Viral evolution model with several time scales

Abstract

In this paper a viral evolution model with specific immune response is considered. By introducing of dimensionless variables and parameters this model can be modified to the singularly perturbed system of partial integro-differential equations with two small parameters. The transition from the initial-boundary value problem of the initial system to the generating problem makes it possible to reduce the dimension of the system and, as a consequence, to reduce the computational volume. The theorem on the passage to the limit is also represented. The presence of several time scales in the models of evolution biology is more a rule than an exception. This is due to the fact that an extremely slow biological evolution process proceeds against the background of significantly faster interactions of different nature. To model such processes with several time scales, systems of differential equations with a small parameter for a part of the derivatives (the so-called singularly perturbed systems of differential equations) are usually used. Numerical analysis of such systems involves a large amount of computation due to the presence of variables that vary with significantly different velocities. Therefore, it becomes relevant to construct simplified (reduced) models of lower dimensionality, but with a high degree of accuracy reflecting the behavior of the original processes. One of the reduction methods for singularly perturbed systems are the integral manifold method, developed in [1-3], and the passage to the limit to the solution of the degenerate system, used in present paper. In this case, the dimension of the systems under consideration is reduced. Below, this approach is used to reduce the dimension in the initial-boundary value problem for a system describing the dynamics of populations of healthy and infected cells and cytotoxic T-lymphocytes.

Keywords

• viral dynamics
• immune response
• specific immunity
• singular perturbations
• initial-boundary value problem
• degenerate system
• passage to the limit

1. Introduction

2. Biological background

3. Model

Mathematical Modeling / A.A. Archibasov die naturally at a rate m, 1 day, and are eliminated by CTL response at a rate  , mm3 virion day . The activation term of CTL response is assumed to be proportional to vt, s with a coefficient q, 1 day , since the number of infected cells has to be different from zero in order to activate the growth of zt, s . After activation of CTL response the activated cells will multiply by cloning (the so-called “clonal expansion”). To model this phenomenon, a logistic term is employed. Random mutations are described by the dispersion with a coefficient  , 1 day . Since vt, s is a distribution, it is natural to assume that vt,  0. The boundary condition at s  0 is the non-flux condition v t,0  0 for convenience. Non-negative initial conditions at t  0 t are u0  u0 , v0, s  v0 s and z0, s  z0 (s) (it is assumed that a host is already infected by a virus).

Without loss of generality, for simplicity, we assume that only  depends on s and that m ,  , q ,  are constant and have common values for all phenotypes.

Although the model is stated for s  0, , the parameters s is usually assumed to belong a finite interval 0, , and the boundary condition vt,  0 is replaced by the condition v t,   0 .

s

4. The dimensionless system

Let us introduce the following notations t  Tt , s  Ss , uTt   Uu (t ) , vTt , Ss   V~vt , s , zTt , Ss   Z~zt , s , and assume ~ ~ ~ ~ ~ that T S 2  1, U  b c , V   qZ , Z  p . Then the initial-boundary value problem for model ( 1 ) takes the form v t ,0  0, s v t ,   0, z0, s   z 0 s , s  Ss  , m  mT , d 

,   pT , u 0  bqS p cu0 b , v 0 s   q v0 Ss  , z 0 s   p 1 cT

t  du (t )  1  u t   s v t , s ds  u t ,

dt 0 v t , s  2v t , s  

s s 2 zt , s 

 v t , s  zt , s 1  zt , s , u 0  u 0 , v0, s   v 0 s , where   ,   c ,  s   pS  cq 

S .



 mv  d s u t vt , s  v t , s ,

The parameter T must be taken so that the inequality  1 holds. For example, T  1  , then S  T  1 . The parameter  is proportional to the mutation probability. For HIV  does not exceed 107 109 1 day , and HIV is known as one of the most rapidly mutating RNA-viruses, so that  is substantially smaller for more slowly mutating RNA-viruses (and so much the more DNA-viruses). As c   , then  1 . Thereby system ( 2 ) is a singularly perturbed system with two small parameters and as result has three time scales. It should be noted that a system with several time scales was considered in the original work [5]. Further to simplify the notation, we omit the bar.

5. Reduction of dimension

Setting   0 in ( 2 ), we obtain the so-called first-order degenerate system

du   1  u  vds  u, dvt 2v 0

  mv  d uv vz, t s2 0  v  z1  z.

The third equation is algebraic and has two roots z1,2  1 1 4v  2 . For the first-order associated system zˆ   zˆ1  zˆ  v,  zˆ1 zˆ zˆ where v enters as a parameter, only one of the roots, namely z   v  1  1  4v  2 , is the asymptotically stable (in the sense of Lyapunov) stationary point, because

  1 4v  0 .

zˆ v

At the initial value of the parameter v , i.e., at v  v0 s , the system ( 5 ) with the initial condition zˆ0, s  z0 (s) has a unique solution zˆ , s for   0 , and besides lim zˆ , s   v0 s s  0,  (see Appendix). Thereby the initial point z0 s of the   ( 2 ) ( 3 ) z0 Ss  p

Mathematical Modeling / A.A. Archibasov first-order associated system ( 5 ) belongs to the domain of attraction of the stable stationary point  v0 s . Thus, for sufficiently small  , problem ( 2 ), ( 3 ) has a unique solution and, for some t1 , the following limiting equalities hold [6]: lim ut, ,   u0 t,  for 0  t  t1,  0 lim vt, s, ,   v0 t, s,  for 0  t  t1, 0  s  , ( 6 )  0 lim zt, s, ,    v0 t, s,  for 0  t  t1, 0  s  ,  0 where ut, ,  , vt, s, ,  , zt, s, ,  are the solutions of the system ( 2 ) and u0 t,  , v0 t, s,  are the solutions of the system ( 4 ). Note that the third limiting equality holds for t  0 , as the solution z   v of reduced system ( 4 ), generally speaking, does not satisfy initial condition for this variable in ( 3 ). The boundary layer phenomenon occurs. Equation ( 5 ) is also called the boundary layer equation. Naturally, there is no boundary layer if the initial point falls on the slow surface [7-9]. The system ( 4 ) has a dimension one less in comparison with ( 2 ).

Let   0 in ( 4 ). Then we obtain the second-order degenerate system 0  1  u vds  u,

0 v

2v  t s2 0  v  z1  z,

 mv  duv vz, where v00 t, s is the solution of the second equation in ( 7 ) with boundary and initial conditions for variable v in ( 3 ). The passage to the limit for u0 is not carried out at point t  0 . As a result, a system of three integro-differential equations reduces to one integro-differential equation. The existence and uniqueness of the solution of the initial value problem for integroparabolic equation in ( 7 ) can be justified with the use of the approach outlined in the monograph [10].

6. Admissibility of the passage to the limit

In work [6] the theorem, that connects the solutions of the singularly perturbed system of partial integro-differential equations with one small parameter, is proved. Generalize this theorem to the case of two small parameters.

Consider the singularly perturbed system of partial integro-differential equations  du  f  u,  gs, vds,

 0  first equation in which is algebraic with respect to u and has a root u  v  1 1  vds . This root is the asymptotically   0  stable (in the sense of Lyapunov) stationary point of the second-order associated system duˆ  1  vdsuˆ 1. ( 8 ) d  0 

The latter equation with the initial condition uˆ0  u0 at the initial value of the parameter v  v0 s has a unique solution uˆ   u0 1 f e f 1 f , f  v0 s  1   sv0 sds , for all   0 and lim uˆ   1 f . Thus, the initial point u 0 of the 0   second-order associated system ( 8 ) belongs to the domain of attraction of the stable stationary point  v0 s . Consequently, for some t2

lim0 u0 t,   v00t, s for 0  t  t2 ,  0 v0 t, s,   v00t, s for 0  t  t2 , 0  s  , lim lim0 z0 t, s,    v00t, s for 0  t  t2 , 0  s  , ( 7 ) ( 9 ) ( 10 ) ( 11 )

 hz, v, t s2 with the initial and boundary conditions are uniformly continuous and bounded in the respective domains 3  z  d, v  c, 4  0  s  , u  a, z  d, v  c.

We assume that system ( 10 ) satisfies the following conditions.

I. The functions f u, x , gs, v , hz, v , and qs, u, z, v , together with their partial derivatives with respect to all variables,   u  a, x  b , 2  0  s  , v  c ,

1

Mathematical Modeling / A.A. Archibasov

II. The equation hz, v  0 has an isolated root z   v in the domain v  c and in this domain function z   v is continuously differentiable.

III. The inequality hz  v, v    0 holds for v  c . This condition implies that the stationary point zˆ   v of the firstorder associated system zˆ  hzˆ, v, ( 12 )  which contains v as a parameter, is Lyapunov asymptotically stable as    uniformly with respect to v , v  c . If assumption III is satisfied, then we say for brevity that the zero of the function  v is stable.

IV. There exist a solution zˆ  of the problem zˆ  hzˆ, v0 s, zˆ0, s  z0 s, (13)  for   0 , 0  s  . Further, this solution tends to the stationary point  v0 s as    , i.e. z0 (s) belongs to the domain of attraction of the stable stationary point  v0 s .

V. The equation f u, x  0 has an isolated root u  x in the domain x  b and in this domain function u  x is continuously differentiable.

 

VI. The inequality fu  x, x    0  x  0 gs, vds holds for v  c , i.e. the stationary point uˆ  x of the secondorder associated system duˆ  f uˆ,  gs, vds, (14) d  0  which contains v as a parameter, is Lyapunov asymptotically stable as    uniformly with respect to v , v  c . If assumption VI is satisfied, then we say for brevity that the zero of the function  x  is stable.

VII. There exist a solution uˆ  of the problem   for   0 . Further, this solution tends to the stationary point    gs,  v0 sds as    , i.e. u 0 belongs to the domain  0  (15) (16) (17) (18) duˆ  f uˆ,  gs, v0 sds, d  0 

uˆ0  u0 , of attraction of the stable stationary point.

VIII. The truncated system v   2v  qs, x, v, v, t s2 u  x, z   v, x   gs, vds,

0 v0, s  v0 s, v t,0  0, s v t,   0, s   lim ut, ,   u t     gs, v t, sds,  00  0  lim zt, s, ,   z t, s   v t, s, 0  t  T , 0  s  ,  00 lim vt, s, ,   v t, s, 0  t  T , 0  s  ,  00

0  t  T ,   has a unique solution v t, s  , u t     gs, v t, sds , z t, s   v t, s .

 0 

Theorem. If conditions I-VII are satisfied, then, for sufficiently small  ,  , problem ( 10 ), ( 11 ) has a unique solution ut, ,  , zt, s, ,  , vt, s, ,  , which is related to the solution u t  , z t, s  , v t, s  of the truncated problem (16), (17) by the limit formulas

  Here T is an arbitrary number such that u    gs, v t, sds , z   v t, s are the isolated stable roots of the equations  0    f u,  gs, v t, sds  0 , h v t, s, v t, s  0 for 0  t  T accordingly.

 0 

The proof of this theorem is the same as one in [6].

Mathematical Modeling / A.A. Archibasov

7. Conclusion

In this paper the procedure that the original system of three integro-differential equations reduces to a single integrodifferential equation is given for a viral evolution model with specific immune response. The theorem on the passage to the limit is also formulated. The limiting equalities for fast variables whose physical meaning is the concentration of populations of healthy cells and killer T-cells are valid only for some segment  ,T  ,   0 , separated from zero. To construct an approximate solution in a neighborhood of the point t  0 Tikhonov-Vasil’eva boundary function method [11] can be applied. In the paper [12] a model of viral evolution without immune response (but this model is described by a system of the same type which this work deals with) was considered. By the method mentioned above the solutions in powers of small parameters were found.

It should be noted that the mathematical models of evolution biology are usually formulated as integro-differential equations and PDE. Thus the same concept and the same techniques can be used to a model of evolution based on any other model virus dynamics.

Acknowledgements

Appendix

The study was supported by the Russian Foundation for Basic Research and Samara region (grant 16-41-630529-p) and the Ministry of Education and Science of the Russian Federation as part of a program to increase the competitiveness of SSAU in the period 2013-2020.

Let us solve the initial value problem successively in the equation of change of variables zˆ  zˆ1  1 v0 s 2 , y  1 zˆ1 , we first bring it to the Bernoulli equation, and then to a linear nonhomogeneous equation of the first order y  y 1  4v0 s  1,     .

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