Introduction - WSEAS



Modeling the interaction between HIV-1 and the Immune System.

F. L. BIAFORE 1, C.E. LUJÁN1, C.E. D’ATTELLIS 1,2

1.- Escuela de Ciencia y Tecnología, Universidad Nacional de San Martín, Alem 3901, 1653-Villa Ballester, ARGENTINA

2.- Departamento de Matemática, Universidad Favaloro, Av. Belgrano 1723, 1093-Buenos Aires, ARGENTINA

Abstract: - The evolution of HIV-1 infection in the untreated individual can be represented by three stages.

Several mathematical models can explain some dynamical aspects of this infectious disease. In this work, it is proposed a four dimensional and nonlinear dynamic model that describes satisfactorily the three stages of the infection. The introduction of a time-dependent parameter in the model influences significantly in the description of the third stage, characterized by the collapse of the immune system.

Key-Words: - HIV, mathematical modeling, AIDS.

1 Introduction

The course of HIV-1 infection in the untreated individual can be viewed as a spectrum of disease progressing through various stages [1-4]:

The initial acute or primary HIV-1 infection: Immediately after infection the amount of virus detected in the blood rises dramatically and there is frequently a marked decrease of CD4 cell count. The CD8 cell count rises initially which may result in a CD4/CD8 ratio of < 1.

The asymptomatic stage: After a few weeks to months the symptoms disappear and the virus concentration falls to a lower level. An immune response to the virus occurs. This response is attributed to the CD8 lymphocyte cells. The viral replication persists and there is a relentless decline of CD4 lymphocytes. This occurs from the very beginning of infection and is permanent. Throughout most of this period, the patient may be entirely asymptomatic. This period can last as long as 10 years.

The final phase of HIV-1 infection: This occurs when a significant number of CD4 lymphocytes has been destroyed. The viral load increases and the symptoms of full-blown AIDS appear. Near the end stage of AIDS, CD8 cells also decline precipitously. The end result is a state of profound immune suppression that renders the infected individual susceptible to a multitude of opportunistic infections.

There are several deterministic models that consider the dynamics of HIV-1 and the host immune system and can explain various dynamical aspects of this infectious disease.

One of those recent models is proposed by De Souza [5] in which their free dynamics describes the acute primary infection and the asymptomatic period characterized by a lower viral load in blood that culminates in an equilibrium of its state variables. This equilibrium depend on the parameters of the system and they don't correspond with the third stage of the disease -described in the previous paragraphs- characterized by the collapse of the immune system -abrupt fall of CD4 and CD8 cell levels -.

The nonlinear dynamic model presented in this work describes the three stages of the infection, using as state variables the CD4 lymphocytes, infected CD4 lymphocytes (virus is produced by infected CD4 cells), CD8 lymphocytes and the viral load.

The description of the third stage of the infection is due to the incorporation of a time-dependent parameter in this one-compartment model (the compartment is the blood).

See references [6] for multi-compartment model.

2 The Model

What is known about the interaction of HIV-1 and the CD4 and CD8 lymphocytes is the following:

• HIV-1 uses the CD4 cells to replicate itself.

• The infected CD4 cell can produce new HIV-1 copies.

• The larger the HIV-1 population, the larger the growth rate of the CD8 population.

• The CD8 cells attack infected CD4 cell, destroying it.

• Some strains of HIV-1 that are often presented late in the course of HIV infection prompt a mass apoptosis of CD8 cells.

Starting from this premises, the following dynamical model is proposed:

• The growth rate of CD4 diminishes when the HIV-1 population grows.

• The growth rate of infected CD4 increases when the HIV-1 and infected CD4 populations grow.

• The growth rate of HIV-1 increases when the infected CD4 population grows.

• The growth rate of CD8 increases when the infected CD4 and CD8 population grows.

The system of nonlinear differential equations will be:

[pic]

where x1 (CD4 cells), x2 (infected CD4 cells), x3 (virus) and x4 (CD8 cells) are the state’s variables, [pic] and [pic] are the equilibrium values of the CD4 and CD8 populations in the absence of the virus and a, b, µ1, p, µ2, k, µ3, c, d, f(t) and µ4 are the parameters of the system. They are all positives.

The human body has a homeostasis that regulates the normal levels of CD4 and CD8 cells. These normal levels are [pic] and [pic]. The parameter a represents the dynamics of this regulating mechanism for CD4 cells. The parameter c plays the same role in the CD8 population dynamics.

When one individual HIV-1 copy meets a CD4 cell, the cell becomes infected. Thus, these encounters negatively affect the growth rate of CD4. The number of those encounters will be proportional to the product of the populations of HIV-1 and CD4 cells. The constant of proportionality is the parameter b (rate of infection).

The CD4 cells have a natural life-span. The parameter µ1 is the death rate per CD4 cell (Eq.1)

When one CD8 cell meets an infected CD4 lymphocyte, the infected CD4 cell will be destroyed. These encounters negatively affect the growth rate of infected CD4 cells. The number of those encounters will be proportional to the product of the population of CD8 and infected CD4 cells. The constant of proportionality is the parameter p.

The infected CD4 cells have a natural life-span. Here µ2 is the death rate per infected CD4 cell. (Eq.2)

Here it is assumed that on average each infected CD4 cell produces k infectious virions per day during your lifetime. The clearance rate constant of HIV-1 is represented by the parameter µ3 . (Eq. 3)

When one CD8 cell meets an infected CD4 lymphocyte, this encounter will trigger a mechanism that increases the production of CD8 cells. The number of those encounters will be proportional to the product of the population of CD8 and infected CD4 cells. The constant of proportionality is the parameter d. The CD8 cells have a natural life-span. The parameter µ4 is the death rate per CD8 cell. (Eq.4)

It has been introduced in the model a time-dependent parameter -f(t)- that represents the time dependence of the probability of the encounters among the CD8 lymphocytes and the viral strains able to cause the death of these cells in the late stage of the infection - see above - The temporary dependence of this probability is represented by a growing exponential function-[pic]- that confers a variable weight to the term on which acts making it important in the final stages of the disease, as it indicates the recent experimental evidence [3].

3 Simulation

The model was simulated using the following values*:

[pic]1000 ,[pic]550 , [pic]0.35 , [pic]0.08 , [pic]0.01 , [pic]2 , [pic]0.7 , [pic]250 , [pic]5 , [pic]0.1 , [pic]0.9 , [pic]0.05 , [pic]0.0001 , [pic]0.65

The parameters were chosen to produce biologically plausible values.

The initial conditions were:

[pic]1000 cells/mm3

[pic]0 cells/mm3

[pic]0.03 (corresponding to 0.3 copies/mm3 )

[pic]550 cells/mm3

The result is shown in Fig.1

[pic]

Fig.1 Evolution of the infection (no drugs)

The graph of the infected CD4 cells (x2 ) was amplified by a constant factor (100) to visualize the solution.

In the graph three stages during the evolution of the infection can be distinguished. The first stage is characterized by an initial overshoot in the viral load and the descent of the amount of CD4 cells and an increase of the CD8 lymphocytes level.

The second stage is characterized by a low level of viral load, infected CD4 cells and a value of CD8 cells that stays in higher level that their corresponding initial value.

A characteristic of this stage is the oscillatory character of all the state variables.

The third stage is distinguished starting from the eighth year characterized by a fast increment of infected CD4 cells and viral load, accompained by a pronounced fall of CD8 and CD4 cell levels.

The length of the primary stage of infection -characterized by the width of the initial viral pulse- is bigger than the reported by experimental evidence (weeks to months).

To achieve an appropriate agreement with this evidence it is necessary to multiply the equations for an appropriate constant and it can be necessary to adjust the time-dependent parameter.

The Fig.2 shows an example where the time scale was adjusted by multiplying all equations by a constant value (5).

[pic] Fig.2 Evolution of the infection (no drugs) with adjusted time scale. The graph of the infected CD4 cells (x2 ) was amplified by a constant factor (100) to visualize the solution.

This example shows a better correlation of the first stage of the disease with the reported clinical data.

Starting from the eleventh year it is observed that the values of the state variables of the model represent a systemic immunologic deficiency, commonly appeared in these cases.

The term introduced in the model to represent the induction to apoptosis of CD8 cells for late strains of HIV-1 in the last stage of the disease, also generates the global descent of the values of CD4 cells.

The action of this time-dependent parameter becomes evident in the following simulation where the omission of the same does not generate the third stage (Fig.3)

[pic] Fig.3 Evolution of the infection -no drugs- without the time-dependent parameter. The graph of the infected CD4 cells (x2) was amplified by a constant factor (100) to visualize the solution.

This term becomes more important in the time course and in the latest stage it affects negatively the rate of growth of the CD8 lymphocytes until its absolute value overcomes to the remaining terms of the respective equation.

The descent of the population of CD8 cells acts positively on the rate of growth of infected CD4 cells, until causing a progressive increase in the number of these cells. As a consequence, the number of viral copies present in the blood plasma increases.

As a consequence of the viral increase, the rate of growth of CD4 cells is affected negatively, until generating the abrupt fall observed in the final stage.

This way, the progressive depletion of CD4 cells is explained starting from a term proposed initially to explain the abrupt fall of CD8 cells in the late stage of the infection.

4 Conclusion

The model presented in this work describes the three stages of the HIV infection satisfactorily in absence of pharmacological control. The parameters that have been adjusted were chosen to achieve values of the state variables that resemble to a real case.

The next step will be the incorporation of nonlinear control methods that represent the action of antiretroviral drugs, with the aim of eradicates the virus of the blood plasma and to compare and to optimize different strategies of drug administration.

To facilitate the applications of control strategies in the different stages of the infection, it is possible to change the time dependent parameter by an appropriate constant in the considerated disease stage.

References:

[1] L. Fainboim, M. Satz, J. Geffner, Introducción a la Inmunología Humana, Edición del autor. 1999.

[2] A. Perelson, P. Nelson, Mathematical Analysis of HIV-1 Dynamics in Vivo, SIAM Review, Vol.41, No.1, 1999, pp. 3-44.

[3] E. Lawrence, How HIV-1 induces cell suicide, Nature Science Updates (medicine), 1998,

[4] C. Hoffmann, B. Kamps, HIV Medicine 2003, Flying Publisher, 2003,

[5] F. M. C. de Souza, Modeling the Dynamics of HIV-1 and CD4 and CD8 Lymphocytes, IEEE Eng. Med. Biol. Mag., Vol. 18, 1999 pp. 21-24.

[6] S. Snedecor, Comparison of three Kinetic Models of HIV-1 Infection: Implications for Optimization of Treatment, J. theor. Biol., No.221, 2002, available online at

Acknowledgement: This work was partially supported by Fundación Antorchas, Buenos Aires, Argentina.

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