Applied Mathematical Sciences, Vol. 8, 2014, no. 48, 2383-2389 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43150 On CTL Response against Mycobacterium tuberculosis Eduardo Ibargüen-Mondragón Departamento de Matemáticas y Est., Facultad de Ciencias Exactas y Nat. Universidad de Nariño, Pasto, Colombia Lourdes Esteva Departamento de Matemáticas, Facultadad de Ciencias Universidad Nacional Autónoma de México, México DF, México Copyright c 2014 Eduardo Ibargüen-Mondragón and Lourdes Esteva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we formulate a mathematical model trying to describe basic aspects into the dynamics of the Mycobacterium tuberculosis infection. The purpose of this study is to evaluate the impact of the response of T cells in the control of Mtb. Mathematics Subject Classification: 34D23, 93D20, 65L05 Keywords: Mathematical model, Tuberculosis, Stability, Innate immunology, Equilibrium solutions 1 Introduction Tuberculosis TB) is an infectious disease whose etiological agent is Mycobacterium tuberculosis Mtb). The World Health Organization WHO) reports 9.2 million new cases and 1.7 million death each year [3, 9]. However, only 10% of infected individuals with Mtb develop the disease in their lifetime [9]. This indicates that in most cases the host immune system is able to control replication of the pathogen. The Mtb bacteria may affect different tissues, but usually develop pulmonary TB. After the entrance of the bacilli into the
2384 E. Ibargüen-Mondragón and L. Esteva lung, phagocytosis of the bacteria by alveolar macrophages takes place. Cell mediated immune response develops within 2 to 6 weeks, this leads to the activation and recruitment of other immune cell populations, such as CD4 + T or CD8 + T lymphocytes. These cells secrete cytokines that help to kill the infected macrophages [1]. In most cases the initial infection progresses to a latent form which can be maintained for the lifetime of the host with no clinical symptoms. The reactivation of the latent infection can be due to aging, malnutrition, infection with HIV, and other factors. The immune response that occurs after the first exposure to Mtb is multifaceted and complex. Animal models have been extensively used to explain the mechanisms involved in this response, however, these models have limitations, since cellular response may vary between species [8]. In this sense, mathematical models have been applied to understand the celular immunology of TB. Among them we have G. Magombedze et al. [2], E. Ibargüen-Mondragón et al. [4, 5, 6]. 2 The basiodel of Mtb infection dynamics with CTL response In this section we formulate a model on cytotoxic T cells CTL) response against Mtb. We consider the following population; uninfected macrophages, infected macrophages, Mtb bacteria and T cells which are denoted by x, y, b and c, respectively. We assume that uninfected macrophages reproduce at constant rate λ, and die at a per capita constant rate μ. Uninfected macrophages become infected at a rate proportional to the product of x and b, with constant of proportionality β. Infected macrophage die at per capita constant rate ν, where ν μ. Mtb bacteria multiply inside infected macrophages up to a limit at which the macrophage bursts, and releases bacteria. For this reason, we assume that infected macrophages produce Mtb bacteria at a rate proportional to the population, ρy. Infected macrophages die at a rate νy and Mtb bacteria are removed from the system at a rate γb. Let us explore the effect of a CTL response, which provides a maximum amount of CTL to eliminate infected macrophages. In the presence of bacteria and infected macrophages, the supply of specific T cells is given by σ 1 c/ax ) y, where σ is the recruitment rate of T cells, and is the maximum T cell population level. Finally, the T cells die at per capita rate δ. The assumptions above lead to the following
On CTL response against Mtb 2385 system of nonlinear differential equations x = λ μx βxb y = βxb αyc νy b = ρy γb c = σ 1 c ) y δc. 1) In this case, the set of biological interest is given by Ω= { x, y, b, c) R + 0 )4 :0 x + y λ/μ, 0 b λρ/γμ, 0 c λσ/δμ }. 2) The following lemma ensures that system 1) has biological sense, that is, all solutions starting in 2) remain there for all t 0. Lemma 2.1. The set Ω 1 defined in 2) is positively invariant for the solutions of the system 1). Proof. We begin adding the first two equations of 1) and using the fact that ν μ we obtain x + y) + μx + y) λ which implies xt)+yt) λ μ + λ μ + x 0 + y 0 ) e µ U t, where x 0 + y 0 λ/μ. In consecuence, xt)+yt) λ/μ for all t 0. Similarly it is proved that 0 bt) λρ/γμ and 0 c λσ/δμ. On the other hand, it can be easily verified that the vector field defined by 1) points to the interior of Ω. Therefore the solutions starting in Ω remain there for all t 0. 2.1 Equilibrium Solutions In this case, before infection, the system is at the equilibrium x =1,y =0, b = 0, and c = 0. Suppose that bacteria enter to the organism. The infection progression will depend of the basic reproductive number, R 0 = βλρ μνγ. 3) The parameter R 0 can be interpreted biologically as the number of secondary infections that arises from a macrophage during its lifetime if all other macrophages are uninfected. The following theorem summarizes the existence results of the equilibria. Proposition 2.2. If R 0 1, then P 1 =λ/μ, 0, 0, 0) is the only equilibrium in Ω. If R 0 > 1, in addition to P 1, there exists an infected equilibrium, P 2 = x 2,y 2,b 2,c 2 ).
2386 E. Ibargüen-Mondragón and L. Esteva Proof. Equilibrium solutions of 1) are given by solutions of the following algebraic system λ μx βxb =0,βxb αyc νy =0ρy γb =0,σ 1 c ) y δc =0, 4) which are the infection-free equilibrium P 1 =λ/μ, 0, 0, 0) and endemic equilibrium λ P 2 =, γ ) μ + βb 2 ρ b σγb 2 2,b 2,, γb 2 + δρ where b 2 is a solution of b 2 + [ β μ + ρβδν λ) ασ + ν)γν ] b δρr 0 ) ασ + ν)γ 3 ν 2 μ = 0. 5) Since R 0 > 1, then b 2 is the unique positive solution of 5). 2.2 Stability of equilibrium solutions In this section we analyze the stability of equilibria. We begin with the stability analysis of the infection-free equilibrium. Proposition 2.3. For R 0 < 1, P 1 is locally asymptotically stable, and for R 0 > 1, P 1 is unstable. Proof. The eigenvalues of the Jacobian of the system 1) evaluated at P 1 are given by μ, δ and the solutions of the quadratic equation ξ 2 +ν + γ)ξ νγr 0 ) = 0. 6) From Routh-Hurwitz criteria we conclude that the roots of the equation 6) have negative real part if and only if R 0 < 1. Actually, we can prove global stability of P 1 when R 0 1. Proposition 2.4. If R 0 1 then P 1 is globally asymptotically stable. Proof. The function U = ρy + νb satisfies UP 1 ) = 0 and UP ) 0 for all P Ω. Since R 0 1 implies λ/μ νγ/ρβ, then its orbital derivative satisfies U = ρβb x νγ ) ρδyc ρβb x λ ) ρδyc 0. ρβ μ
On CTL response against Mtb 2387 In consequence UP ) 0 for all P Ω 1. From inspection of system 1) we can see that the maximum invariant set contained in the set U = 0 is the plane y =0,b = 0. In this set, system 1) becomes x = λ μx, y =0, b =0, c = δc. Which implies that the solutions starting there tend to equilibrium P 1 as t goes to infinity. Therefore, applying the LaSalle-Lyapunov Theorem see [7]) we have that P 1 is globally asymptotically stable. In the following we will prove the asymptotic stability of P 2 when R 0 > 1. For this, we use the following Lyapunov function )] [ )] x y V = a 1 [x x 2 x 2 ln + a 2 y y 2 y 2 ln x 2 y )] [ 2 )] b c +a 3 [b b 2 b 2 ln + a 4 c c 2 c 2 ln, where a 1 is a positive constant and b 2 a 2 = a 1,a 3 = βb 2x 2 a 1 ρy 2,a 4 = c 2 αc 2 y 2 a 1 σy 2 1 c 2 / ). 7) To prove the global stability of P 2 using Lyapunov direct method we have to show that V P ) 0 and V P ) < 0 for all P Ω. For this end we need next results. Proposition 2.5. The orbital derivative V of V is equal to V = f where f is given by [ f = a 1 μx 2 w 1 + 1 ) 2 + βx 2 b 2 w 1 w 3 + 1 )] w 3 w 1 w [ ) 1 ] w1 w 3 +a 2 βx 2 b 2 + w 2 w 1 w 3 + αy 2 c 2 1 + w 2 w 4 w 2 w 4 ) w 2 ) ) w2 w2 +a 3 ρy 2 + w 3 w 2 + a 4 σy 2 + w 4 w 2 w 3 w 4 σ +a 4 y 2 c 2 w 2 w 4 +1 w 2 w 4 ), 8) where w 1 = x/x 2, w 2 = y/y 2, w 3 = b/b 2 and w 4 = c/c 2. Proof. The orbital derivative of V is given by V = a 1 1 x 2 x +a 3 1 b 2 b ) λ μx βxb)+a 2 1 y ) 2 βxb αyc νy) y ) ρy γb)+a 4 1 c ) [ 2 σ 1 c ) ] y δc. c 9)
2388 E. Ibargüen-Mondragón and L. Esteva From the equilibrium equations 4) we have λ = μx 2 + βx 2 b 2,ν= βx 2b 2 αy 2c 2,γ= ρy 2,δ= y 2 y 2 b 2 Replacing λ, ν, γ and δ in 9) we obtain x V = a 1 [μx 2 + x ) 2 xb x 2 x 2 + βx 2 b 2 + x 2 x 2 b [ 2 xby2 a 2 βx 2 b 2 x 2 b 2 y + y xb ) y 2 x 2 b 2 b2 y a 3 ρy 2 + b y ) by 2 b 2 y 2 c2 y a 4 σy 2 + c y ) cy 2 c 2 y 2 ) σ 1 c 2 y 2. c 2 x b )] b 2 + αc 2 y 2 cy c 2 y 2 +1 y y 2 c c 2 σ cy a 4 y 2 c 2 +1 c y ). c 2 y 2 c 2 y 2 )] 10) In the variables w 1 = x/x 2, w 2 = y/y 2, w 3 = b/b 2 and w 4 = c/c 2, we have V w 1,w 2,w 3,w 4 )= fw 1,w 2,w 3,w 3 ). Proposition 2.6. The function f is nonnegative. Proof. From the following equalities σ a 1 βx 2 b 2 = a 2 βx 2 b 2 = a 3 ρb 2,a 4 σy 2 = a 3 αc 2 y 2 + a 4 y 2 c 2, we obtain the constants defined in 7). Replacing the equilibrium equation ρy 2 = γb 2 and the constants 7) in the function f we have fw 1,w 2,w 3,w 4 ) = a 2 μx 2 w 1 + 1 ) ) 1 2 + a 4 σy 2 w 2 + w 4 2 w 1 w 4 w1 w 3 +a 2 + a 3 )βx 2 b 2 + w 2 + 1 ) 3, 11) w 2 w 3 w 1 It can be seen that the expressions inside the parenthesis of 11) are nonnegative, and therefore f is nonnegative. Theorem 2.7. If R 0 > 1 then nontrivial equilibrium P 2 is globally asymptotically stable. Proof. It is clear that V P 2 ) = 0 and V P ) 0 for all P Ω. From Proposition 2.5 we have V = f and from Proposition 2.6 we have f is nonnegative, therefore V P ) 0 for all P Ω. Further V = 0 if and only if x = x 2, y = y 2, b = b 2 and c = c 2 which implies all trajectories inside Ω approach P 2 when t goes to infinity.
On CTL response against Mtb 2389 3 Discussion In this paper we formulated a mathematical model on the immune response to Mtb in order to evaluate the efectiveness of T cells in controlling TB. Although our model is quite simple compared to the complexity of the immune response to Mtb, it predicts in terms of the basic reproductive number R 0, when the bacteria is cleared or infection progresses to disease. References [1] AM. Gallegos, EG. Pamer, MS. Glickman, Delayed protection by ESAT- 6-specific effector CD4+ T cells after airborne M. tuberculosis infection. J. Exp. Med. 2008 Sep 29;20510):2359-68. [2] G. Magombedze, W. Garira, E. Mwenje, Modellingthe human immune response mechanisms to mycobacterium tuberculosis infection in the lungs, J. Mathematical Biosciences and engineering, 32006) 661-682. [3] Global tuberculosis control: surveillance, planning, financing: WHO report 2008. WHO/HTM/TB/2008.393. [4] E. Ibargüen-Mondragón, L. Esteva, L. Chávez-Galán, A mathematical model for cellular immunology of tuberculosis. J. Mathematical Eiosciences and Engineering, 82011) 976-986. [5] E. Ibargüen-Mondragón, L. Esteva, L., Un modelo matematico sobre la dinamica del Mycobacterium tuberculosis en el granuloma. Revista Colombiana de Matematicas, 462012)1 39-65. [6] E. Ibargüen-Mondragón, L. Esteva, L., On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics. Mathematical Biosciences, 246 2013) 8493. [7] J. Hale, Ordinary Differential Equations, Wiley, New York, 1969. [8] M. Tsai, S. Chakravarty, G. Zhu, J. Xu, K. Tanaka, C. Koch, J. Tufariello, J. Flynn and J. Chan, Characterization of the tuberculous granuloma in murine and human lungs: cellular composition and relative tissue oxygen tension, Cell Microbiology, 8 2006), 218 232. [9] Palomino-Leo-Ritacco, Tuberculosis 2007, From basic science to patient care. TuberculosisTextbook.com, first edition. Received: March 4, 2014