ANALYSIS OF SOLUTIONS AND DISEASE PROGRESSIONS FOR A WITHIN-HOST TUBERCULOSIS MODEL

. Mycobacterium tuberculosis infection can lead to diﬀerent disease outcomes, we analyze a within-host tuberculosis infection model considering interactions among macrophages, T lymphocytes, and tuberculosis bacteria to understand the dynamics of disease progression. Four coexisting equilibria that reﬂect TB disease dynamics are present: clearance, latency, and primary disease, with low and high pathogen loads. We also derive the conditions for backward and forward bifurcations and for global stable disease free equilibrium, which aﬀect how the disease progresses. Numerical bifurcation analysis and simulations elucidate the dynamics of fast and slow disease progression.


Introduction
Mycobacterium tuberculosis (Mtb) is a bacterium that causes an ancient and deadly infectious disease in humans, called tuberculosis (TB) [9]. Currently, TB affects approximately one third of the world's population [10,6]. In 2018, the World Health Organization (WHO) estimated approximately 10 million infections globally, and 1.2 million deaths among HIV-negative people [12]. It has also been found that TB susceptibility and disease are increased in HIV-AIDS infected individuals, resulting in higher mortality rates [8,1,14,16].
The pathological outcomes of TB infection include clearance, latent infection, and primary disease with fast or slow progression [13]. After initial infection, 5−10% of infected subjects can clear the disease. Of the remaining individuals, 5 − 10% will progress to primary disease, and the rest will remain in a latently infected state with no clinical symptoms, with the possibility of re-activation to primary disease later in their life. A large number of mechanisms have been proposed to explain TB disease progression considering individual factors, including bacterial and immune response mechanisms. However, the most influential factors for TB outcomes are not currently known. Motivated by this, we analyze a TB host-pathogen model first proposed in Ref. [3]. The model incorporates known mechanisms of host-pathogen interaction in TB dynamics, and includes all realistic disease outcomes. Analysis is performed to determine the driving factors behind disease progression and outcome, especially fast or slow progression to primary disease.
The paper is organized as follows. In Section 2, we introduce the established tuberculosis progression model. In Section 3, model dynamics are shown through the proofs of the well-posedness of solutions, the existence of equilibrium solutions, and analyses of the disease free equilibrium. The basic reproduction number R 0 and the vector field on the center manifold for the disease free equilibrium when R 0 = 1 are derived analytically. The conditions for the occurrence of the backward and forward bifurcations are also derived. In Section 4, numerical continuations are carried out for the infected equilibrium to confirm the existence of a backward bifurcation. The corresponding numerical simulations show the fast and slow disease progressions to latency and primary diseases. Finally, conclusions are drawn in Section 5.

Model
The 4-dimensional model (2.1) includes the MTb ideal target cell population, macrophages (their uninfected M u and infected M i populations). It also includes the Mtb bacterial population B, and a population of CD4 T cells, which aid in TB clearance. The model is as follows: Briefly, uninfected macrophages M u enter the system with constant rate s M , and can die naturally (µ M ), or be infected by the pathogen B (βM u B). It is assumed that infected macrophages can release new bacteria into the system in two different ways: (1) through cell death and bursting b, producing N 1 new bacteria, and (2) through cytotoxic T-lymphocyte killing (represented by the ratio T /M i ) with rate γ and saturating factor c, which releases N 2 new bacteria into the system. It is assumed that the bacteria population can divide (δB(1 − B/K) and that bacteria can be lost due to interaction with macrophages. This occurs through immune system neutralization ηBM u or macrophage infection βBM u involving, on average, N 3 individual bacteria. Finally, it is assumed that T-cells are produced at a constant rate s T by the thymus, can be stimulated to proliferate through interactions with the infected macrophage c M M i T /(e M T + 1) and bacteria c B BT /(e B T + 1), and can die naturally, with rate µ T . Infection is initiated with an initial pathogen load. We refer the reader to Du et al. [3] for more detail on the biology and model assumptions. Parameters and their values are listed in Table 1.
In previous work, Du et al. [3] found four biologically realistic equilibria and determined the basic reproduction number. Note that, in the original contribution, there is no mention of the driving factors behind the different outcomes of disease (namely, clearance, latency, and primary disease with fast or slow progression) and only an asymptotic version of the model that neglects the effects of the CD4 T-cell population is used/analyzed. In the following, we expand and elaborate on the four disease outcomes and other interesting aspects of the model using the full model system Eq. 2.1.

ANALYSIS OF SOLUTIONS AND DISEASE PROGRESSIONS FOR A WITHIN-HOST TUBERCULOSIS MODEL 41
Proposition 3.1. Under the flow of (2.1), there exists a positive invariant set D that attracts all solutions in R 4 + as time moves forward.
Proof. The smoothness of the right hand side of model (2.1) guarantees the local existence and uniqueness of the solution of the initial value problem of model (2.1). The trajectories starting from positive initial values never cross the boundary of R 4 Next, we show that positive solutions are bounded. Due to the positiveness, we have Moreover, where C 0 is determined by initial condition and C 0 + t > 0 for sufficiently large t. We have Then, the last equation in (2.1) satisfies It hence follows that and the proposition is proven.
The corresponding steady states are derived as follows: Considering the preceding results (3.2) and (3.3), we obtain This indicates that the equilibrium does not exist. Therefore f 3 = 0 only implies f 3b = 0 followed bȳ , B,T (B)) = 0, the following is true: Then, we find the infected equilibrium ). We note that there could be more than one solution, and up to three feasible infected equilibria. .
This is followed by f 4 (M u (B),M i0 ,B 0 ) = 0, which yields We thus find the disease free equilibrium (DFE) 3.3. Analysis of the disease free equilibrium.

3.3.1.
Calculation of the basic reproduction number. Following the next-generation matrix approach in Ref. [11], the basic reproduction number R 0 is the spectral radius of F V −1 , where

ANALYSIS OF SOLUTIONS AND DISEASE PROGRESSIONS FOR A WITHIN-HOST TUBERCULOSIS MODEL 43
The Jacobian matrix of model (2.1) at the disease free equilibrium is: and gives the following characteristic equation where Equation (3.11) admits at least two negative roots, z = −µ T and z = −µ M . The third root, which is equivalent to R 0 = 1.

3.3.2.
Existence of a backward bifurcation. Following Theorem 4.1 in Ref. [2], we first shift the disease free equilibrium to the origin by letting .
Then we compute the approximated center manifold for the system near the origin with one simple zero eigenvalue at R 0 = 1, and three negative eigenvalues. We choose a right eigenvector associated with the simple zero eigenvalue, w, and the left eigenvector, v, satisfying vw = 1 as follows: Further, the flow of the center manifold y(t) truncated at the quadratic term is written aṡ  The preceding equations are followed by The occurrence of a backward bifurcation destabilizes the globally stable disease free equilibrium E 0 under the condition R 0 < 1 and an extra condition to regain stability is needed, i.e. b + γ + s M µ M (N 3 β + η) > δ, as shown in Theorem 3.3. In the next section, we verify the existence of a backward bifurcation computationally and investigate the associated dynamical behaviors by numerical simulations.

Bifurcation analysis and numerical simulations
Consider the n-dimensional nonlinear system with m parameter values The equilibrium solutions x e = x e (p) are derived from the equilibrium condition The local stability of the equilibrium points x e (p) is determined by the eigenvalues of the Jacobian J(p) = [∂f i (x e (p), p)/∂x j ], which are the roots of the corresponding characteristic polynomial equation P n (λ) = det[λI − J(p)] = λ + a 1 (p)λ n−1 + a 2 (p)λ n−2 + · · · + a n−1 (p)λ + a n (p). (4. 3) The necessary and sufficient conditions for zero-eigenvalue bifurcation (zero-singularity) are given in Ref. [15].
Theorem 4.1. The necessary and sufficient conditions for system (4.1) to have a k-zero singularity at a fixed point (equilibrium), x = x e (p), of the system are given by a n (p) = a n−1 (p) = · · · = a n+1−k (p) = 0, (4.4) which a i (p)'s are the coefficients of the characteristic polynomial (4.3). Further, if the remaining coefficients a 1 , a 2 , . . . a n−k still obey the Hurwitz conditions for order n − k, then all the remaining eigenvalues of the Jacobian have negative real parts.
Based on the results of the uncertainty and sensitivity analysis in Ref. [3], the model is significantly affected by the change of macrophage loss rate b, the infection rate β, cell-mediated immunity rate γ, and bacterial killing rate η. We thus choose the macrophage loss rate b as a bifurcation parameter to verify the analytical result for the backward bifurcation discussed in Theorem 3.1. The other parameter values are fixed and shown in Table 1.
Using  Figure 1(B). These examples of fast and slow disease progression dynamics seem to confirm the numerical findings in Ref. [3] and are the object of current investigation.

Conclusion
In this paper, we analyze a four-dimensional within-host model (2.1) for tuberculosis infection, which has been previously proposed and studied numerically in Ref. [3].
We carry out analyses for the well-posedness and boundedness for solutions, existence of the disease free and infected equilibriums and local and global stability analysis. A bifurcation analysis for the disease free equilibrium is also conducted, and a numerical continuation for the infected equilibrium shows when a backward bifurcation occurs. Numerical simulations finally show how fast and slow disease progressions take place close to the bifurcation, with examples of bistability behaviour. This is important because different initial infections can lead to different disease progressions, with considerable differences among latency times.
An in-depth analysis of the bifurcation scenario of this model is currently under progress, with the aim of characterising the different, possible behaviours towards infection that TB shows.