HIV / AIDS Model Based on Local Activity of CNN

A Cellular Neural Network (CNN) model of two strain (antiretroviral sensitive and resistant) Human Immunodeficiency Virus (HIV) is established, and then analyzed and simulated based on the local activity of CNN with five local state variables and one port. Numerical simulations exhibit that this CNN model may explain some complex phenomena during antiretroviral therapy which make it possible to judge the curative effect of long-time antiretroviral therapy and social influence. All these imply that the local activity of CNN provides a practical tool for the study of the complex dynamics of some coupled nonlinear systems, especially life systems.


Introduction
Coupled nonlinear dynamical systems have been widely studied in recent years.However the dynamical properties-the determination, prediction, and control of these systems are difficult to deal with.Nature abounds with complex patterns and structures emerging from homogeneous media, the local activity is the origin of these complexity (Chua, 1997;Chua 2005).The Cellular Neural Network (CNN), firstly introduced by Chua (Chua, 1988), has been widely studied for image processing, robotic, biological versions, higher brain functions and so on (Chua, 1988).It can model and study many coupled nonlinear systems (Chua, 1999).The local activity of CNN proposed by Chua (Chua, 1997) asserts that a wide spectrum of complex behaviors may exist if the cell parameters of the corresponding CNN are chosen in or nearby the edge of chaos (Chua, 1999;Chua, 2005).This theory has been successfully applied to the research on many complex systems with physical, biological, and chemical backgrounds.
Life systems consist of locally coupled homogeneous media.Virtually, dynamics of life systems are suitable to be described via locally connected CNNs.It may be expected that CNN will become a promising candidate for modeling life phenomena.This theory has been successfully applied to life systems, such as the biochemical model (Min, 2000), coupled excitable cell model (Min, 2002), tumor growth and immune model (Min, 2003), HBV infection model (Min, 2006), T cells (Ji, 2008), and so on.
HIV, the Human Immunodeficiency Virus, is the etiological agent of Acquired Immune Deficiency Syndrome (AIDS).Deaths due to AIDS are more than two million people per year recently, making it one of the destructive epidemics in history and the leading cause of death in the world (WHO, 2009).A number of theoretical studies have focused on the mathematical modeling of HIV/AIDS (Mukandavire, 2006;Martcheva, 2007;Bhunu, 2009).But these models were analyzed simply because of the complexity usually.In (Bhunu, 2009), a two strain (antiretroviral sensitive and resistant) HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of nonlinear differential equations with five variables.
In this paper, the model in (Bhunu, 2009) is mapped into a CNN form and the equilibrium points are calculated and analyzed based on the local activity of CNN with five local state variables and one port (Dong, 2009).The bifurcation of this CNN has been calculated and analyzed.Numerical simulations show that this CNN model may explain some complex phenomena during HIV/AIDS treatments.The quantitative understanding of HIV/AIDS dynamics will make it possible to judge the curative effect of long-time antiretroviral therapy and social influence.

The HIV/AIDS Model and its CNN model
In (Bhunu, 2009), a two strain (antiretroviral sensitive and resistant) HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear differential equations, which classifies people into five classes: susceptible (S ), antiretroviral sensitive HIV infected (I 1 ), AIDS individuals with antiretroviral sensitive HIV (A 1 ), antiretroviral resistant HIV infected (I 2 ) and AIDS individuals with antiretroviral resistant HIV (A 2 ).The formulation of HIV/AIDS model is where, individuals are reproduced into the susceptible class at constant rate a, and u is a natural death rate in each human subgroup.The antiretroviral sensitive HIV infected becomes AIDS individuals with antiretroviral sensitive HIV (I 1 → A 1 ) at rate q 1 .Individuals in A 1 have an additional AIDS-induced death rate d 1 , and are given antiretroviral therapy at rate e.A proportion y of individuals in A 1 given antiretroviral therapy respond well and move into I 1 , and the complimentary proportion 1 − y develops resistance and enters A 2 .The antiretroviral resistant HIV infected becomes AIDS individuals with antiretroviral resistant HIV (I 2 → A 2 ) at rate q 2 .Individuals in A 2 have an additional AIDS-induced death rate d 2 .The force of infection y i is given as is the product of the effective contact rate and the transmission probability of antiretroviral sensitive and resistant HIV infection per contact, respectively.n > 1 is the modification parameter which models the fact that individuals in the AIDS stage are more infectious than the corresponding HIV infected not yet in the AIDS stage.
The HIV/AIDS CNN model has the form: where

Analysis of Equilibrium Points
Let Eq.(2) be zero (where D 1 = 0) and solve it, we can get the three equilibrium points: where The equilibrium Q 1 , Q 2 , Q 3 stand for the disease-free equilibrium, retroviral resistant HIV strain only equilibrium and both HIV strain coexist equilibrium, respectively.But we cannot find the antiretroviral sensitive HIV strain only equilibrium in chapter 3.2.1 in (Bhunu, 2009).
Obviously, the numbers of the equilibrium points could be greater than zero, then, we can get: where R 1 , R 2 are equivalent to " R 1 , R 2 " in chapter 3.2.1 in (Bhunu, 2009)(See Page 366).
Consequently, the Jacobian matrix at the equilibrium points Q i , i = 1, 2, 3 is where

The Bifurcation Diagram of Equilibrium Points
Take parameters from Table 1 and let b 1 , b 2 ∈ (0, 1) are variables, we can calculate the bifurcation of the CNN model Eq.( 2) at the equilibrium points Q 1 , Q 2 and Q 3 based on the local activity of CNN with five local state variables and one port (Dong, 2009), see Fig. 2. In Fig. 2, the domains are coded as follows: edge of chaos (locally active and stable) domain (shown red), locally active and unstable domain (shown green) and locally passive domain (shown blue).

Simulations and Analysis
Take parameters from Table 1 and let b 1 , b 2 be different numbers, we model the dynamic trajectories of Eq.( 2) using MATLAB, see Table 2.
From EQ.(3) and Eq.( 4) and take parameters list in Table 1, we can get From Fig. 2, Table 2, and Eq.(3)-Eq.(6), we can conclude: 1.When b 1 , b 2 are both less than 0.0941, b 1 , b 2 is located in the red domain of Fig. 2(a)(the Edge of Chaos domain), max{R 1 , R 2 } < 1, Eq.( 2) converges to the equilibrium point Q 1 (disease-free) neglecting initial values which implies that there is no any HIV infected and AIDS individuals.
2. When b 1 or b 2 is greater than 0.969, Eq.( 2) converges to the equilibrium point Q 2 or Q 3 , which implies that there are HIV infected and AIDS individuals.2) converges to the equilibrium point Q 3 and I 2 > A 2 neglecting initial values which implies that antiretroviral sensitive and resistant HIV infected and AIDS individual coexist and the number of antiretroviral resistant AIDS individual is less than the number of HIV infected.This suggests that the treatment intends to prolong the incubation period only, but does not reduce infectiousness and take more HIV infections, thus may not benefit the community.2) converges to the equilibrium point Q 2 (retroviral resistant HIV strain only equilibrium) and I 2 > A 2 neglecting initial values which implies that there are no antiretroviral sensitive HIV infected and AIDS individual, but antiretroviral resistant HIV infected and AIDS individual still exist.This suggests that the treatment is very effective to antiretroviral sensitive HIV infected and AIDS individual, but the antiretroviral sensitive HIV infected and AIDS individual are more difficult to be treated, which may not benefit the community.
In the following discussions, we select some parameters list in Table 2, the simulation results are shown in Fig. 3 and Fig. 4. Fig. 3 model the convergence to equilibrium point Q 1 , and Fig. 4 model the situation when b 1 = 0.6 and b 2 varies from 0.011 to 0.095.

Conclusion and Future Works
A CNN model of two strain (antiretroviral sensitive and resistant) HIV is established, and then analyzed and simulated based on the local activity of CNN with five local state variables and one port (Dong, 2009).Numerical simulations exhibit that this CNN model may explain some complex phenomena during antiretroviral therapy which make it possible to judge the curative effect of long-time antiretroviral therapy and social influence.
Practically, the dynamic behaviors of HIV/AIDS infection and therapy are very complex and puzzling.Then, more and accurate experimental data are needed for modeling the dynamics of HIV/AIDS.Further research for HIV/AIDS dynamics is promising.More complex the model is, more approximate to the reality, but more difficult to be analyzed.The local activity of CNN provides a possible method for some of them.
All these imply that the local activity of CNN provides a practical tool for the study of the complex dynamics of some coupled nonlinear systems.Especially, it may be expected that CNN will become a promising candidate for modeling life phenomena.
(a) when b 1 > 0.0969 and b 2 is more less than b 1 , b 1 , b 2 are located in the blue domain of Fig.2(c) (the locally passive domain), Eq.(2) converges to the equilibrium point Q 3 (both HIV strain coexist equilibrium) and I 2 < A 2 neglecting initial values which implies that antiretroviral sensitive and resistant HIV infected and AIDS individual coexist and the number of antiretroviral resistant AIDS individual is larger than the number of HIV infected.This suggests that the treatment intends to help the community by lengthening the lives of AIDS patients and reducing HIV/AIDS mortality and hence reducing the number of orphans in affected communities.(b) when b 1 > 0.0969, and b 2 < b 1 but near b 1 , b 1 , b 2 are located in the red domain of Fig.2(c) (the Edge of Chaos domain), Eq.( (c) When b 1 > 0.0969 and b 2 ≥ b 1 , b 1 , b 2 are located in the red or blue but near red domain of Fig.2(b), Eq.(

Figure 1 .
Figure 1.Structure of the HIV/AIDS model

Table 2 .
Cell parameters and corresponding dynamic properties of the CNN of HIV/AIDS model