate exogenous re-infection, partial immunity to reinfection and primary progression. However, neither group examined how 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: susceptible individuals (S), exposed individuals (E) (these are infected individuals who are not able to transmit infection), individuals with active TB (I) (who manifests symptoms and are able to pass on the infection), and recovered/treated individuals (R). Individuals have the potential to move through these four classes, as for example in the loop S → E → I → R → I or S → E → I → R → E, upon contact with the disease.

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Fig 3. Schematic diagram of the main processes involved in TB infection according to model Eq (1).

https://doi.org/10.1371/journal.pone.0194256.g003

Susceptible individuals are generated by recruitment through births and immigration at a rate Λ. Upon contact with an infective, a small proportion q of infected susceptible individuals follow the fast primary progression route (i.e they move directly to the infective class) while the rest (1 − q) move to the exposed class where they pass through a long latency period before reactivation and becoming infectious. 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 3:

1. (i) Path A = pλE, where exposed individuals, who have partial immunity against exogenous reinfection (See [23]), become reinfected and progress to active TB,
2. (ii) Path B = (1 − σ)θλR, where recovered individuals become reinfected and progress to the exposed sub-population and
3. (iii) Path C = σθλR, where recovered individuals become reinfected and progress to active TB.

In our model, the parameter p measures the degree of partial protection against TB among latently infected individuals and pλE is the exogenous reinfection incidence rate. σ measures the probability of fast progression to the infectious class after reinfection. Following the studies of [22, 23, 25, 26, 29], p = 1 implies that the body does not render protection against exogenous reinfection while 0 < p < 1 implies the body is partially immune against exogenous reinfection (i.e latent infection provides immunity against new infections) [27, 30]. Note that p > 1 would imply that an individual with latent TB infection has increased susceptibility to become newly infected, as compared to the susceptibility of the general population [27]. 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 parameter θ (0 < θ < 1) quantifies the amount of exogenous reinfection among TB recovered individuals via paths B and C while σ represent the probability of fast progression after reinfection amongst recovered individuals. θ < 1 indicates that recovered individuals have acquired some degree of partial protective immunity to TB, while θ > 1 indicates increased susceptibility.

The full model equation is given in terms of the rates of change of each of the sub-populations S, E, I, and R namely: (1)

Note that the total population at time t is given by N(t),

The model assumes a frequency dependent incidence rate. Here we have followed the convention of working with the so-called ‘force of infection’ λ defined as (2) where c represents the host-host contact. The parameter β is the probability of a contact being infectious [31] and I/N denotes the likelihood that the encounter is with an individual with active TB (assuming random mixing) [32]. 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 [33, 34]. It is important to note that although a large number of previous TB models utilised frequency dependence incidence rates, most assume that the disease-induced death (μ_d) is negligible for the purpose of mathematical simplification. 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. [23] and Porco et al. [7] resorted to this assumption to ensure that the total population N(t) remained constant, even though this was not true of their more general model formulation. In the analysis here, we make no such assumptions or simplifications and consider the total population to be varying in time.

In more detail, as Eq (1) shows, susceptible individuals move to the latently infected class E upon an effective contact with an infected individuals (I) at a rate λS. The exposed sub-population (E) increases with the infection of susceptible individuals at a rate (1 − q)λS and reinfection (recurrent TB) of recovered individuals at a rate (1 − σ)θλR. It decreases by exogenous re-infection (pλEI), and endogenous reactivation (kE) upon which exposed individuals move to the infectious class. The infected sub-population is generated by fast primary progression of TB susceptibles (qλS), exogenous reinfection amongst exposed sub-population (at rate pλE), exogenous reinfection of recovered individuals (σθλR) and endogenous reactivation (k). The sub-population is decreased by per-capita recovery due to treatment r, and by disease induced death at rate μ_d. Finally the recovered sub-population (R) is generated by the recovery of infected individuals and exogenous reinfection (at rate θλR). Note that we model natural death which affects all classes of the population at the same rate via the mortality parameter μ.

Our model assumes all immigrants are susceptible. While this excludes the realistic possibility of immigration of infectives (either latent individuals or individuals with active TB), it 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 [35, 36]. 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 [23, 37, 38]). Feng et al. [39] investigated the impact of variability in latency using arbitrary continuous distributions and found that such generalization did not result in qualitative differences in terms of the model dynamics. Before the Feng et al. [23] study, Blower et al. [37] formulated a differential equation model with two latent cohorts: one cohort consisted of those who rapidly develop TB after primary infection, while the second cohort involved individuals who develop the infection slowly through endogenous reactivation. Feng et al. [39] showed that the artificial divisions (as in Blower et al. [37]) play no role in the qualitative dynamics.

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 very 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. The paper by [18] that criticised Feng et al. [23] also had a single latent compartment. Since our goal is to investigate how recurrent TB (reinfection of recovered individuals) can impact the backward bifurcation phenomenon found by Feng et al. [23], to keep the same single latent compartment structure.

Table (1) provides a detailed list and description of model parameters as well as typical parameter values used here, as obtained from the literature.

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Table 1. Description of variables and parameters of model (1).

https://doi.org/10.1371/journal.pone.0194256.t001

Model analysis

No recurrent TB (i.e θ = 0): Quadratic P₂(λ)

We first examine the important particular parameter subset in which σ = θ = 0, that is, in the absence of recovered individuals becoming reinfected. This model has the same basic reproduction number as for the case σ, θ > 0, as σ and θ do not appear in the R₀ expression (3). For σ = θ = 0 the third degree polynomial Eq (9) collapses to the quadratic: (10) where

In the above, we see that c₀ < 0 corresponds to R₀ > 1 and vice versa. Thus, in the absence of recurrent TB we deduce c₁ < 0, R₀ < 1 and (see Theorem 1 in S1 Appendix) which indicates conditions for the existence of a backward bifurcation, based on the roots of the quadratic Eq (10).

Note that Case (iii) of Theorem 1 in S1 Appendix stipulates the condition that which means there are two real positive endemic equilibria as required for a backward bifurcation to appear. In fact △ = 0 provides the critical point for the backward bifurcation where the two positive endemic equilibria collide and annihilate each other leaving the DFE as the only equilibria.

By setting △ = 0 we can determine the critical value of the transmission coefficient denoted by β_c. For mathematical convenience let

Now the discriminant △(β) may be expressed in terms of β. Let β_c be the critical value of β for which the discriminant equals zero i.e.,

Some algebraic rearrangement gives (11)

The critical value of basic reproduction number denoted by R_c is obtained by replacing parameter β in R₀ with β_c which yields (12) where the right-hand term in large brackets is just β_c.

In fact R_c defines a sub-threshold domain of bistable equilibria of the model system (1) in the sense that within the region R_c < R₀ < 1 the model Eq (1) has two positive endemic equilibria simultaneously existing with a stable disease free equilibrium. Thus, the backward bifurcation for Eq (1) occurs for values of the basic reproduction number R₀ that lie between R_c < R₀ < 1. The associated backward bifurcation for the model without the reinfection pathways A and B (i.e, σ = θ = 0) shown in Fig 4 is obtained by plotting λ as a function of β. Fig 4 shows that model (1) has a disease free equilibrium which corresponds to λ = 0 and two non-trivial endemic equilibria which, according to numerical simulations, one is locally asymptotically stable (LAS) and the other is unstable (saddle). Shortly we will apply center manifold to examine stability and confirm coexistence of these three equilibria (see S2 Appendix). We thus find coexistence of two positive equilibria when R₀ < 1, hence confirming that the model exhibits the phenomenon of backward bifurcation for R_c < R₀ < 1.

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Fig 4. Illustration of backward bifurcation when there is no recurrent TB (i.e. θ = 0).

Parameters are defined in Table (1) except p = 0.09 > p_c = 0.0658, k = 0.0002, q = 0.05, c = 60, β ∈ {0.4, 0.7} and β_c = 0.5099 corresponding to R_c = 0.8852. The blue solid line represent the stable equilibria while the red dotted line represent unstable equilibria.

https://doi.org/10.1371/journal.pone.0194256.g004

It can be seen from Eqs (11) and (12), that when exogenous reinfection parameter p increases, R_c decreases, and vice-versa. Hence, we can deduce from expression (12) that R_c is inversely proportional to the level of exogenous reinfection p. This observation is confirmed by Fig 5 which illustrates the effect of increasing exogenous reinfection p on R_c. That is, with low values of exogenous reinfection the critical value R_c is high implying that the extent of the backward bifurcation regime becomes smaller as R_c becomes closer to unity (R₀ = 1). The threshold implies that TB can be eliminated from the community if the basic reproduction number is maintained below R_c (i.e R₀ < R_c). More formally we have the following lemma:

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Fig 5. A plot of the critical value of R_c as a function of the level of exogenous reinfection (p).

NBB and BB respectively denote no backward bifurcation and backward bifurcation regions. In the region denoted by NBB, the level of exogenous reinfection is too low to induce backward bifurcation while in region denoted by BB the level of exogenous reinfection is sufficient to cause multiple equilibria. Parameters used remain as shown in Table (1).

https://doi.org/10.1371/journal.pone.0194256.g005

Lemma 1 For model Eq (1), when σ = θ = 0,

1. (i) If R₀ > 1 the model has one positive endemic equilibrium point,
2. (ii) If R_c < R₀ < 1 the model has two positive endemic equilibria,
3. (iii) If R₀ < R_c has only disease free equilibrium.

Recurrent TB: Model with all reinfection pathways (A, B and C) θ > 0: Cubic P₁(λ)

We now return to the fully general model with all parameters p, σ, θ positive. Recall that the sign of the roots of the cubic polynomial (Eq (4)) P₁(λ), tell us the signs of the equilibrium populations for the number of infected individuals (via Eq (2)). Observe that the coefficients in the cubic polynomial a₃, a₂, a₁ and a₀ (see Eq (5)) are all real numbers. For any non-negative model parameters, a₃ is always positive while a₂, a₁ and a₀ can be either positive, zero or negative depending on β₀, β₁ and β_R, respectively (see Eqs (6)–(8)). A comprehensive analysis of the roots of the cubic (9) may be carried out using Descartes rule of signs as described in S4 Appendix. A simpler more intuitive approach is to examine the roots at the transcritical bifurcation point R₀ = 1. Such an analysis gives an understanding of the type of bifurcation that is likely to occur in the vicinity of R₀ = 1, and provides conclusions that coincide with the more detailed analysis based on Descarte’s rule of signs (S4 Appendix). Conveniently when R₀ = 1 the cubic polynomial (9) reduces to the quadratic equation

Keeping in mind Eq (5), a simple study of the roots shows that if either a₂ < 0 and a₁ < 0 (i.e., if β > β₀ and β > β₁) or if β₁ < β < β₀, the quadratic equation has one positive endemic equilibria. As seen in Fig 6, this is the signature of a backward bifurcation. Namely, when R₀ is slightly below unity, the model Eq (1) has two positive endemic equilibria but only one when R₀ ≥ 1. Furthermore, if a₂ > 0 and a₁ > 0 (i.e β < β₀ and β < β₁) then at the point R₀ = 1, the model has no positive endemic equilibria. This characteristic is indicative of forward bifurcation as seen in Fig 7. However, if R₀ is increased slightly above unity then model Eq (1) has one positive endemic equilibrium point. It can be shown that if β₀ < β < β₁, then the reduced equation has two positive real roots, indicating that the model Eq (1) exhibits hysteresis (see Figs 8 and 9), as will be discussed in detail shortly.

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Fig 6. Infected population, I* at equilibrium plotted as a function of the parameter β.

Illustrates backward bifurcation when recurrent TB parameters are included. Parameters used include; p = 0.09 > p_c = 0.0639., σ = 0.05, θ = 0.3, μ = 0.0167, μ_d = 0.1, r = 2, q = 0.05, Λ = 100 c = 60, k = 0.0002 and β ∈ {0.3, 0.8}. Note that the blue solid line denote stable equilibria while dotted red line denote unstable equilibria. To aid visualisation, bifurcation structure is plotted with semi-log axes.

https://doi.org/10.1371/journal.pone.0194256.g006

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Fig 7. Infected population, I* at equilibrium plotted as a function of the parameter β.

Shows forward bifurcation when recurrent TB parameters are included. Parameters used are same as in Fig 6 except p = 0.06 < p_c = 0.0639. To aid visualisation, bifurcation structure is plotted with semi-log axes.

https://doi.org/10.1371/journal.pone.0194256.g007

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Fig 8. Shows forward bifurcation with hysteresis where the multiple equilibria are strictly to the right of R₀ = 1.

Parameters: μ = 0.0167, μ_d = 0.1, k = 0.0002, θ = 0.5, r = 2, q = 0.05, σ = 0.2, c = 60, Λ = 100, β ∈ (0.2, 0.8) and p = 0.057 < p_c = 0.0639. Note that the blue solid line denote stable equilibria while dotted red line denote unstable equilibria. To aid visualisation, the bifurcation structure is plotted with semi-log axes.

https://doi.org/10.1371/journal.pone.0194256.g008

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Fig 9. Shows forward bifurcation with hysteresis where there are multiple equilibria to the left and to the right of R₀ = 1.

Parameters used are similar to Fig 8 except p = 0.058 < p_c = 0.0639. For a clear view, the bifurcation structure is plotted with semi-log axes.

https://doi.org/10.1371/journal.pone.0194256.g009

This discussion based on the point R₀ = 1 is summarized in Lemma 2.

Lemma 2 At the point R₀ = 1, where β = β_R the model Eq (1) has:

1. (i) Two positive endemic equilibria if β₀ < β_R < β₁, which signals hysteresis,
2. (ii) One positive endemic equilibria if β_R > β₀ and β_R > β₁, or if β₁ < β_R < β₀, either of which signals backward bifurcation,
3. (iii) No positive endemic equilibria if β_R < β₀ and β_R < β₁, which signals forward bifurcation.

Hysteresis

For the usual forward bifurcation (eg., Fig 4), a model has two locally stable branches at the transcritical point R₀ = 1: i) an infection free equilibrium that is locally asymptotically stable when R₀ < 1 and ii) an endemic equilibrium which is stable for R₀ > 1. However, the scenario where there is only one endemic equilibria when R₀ > 1 may not always be the case. For example, Reluga et al. [52] noted in their study of epidemic models with structured immunity that it is possible that more than one endemic equilibria may coexist even though the basic reproduction number R₀ > 1. This leads to an unusual phenomenon of forward bifurcation with hysteresis, which can be triggered in our TB model when reinfection is taken into account. Thus Eq (1) exhibits a hysteresis effect where multiple endemic equilibria coexist when R₀ > 1, as shown in Fig 8. The two ‘outer’ equilibria are stable while the interior equilibrium (red) is unstable. Table 2 clarifies that three endemic equilibria coexist for R₀ > 1 if β₀ < β < β₁. For some parameter regimes with hysteresis, the endemic equilibria may also be found in the region where R₀ < 1 and where disease is not expected, as shown in Fig 9. In this scenario, (similar to a backward bifurcation) TB persists for R₀ < 1 even though the bifurcation at R₀ = 1 is a forward bifurcation. We know of no other epidemic modelling study reporting this feature. Hysteresis loops to the left of the epidemic threshold R₀ = 1 have epidemiological implication in that, although there is no backward bifurcation (as ascertained by the fact that the hysteresis effect occurs when (p < p_c)) policy makers need to reduce R₀ below another threshold R_c to eradicate TB. That is, reducing R₀ below unity will be necessary but not sufficient in eradicating TB within the community.

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Table 2. Generalization of the model equilibria of Eq (1).

https://doi.org/10.1371/journal.pone.0194256.t002

No reinfection path A, (i.e p = 0, θ > 0)

The case when there is no reinfection among recovered individuals but reinfection of exposed individuals (i.e p > 0 and θ = 0) was studied by Kar et al. [22], where an exogenous reinfection threshold was established. In this section we focus on the scenario when there is no exogenous reinfection among exposed individuals (i.e p = 0, reinfection path A is omitted). Our goal is to demonstrate that in such a situation recurrent TB due to reinfection of recovered individuals only (θ > 0) can induce the phenomenon of backward bifurcation.

In model system (1) setting p = 0 we obtain the following subsystem: (18)

The model system (18) equilibrium points (S*, E*, I*, R*) can be expressed in terms of force of infection λ*, obtained by solving the following quadratic equation; (19)

Note that subsystem (18) has the same basic reproduction number as model (1).

It is easy to see that R₀ = 1 implies b₀ = 0. Thus, the following equality is satisfied; (20)

This combined with the condition b₁ < 0, which is necessary for backward bifurcation to occur, and with some algebraic manipulation leads to the criterion for backward bifurcation: (21)

Thus, we arrive at the following Theorem;

Theorem 2 The model subsystem (18) at R₀ = 1 has:

1. (i) Backward bifurcation if θ > θ_c
2. (ii) Forward bifurcation if θ < θ_c

Furthermore, a similar condition to (21) can be obtained by setting p = 0 in the center manifold results of S2 Appendix, hence corroborating Theorem 2. Thus, if θ > θ_c = 5 model (18) will exhibit backward bifurcation. However, if θ < θ_c = 5 model (18) does not exhibit backward bifurcation (i.e has only forward bifurcation) for the parameters given in Fig 10 caption.

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Fig 10. Shows forward bifurcation in a scenario where reinfection path A is omitted (i.e no reinfection of exposed individuals, p = 0).

Parameters used include: μ = 0.0167, μ_d = 0.1, k = 0.0002, r = 2, q = 0.05, σ = 0.2, c = 60, β ∈ (0.5, 0.65) and θ = 4 < θ_c = 5. Semi-logarithmic scales are used to aid visualisation.

https://doi.org/10.1371/journal.pone.0194256.g010

With the existing evidence that recovered individuals have increased susceptibility to reinfection, four times higher than that of new TB [13, 27], Theorem 2 suggests that the contribution of recurrent TB in the general TB burden can significantly alter TB dynamics especially in a scenario where recurrent TB independently triggers the phenomenon of backward bifurcation.

It is important to note that although previous TB models have attempted to incorporate recurrent TB pathways they do not investigate whether recurrent TB alone can trigger bi-stability, but rather concentrate on backward bifurcation caused by exogenous reinfection of exposed individuals (i.e p > 0). Selecting value of parameter θ from the estimated interval, i.e θ ∈ 3.87[1.61, 7.79] [41], we verify the threshold given in Theorem 2. Fig 10 is a bifurcation diagram corresponding to the case θ < θ_c and indicates a forward bifurcation, as predicted by Theorem 2 Case (ii). Similarly, Figs 11 and 12 indicate backward bifurcation since θ > θ_c as predicted by Case (i).

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Fig 11. Shows backward bifurcation structure when reinfection path A is excluded.

Parameters used are the same as in Fig 10 except θ = 6 > θ_c = 5. The solid blue line represent stable equilibrium while the dotted red line represent unstable equilibrium. Semi-logarithmic scales are used to aid visualisation.

https://doi.org/10.1371/journal.pone.0194256.g011

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Fig 12. Shows backward bifurcation structure when reinfection path A is excluded.

Parameters used are the same as in Fig 10 except θ = 7.5 > θ_c = 5. The solid blue line represent stable equilibrium while the dotted red line represent unstable equilibrium. Semi-logarithmic scales are used to aid visualisation.

https://doi.org/10.1371/journal.pone.0194256.g012

Impact of incorporating recurrent TB parameters

Recall that recurrent TB due to reinfection is denoted by reinfection pathways B and C in Fig 3. Given that reinfection path A does induce backward bifurcation when p > p_c it is of interest to investigate how the additional recurrent reinfection paths B and C can impact the backward bifurcation. In Fig 13 the endemic equilibrium I* is plotted as a function of the basic reproduction number R₀ for scenarios with different recurrent TB contributions. The figure shows that recurrent TB due to reinfection among treated/recovered individuals shifts the backward bifurcation to the left as well as widens the gap between bifurcation curves. In summary we report the following important implications that arise from our analysis. Raising the intensity of recurrent TB:

1. (i) Widens the gap between the bifurcation curves,
2. (ii) Reduces the critical value R_c,
3. (iii) Increases the number of infected individuals (TB burden) (see Fig 14),
4. (iv) Reduces the reinfection threshold p_c that induces the phenomenon of backward bifurcation.

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Fig 13. Shows the effect of recurrent TB on backward bifurcation due to incorporation of reinfection pathways B and C.

Parameter values: μ = 0.0167, μ_d = 0.1, k = 0.0002, r = 2, q = 0.05, σ = 0.05, p = 0.09, c = 60, Λ = 100, β ∈ [0.45, 0.7]. With no recurrent TB (θ = 0), p = 0.09 > p_c = 0.0658 while with recurrent TB (θ = 0.3). Parameters used are the same but p_c is altered, i.e p = 0.09 > p_c = 0.0647. The blue solid line represent stable equilibria while the dotted red line represent unstable equilibria. Semi-log scales are used for a clear view.

https://doi.org/10.1371/journal.pone.0194256.g013

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Fig 14. Contour plots of force of infection at equilibrium as a function of θ and p.

Increasing recurrent TB increases TB prevalence.

https://doi.org/10.1371/journal.pone.0194256.g014

Discussion

Until just before the turn of the century the role of exogenous reinfection in the transmission of TB was usually believed to be minimal. van Rie et al. [50] wrote: “For decades it has been assumed that postprimary tuberculosis is usually caused by reactivation of endogenous infection rather than by a new, exogenous infection.” However, these views are no longer considered accurate and our understanding of the role of exogenous reinfection has been completely revised. Warren et al. [53] refuted the unitary concept of pathogenesis of tuberculosis proposed in 1960s: that is, tuberculosis results from a single infection with a single Mycobacterium tuberculosis strain, and such infections were thought to confer protective immunity against exogenous reinfection. Thus, exogenous reinfection was thought to be uncommon. Murray et al. [19] found that their data for exogenous reinfection among US immigrants strongly suggested that reinfection likely plays a major role in high-incidence TB areas.

Lipsitch and Murray [18] argued that backward bifurcation should not occur when taking into account biologically realistic parameter values. Their argument was built on the premise that exogenous reinfection among exposed individuals should be less than the probability of progressing to clinically active TB due to endogenous reactivation, which translates in mathematics to p < P_L = k/(μ + k). But Feng et al. [23] found that backward bifurcation can only take place if p > P_feng > P_L. For this reason, Lipsitch and Murray [18] concluded that backward bifurcations are unlikely to be relevant in the context of TB despite non-existence of a data to support their claim. The argument of Lipsitch and Murray [18] was based on the Feng et al. [23] TB model which as mentioned, failed to incorporate key TB pathways such as primary progression and recurrent TB due to reinfection where some recovered/treated individuals revert directly to the infective stage. Yet, these pathways are critical to TB epidemiology, an aspect that was even pointed out by Lipsitch and Murray [18] as a weakness of the Feng et al. [23] model. Lipsitch and Murray [18] called for further research that would account for these omitted pathways. However, modern research no longer seems to support Lipsitch and Murray’s [18] argument since exogenous reinfection among individuals with latent TB can outweigh endogenous reactivation [20, 21]. And this implies that the protection provided by latent TB infection is not strong enough to prevent individuals becoming reinfected. This is supported by the fact that majority of new TB cases (about ninety percent) are as a result of reinfection rather than endogenous reactivation [13, 19–21]. Moreover, studies of TB reinfection in high-HIV burden countries have demonstrated the strong negative impact of HIV on immunity, which is likely to outweigh the possible protection conferred by latent infection [43]. Backward bifurcation is likely to occur in this scenario, since reinfection will be greater than endogenous reactivation (i.e, p > k/(μ + k)).

The analysis conducted here has shown that when exogenous reinfection is significant (resulting in relatively small p_c—see S5 Appendix), as is currently understood to be not atypical, backward bifurcation can indeed occur for relatively low values of the reinfection parameter p. Furthermore, we observed that if we omit the exogenous reinfection path A required to cause backward bifurcation in the study of Feng eta al. [23], then recurrent TB alone (i.e., paths B and C in our model) may still yield backward bifurcations. For instance, if p = 0 (thereby omitting pathway A), and the recurrent TB rate parameter θ exceeds a certain threshold θ > θ_c then backward bifurcation can indeed occur (see Figs 11 and 12). To the best of our knowledge, this result has not been observed in previous TB modelling studies. In addition, recurrent TB can induce forward bifurcation with hysteresis, rather than just the usual forward bifurcation. Interestingly, the hysteresis loop depends on reinfection parameters, and this may lead to an unusual scenario where the hysteresis crosses the threshold R₀ = 1 thus entering the region where only backward bifurcation is expected (see Fig 9). Epidemiologically this implies that TB will persist when R₀ < 1 even though we have only a forward bifurcation.

In the literature it has been observed that individuals who previously have had active TB and who were successfully treated are more likely to gain active TB another time [13]. However there is little understanding as to why this occurs. Gomes et al. [41] suggested two alternative mechanisms that might explain this: a) previous infection increases susceptibility of individuals to reinfection; b) population heterogeneity, in which some individuals are more at-risk than others, might lead to this conclusion. In this paper, we have almost exclusively explored the former possibility. However the latter possibility may also be at play as examined by Gomes et al. [41]. In the latter case, heterogeneity would be less likely to create changes in reinfection parameters such as θ, and thus might not lead to the same bifurcation phenomena found in this study. Nevertheless, a wealth of theoretical studies do suggest that heterogeneous infection processes are often involved in creating backward bifurcations, and thus we similarly expect complex dynamical phenomena [54–56].

In future work, it would be interesting to additionally consider the possibility that some infected individuals who are treated do not become completely cured. This situation would lead to another cohort of individuals characterised by the fact that treatment has failed, and thus require adding an extra compartment which distinguishes complete recovery and incomplete recovery (of individuals who are infectious). The extra infectives from the latter compartment will tend to increase TB prevalence and thus widen the bifurcation curves. While this possibility is of importance, it falls outside the main scope of the present article, and would lead to a model that is very difficult to analyse mathematically.

Supporting information

Acknowledgments

All authors contributed equally and have no any competing interests. We also gratefully acknowledge the support of the Australia Research Council for providing support though ARC Discovery Project grant DP170102303.