NARMA-L2-based Antiviral Therapy for Infected CD4+ T Cells

in a Nonlinear Model for HIV Dynamics:

Protease Inhibitors-based Approach

C. A. Pe˜na Fern´andez

, A. B. B. F. Cunha

and M. A. Alves

Electrotechnical Department at Federal Institute of Bahia, Rod. BR 324, KM 102, Feira de Santana, Brazil

Computer Science Department at Federal Institute of Bahia, Rod. BR 324, KM 102, Feira de Santana, Brazil

Keywords:

HIV, Neural Networks, Dynamic Backpropagation, Protease Inhibitors.

Abstract:

The present paper uses the learning ability of neural networks (NN) to design a nonlinear model and a nonlinear

controller that reduces the number of infected/uninfected CD4+ T cells into the HIV dynamic when an antiviral

therapy based on protease inhibitors is applied. The dynamic of the closed-loop system based on such therapy

is analyzed to understand the stability of infected/uninfected CD4+ T cells according to a global feedback law

that regards un-modeled dynamic terms. To this end, a robust control scheme based on NARMA-L2 approach

and a modiﬁed version of an already existing dynamic backpropagation algorithm is used to improve the

antiviral therapy performance (strongly related to the tracking error). The robustness of the proposed model

shows that antiviral therapy performance guarantees less infected CD4+ T cells.

1 INTRODUCTION

The study on the behavior of sensitive and resis-

tant cells infected by AIDS, a communicable disease

caused by the human immunodeﬁciency virus (HIV),

by using retroviral drugs and the study of responses

of these cells to certain treatments have been a high-

light in last years (Althaus and Boer, 2011; Hattaf

et al., 2009; Luo et al., 2012; Magnus and Regoes,

2011; Roy and Wodarz, 2012; Wang et al., 2014; Wil-

son, 2012; Wodarz and Hamer, 2007; Wodarz and

Hamer, 2007). The treatment for HIV/AIDS relies

on anti-retroviral drugs that suppress HIV viral load

below the limit of detection. Some drug classes are

reverse transcriptase inhibitors (RTI) and protease in-

hibitors (PIs). HIV is a deadly disease in a lack of

treatment. Time since the initial infection till death

in an untreated patient is 9-10 years. HIV infec-

tion dilapidates the patients’ health since it attacks

three extremely important cell populations, the CD4+

T cells, macrophages, and dendritic cells. In recent

works, the main goal has been to understand the be-

havior of an infected CD4+ T cells population or

the rate of infected CD4+ T cells when therapy is

applied (Pinto and Carvalho, 2015b; Pinto and Car-

valho, 2015a; Gumel et al., 2001; Karrakchou et al.,

https://orcid.org/0000-0003-0934-5761

2006; Min et al., 2008; Allali et al., 2018; Loudon and

Pankavich, 2016; Korpusik, 2017; Wang et al., 2019).

However, the correctness of any HIV model depends

on suitable approaches whereas by using suitable pa-

rameters for deterministic models or by ﬁnding a suit-

able performance for learning algorithms in compu-

tational methods (Wang et al., 2019). Nevertheless,

the adjusting process of the parameters and the per-

formance of the computational model yields a big

complexity on the better way to ﬁnd the optimal solu-

tion in nonlinear programming problems. Such com-

plexity can be deﬁned in terms of the constraints for

stochastic models as well as choosing the right parti-

cles to model (Liu et al., 2019; Pandit and Boer, 2019;

Cheng et al., 2019; Shi and Dong, 2019). Artiﬁcial in-

telligence (AI) techniques have been used extensively

in numerous approaches to model HIV around sev-

eral CD4+ T cells (Xiang et al., 2019; Zhang et al.,

2019; Wang et al., 2019; Gupta et al., 2019). Amongst

AI numerous techniques, the Artiﬁcial Neural Net-

works (ANN) has dominated in its capability and ap-

plicability in different ﬁelds (Cybenko, 1989). In

the work reported here, it will be presented an ap-

proach to model the behavior of uninfected/infected

CD4+ T cells based on NARMA approach, which

will be implemented with neural networks and known

as NARMA-L2. In this approach, the performance

Fernández, C., Cunha, A. and Alves, M.

NARMA-L2-based Antiviral Therapy for Infected CD4+ T Cells in a Nonlinear Model for HIV Dynamics: Protease Inhibitors-based Approach.

DOI: 10.5220/0008980606750683

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 2, pages 675-683

ISBN: 978-989-758-395-7; ISSN: 2184-433X

675

of the dynamic model is mainly composed by using

an improved learning rule in neural networks based

on sigmoid-functions, as activation function, and the

dynamic backpropagation of the training error. The

controller developed on this approach uses the in-

verse model principle to synthesize an antiviral ther-

apy based on PI whose adjusting method updates the

value of weights of the neurons that model the HIV

dynamics. Next, it is hoped that a good adjusting

of those values implies a better efﬁciency of antiviral

therapy according to the response of resistant CD4+

T cells.

This paper presents the proposal as follows: in

Section 2 will be reviewed the stochastic model based

on the approach in (Wang et al., 2014). It will be

assumed that all antiviral therapy based on PI can

be decomposed in decoupled inputs such that the in-

fected/uninfected CD4+ T cells have independent re-

sponses. In order to impose a performance on the

antiviral therapy and the decreasing/increasing of in-

fected/uninfected CD4+ T cells, an optimal control

law based on the NARMA-L2 approach will be pro-

posed. To this end, in Section 3 will be synthesized

a control scheme with the aid of NNs and suitable

learning algorithms. In Section 4 are shown differ-

ent parametric structures for the NARMA-L2 model

by means different learning rules in order to adjust the

synaptic weights of NNs into NARMA-L2 scheme as

well as the performance of the antiviral therapy repre-

sented by the controller. Finally, closing remarks are

made in Section 5.

2 THEORETICAL BACKGROUND

ON HIV

As well known, a deterministic model can be based

on a system of ordinary differential equations that

represent the dynamic of uninfected CD4+ T cells,

drug-sensitive infectious virus, drug-resistant infec-

tious virus, drug-sensitive infected CD4+ T cells,

and drug-resistant infected CD4+ T cells. In (Wang

et al., 2014) such dynamics is deﬁned by ODE system

(1a)-(1e), where T, V

, V

, T

and T

represent unin-

fected CD4+ T cells, drug-sensitive infectious virus,

drug-resistant infectious virus, drug-sensitiveinfected

CD4+ T cells, and drug-resistant infected CD4+ T

cells, respectively. According to the epidemiology of

the disease, the uninfected CD4+ T cells (T) are pro-

duced at a rate of λ and die at a rate of d. These cells,

when in contact with HIV, are infected at a rate k

, by

drug-sensitive viruses and move τ time units later, to

the T

class. Furthermore, T cells may be infected at

a rate of k

by drug-resistant viruses and move to the

class after τ time units later. The term e

−mτ

is the

probability of T

and T

cells surviving in the inter-

val τ, where

represents the average life of infected

CD4+ T cells before they become actively productive.

The terms 1− n

and 1− n

represent the proportion

of T

and T

cells eliminated by reverse transcriptase

inhibitors (RTIs). The infected CD4+ T cells and the

virus particles die at rates δ and c, respectively. TheV

and V

particles are produced by the infected CD4+ T

cell populations with bursting sizes of drug-sensitive

strain (N

) and of drug-resistant strain (N

). Through-

out the infection, a proportion s (0 < s < 1) of T

cells

can become T

cells. The proportion of virus particles

that are not eliminated by PIs is represented by 1− n

and 1 − n

, where n

is the efﬁcacy of PIs for wild

type strain and n

is the efﬁcacy of PIs for mutants.

According to (Pinto and Carvalho, 2015b), on

stochastic approach for model (1a)-(1e), it is believed

that the death rates d, δ and c can be substituted

by random parameters, more speciﬁcally, d + σ

δ+ σ

and c+ σ

, respectively, where B

(t) and

(t) are independent standard wiener processes. In

the next sections, such wiener processes will be used

as exogenous variables into the learning algorithms.

2.1 HIV Model by Including PI

By assuming that the constant recruitment number of

new uninfected cells T and that the rate of infection

of CD4+ T cells k

by a free virus has been saturated

probably because of overcrowding of free virus the

model (1a)-(1e) can be rewritten as

h = G(h) (2)

where h = [T T

]

∈ IR

is the state vector

and G(h) = [G

(h) G

(h)]

∈ IR

is assumed continuouslydifferentiable on the parame-

ters (T,T

,t) ∈ D

×D

[0,t], being D

⊂ IR

open and

convex sets, and each G

(h), for i = 1,.. .,5, repre-

sents the right-hand side of (1a)-(1e), respectively.

Now, let p = (p

, p

) be an equilibrium

point of the nonlinear system (1a)-(1e) and suppose

that the functions G

, for i = 1, ...,5, are continuously

differentiable. Expanding the right-hand side of (2)

into its Taylor series on the point p yields

h = M(p)h+ ∆(p)u + N(p) (3)

where M(p) is the Jacobian matrix at equilibrium

point p, deﬁned by M(p) = [∂G(h)/∂h

] (for i =

1,.. . ,5), u ∈ IR

is the antiviral therapy deﬁned by

u = [n

]

, ∆(p) is the input matrix with full

rank, deﬁned by ∆(p) = [∂G(h)/∂u

] (for i = 1,2)

and N(p) = N(h)|

h=p

∈ IR

represents the higher-

order terms of expansion. In the antiviral therapy,

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

676

= λ − dT(t) − K

(1− n

(t)T(t) − k

(1− n

(t)T(t) (1a)

= (1− s)k

−mτ

(1− n

(t − τ)T(t − τ) − δT

(t) (1b)

= N

δ(1− n

(t) − cV

(t) (1c)

= sk

−mτ

(1− n

(t − τ)T(t − τ) + k

−mτ

(1− n

(t − τ)T(t − τ) − δT

(t) (1d)

= N

δ(1− n

(t) − cV

(t) (1e)

Figure 1.

the efﬁcacy of PI, n

and n

, are considered bounded

Lebesgue integrable functions, i.e.,

∞

∑

j=1

p, j

< ∞ and

∞

∑

j=1

p, j

< ∞.

If, for instance, n

= 1, the blockage is 100% effec-

tive. On the other hand, if n

= 0, there is no block-

age.

The term N(h)|

h=p

is associated with an un-

modeled dynamic and it has an important role to mea-

sure the performance of the alternative model based

on neural networks, which will be shown in the next

sections.

2.2 Analysis on Inner Close-loop

In the work reported here, it will be assumed that

there exists an initial quantity of uninfected CD4+ T

cells, T

∗

> 0, such that the antiviral therapy guaran-

tees a number of uninfected CD4+ T cells close to T

∗

From (3), if T

∗

> δ

(t) > 0, being δ

(t) a known 1-

diffeomorphism, and h is assumed to be the control

input of the inner close-loop system associated to (3)

then the error converges exponentially to small ball

in W containing the origin whose radius will be ad-

justed by any known constant. The following lemma

explains this idea:

Lemma 1. Let e be the error associated with in-

ner close-loop system deﬁned by e = Π

(h), where

(h) ∈ IR

is a known matrix. If T

∗

> δ

(t) > 0 and

h = ρ, then e converges exponentially to a small ball

containing the origin whose radius can be adjusted

by a k-th known constant M

∗

< 0, for k = 1,... , 5,

∀t ≥ 0, being ρ = [T

∗

···]

= Π

(e) ∈ IR

such that

(e) < M and

M = [M

∗

,... ,M

∗

]

being M

∗

< 0 a known constant.

Proof: Let the Lyapunov function

V =

∑

i=2

∑

i=2



(h)



differentiating it along the close-loop deﬁned by e =

(h), h = ρ and ρ = Π

(e) one obtains

V =

∑

i=2

∑

k=1

∂Π

∂h

(h)

∑

i=2

∑

k=1

∂Π

∂h

(e)

(h)

∑

i=2

∑

k=2

∂Π

∂h

(e)Π

(h) +

∂Π

∂h

(h)

∑

i=2

∑

k=2

∂Π

∂h

(e)Π

(h)

= −

∑

i=2

∑

k=2

∂Π

∂h

∗

(h)

< 0.

where M

∗

is the k-th element of M, Π

(h) and Π

(e)

are the i-th and k-th elements of Π

(h) and Π

(e),

respectively.

Noting T

∗

≥ δ

(t) > 0 (i.e.,

(t) < 0 ) it can be

seen that V exponentially converges to a small ball

containing the origin. The convergence rate is at least

deﬁned by Π

(h). The radius of the ball can be ad-

justed by a constant M

∗

, for k = 1, ...,5. Therefore,

e converges exponentially to the small ball contain-

ing the origin, and the radius of the ball can also be

adjusted by any constant M

∗

. This ends the proof .

So, the control problem considered in this paper is

to ﬁnd an improved version of a global feedback con-

troller that represents the antiviral therapy such that

the inner close-loop of system (3) guarantees that the

number of uninfected CD4+ T cells remains close to

∗

. To this end, in the next sections, it will be used

neural networks (NNs) to adjust this optimal global

feedback law.

NARMA-L2-based Antiviral Therapy for Infected CD4+ T Cells in a Nonlinear Model for HIV Dynamics: Protease Inhibitors-based

Approach

677

3 NARMA-L2-BASED THERAPY

The optimal control problem will an objective func-

tional, a set of state variables h and a set of control

variables u in time t, 0 ≤ t ≤ t

such that there is a

generic global feedback control u = λ

(h,h

) where

is the target response, λ

: IR

× IR

→ IR

is a

known nonlinear function such that there is an opti-

mal control law u , λ

∗

that satisﬁes J(λ

∗

(h,h

)) =

max

∗

(h,h

)∈U

J(u) being U ⊆ IR

and

J(u) =



T(t) − u



for t

> 0 and any symmetric deﬁnite positive ma-

trix R. Our goal is to seek to maximize the ob-

jective functional J(u) by increasing the population

of the uninfected CD4+ T cells, reducing viral load,

and minimizing the cost treatment. To this end, the

nonlinear model (1a)-(1e) will be designed by using

a robust adaptive neural network (NN) with the aid

of NARMA-L2 approach and its learning ability. In

this way, the maximization problem will be associated

with the nonlinear programming problem restricted to

the learning rule of NNs.

Let

h = h− ρ be the error between the state vec-

tor h and the state vector deﬁned by Lemma 1 on the

close-loop system. Thus, for

h one obtains

h+ M(p)

h = ∆(p)u− Γ(p,ρ) (4)

where

Γ(p,ρ) ,

ρ+ M(p)ρ + N(p).

Noting the learning ability of NN, the term Γ(p, h)

can be approximated on-line by using NNs whose ac-

tivation function is a sigmoid. Let σ(p,h) ∈ IR

∗

a vector of continuous sigmoid functions, according

to the approximation property of neural networks in

(Cybenko, 1989), if Γ(p,ρ) is a continuous function

of p and ρ then

Γ(p,ρ) = ψ

σ(p,ρ) + ε

(p,ρ), (5)

where ψ , {ψ

} ∈ IR

∗

×3

is an unknown optimal

constant weight vector, previously calculated, and de-

ﬁned by

ψ = arg min

ς∈IR

∗

×3



max

(p,ρ)∈Ω

kΓ(p,ρ) − ς

σ(p,ρ)k



where Ω is a compact set and ε

is the error associated

with reconstruction of the optimal weight vector. The

approximations results for NN indicate that if m

∗

sufﬁciently large then the reconstruction error can be

arbitrarily small on Ω. In this way, it will assumed

that kε

(p,ρ)k ≤ ϕ

∗

, for ∀(p,ρ) ∈ Ω, where ϕ

∗

is an

unknown constant.

TDL

...

TDL

NN I

NN II

...

(x5)

E(ψ)

∗

h = M(p)h

+∆(p)u+ N(p)

Figure 2: Closed-loop system by using NARMA-L2 ap-

proach in antiviral therapy.

Without loss of generality, by using (5), the equa-

tion (4) can be rewritten as

h+ M(p)

h = ∆(p)u− ψ

σ(p,ρ) − ε

(p,ρ)

s.t. kε

(p,ρ)k ≤ ϕ

∗

(6)

where ψ and ϕ

∗

must be estimated in order to guaran-

tee the robustness of the model (4). Thus, let

ψ and

∗

be the estimates of ψ and ϕ

∗

, respectively. With

aid of Lemma 1 the following theorem is obtained:

Theorem 1. Assuming T

∗

> δ

(t) > 0, the global

feedback law

u = ∆

−1

(p)



−K

σ−

∗



, (7)

guarantee that e,

h, (

ψ−ψ) and (

∗

−ϕ

∗

) converge to

a ball containing the origin whose radius can be made

as small as possible by selecting a suitable value for

, γ

, δ

, ψ

, and ϕ

, where

ψ = −γ

− δ

(

ψ− ψ

) (8)

∗

= γ

h) − δ

(

∗

− ϕ

) (9)

are update laws, K

is a positive deﬁnite constant ma-

trix, T(

h) is a known sigmoid function and γ

, δ

(for

i = 1,2) are positive constants.

Proof: Let the Lyapunov function

V =

∑

i=2

−1

(

ψ− ψ)

(

ψ− ψ)

−1

(

∗

− ϕ

∗

)

and differentiating it along of the closed-loop repre-

sented by Lemma 1 and (7), one obtains

V =

∑

i=2

∑

k=1

∂Π

∂h

+ γ

−1

(

ψ− ψ)

ψ+ γ

−1

(

∗

− ϕ

∗

)

∗

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

678

0 10 20 30 40

-40

-20

BFGS

0 10 20 30 40

-500

500

1000

1500

2000

0 10 20 30 40

-1500

-1000

-500

500

1000

1500

2000

0 10 20 30 40

-2000

-1000

1000

2000

0 10 20 30 40

-2000

-1500

-1000

-500

500

E(ψ) for T

E(ψ) for V

t (s)

t (s)t (s) t (s)

t (s)

Figure 3: Training error for B

(t), E(ψ), of NN II by using three algorithms: Levenberg-Marquardt (LM), gradient descent

(GD), Broyden-Fletcher-Goldfarb-Shanno (BFGS).

0 10 20 30 40

-2

-1

0 10 20 30 40

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0 10 20 30 40

-0.6

-0.4

-0.2

0.2

0.4

0.6

0 10 20 30 40

-1

-0.5

0.5

0 10 20 30 40

-40

-20

100

E(ψ) for T

E(ψ) for V

t (s)

Figure 4: Training error for B

(t), E(ψ), of NN II by using three algorithms: Levenberg-Marquardt (LM), gradient descent

(GD), Broyden-Fletcher-Goldfarb-Shanno (BFGS).

or, by using (7) into (4),

V =

∑

i=2

∑

k=1

∂Π

∂h



−K

σ−

∗

− Γ(p,ρ) + M(p)



+ γ

−1

(

ψ− ψ)

+ γ

−1

(

∗

− ϕ

∗

)

∗

< −c

V + c

where c

is a positive constant which depends on K

, γ

, δ

, and

−1

(

ψ− ψ

)

(

ψ− ψ

)

−1

(

∗

− ϕ

)

+ 2b

∗

being b

∗

a constant which satisﬁes b

∗

= e

−(b

∗

+1)

[i.e.,

∗

= 0.2785]. Without loss of generality, it can be

noted that V converges exponentially to a small ball

containing the origin and the radius of the ball de-

pends on γ

, γ

, r, δ

, δ

, ψ

and ϕ

. Therefore, e,

(

ψ− ψ) and (

∗

− ϕ

∗

) will converge exponentially to

the small ball of the origin. This ends the proof .

The NN-based controller (7) uses the update laws

(8)-(9) to approximate un-modeled dynamics. In this

way, the laws (8)-(9) represent the learning ability of

the NN responsible to optimize the synaptic weights

(

σ) that indirectly will ensure that the reconstruc-

tion and tracking errors converge toward a small ball

containing the origin in the sets Ω and W . Therefore,

in the work reported here, the global feedback law in

0 10 20 30 40

× 10

-4

-2

0 5 10 15 20 25 30 35 40 45

35 40 45

0.1

0.2

t (s)

h for T

k+2

for T

Figure 5: Validation of NARMA-L2 model by comparing

(y) of NN II with output (h) of model deﬁned by (3), with

exogenous brownian noise B

(t).

Theorem 1 will be based on an alternative learning

rule that combines the laws (8)-(9) with an already

existing algorithm for dynamic backpropagation.

3.1 NARMA-L2 Training Algorithm

The NARMA-L2 scheme to design the antiviral ther-

apy u is based on two NNs, the NN II is composed

of ﬁve NNs and it will be used to model the system

(3). The NN I will be used as antiviral therapy by

NARMA-L2-based Antiviral Therapy for Infected CD4+ T Cells in a Nonlinear Model for HIV Dynamics: Protease Inhibitors-based

Approach

679

5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

-0.5

0.5

1.5

t (s)

h for T

k+2

for T

Figure 6: Validation of NARMA-L2 model by comparing

(y) of NN II with output (h) of model deﬁned by (3), with

exogenous brownian noise B

(t).

means of the global feedback law in Theorem 1 (see

Fig. 2). The input T

∗

represents a reference for T(t),

and with this last one the main objective of the pro-

posal reported here is to ﬁnd an antiviral therapy that

guarantees a better decreasing of drug-sensitive infec-

tious virus V

, drug-resistant infectious virus V

and

drug-resistant infected CD4+ T cells T

In order to verify the effectiveness of the

NARMA-L2, the NN II is trained by using three algo-

rithms: Levenberg-Marquardt, Gradient Descent, and

BFGS

. All these, based on the error backpropagation

principle for off-line mode. It will be chosen the al-

gorithm more suitable to model each state variable of

the h vectorin order to guarantee a better performance

between the system and the controller represented by

NN I. On the other hand, the NN I is trained by us-

ing only dynamic backpropagation and to this end,

the BFGS approach is modiﬁed according to feedback

law in Theorem 1.

Since the inputs associated with PIs were assumed

to be mutually decoupled, the NARMA-L2 model

was designed individually for each state variable. A

standard NARMA-L2 model can be deﬁned by

k+2

[TDL(y

,n),TDL(u

,n)]

+ ¯g

[TDL(y

,n),TDL(u

,n)]u

where TDL(p

,n) , p

,... , p

k−n+1

represents a

tapped delay line with n regressors (see TDL blocks in

Fig. 2);

F and ¯g

, ∂

F/∂u

are evaluated around

TDL(y

,n), TDL(u

,n) (with u

= 0) for any nonlin-

ear function

F(·) (Narendra and Parthasarathy, 1990).

Broyden-Fletcher-Goldfarb-Shanno.

To guarantee the laws

ψ and

∗

, a segmented

training based on BFGS algorithm will be used to

compute the synaptic weights of NN I from a mod-

eling error between the system (3) and NN II. Now,

the learning rule proposed here can be described by

using kε

(p,ρ)k , kε

(ψ)k as goal function accord-

ing to next steps:

s1. Init B

= I, kε

(ψ

)k = n

and

∗

= n

where B

is the initial condition for a Hessian ma-

trix of ε

(ψ), n

≥ 100 and n

< 0.001. From (9),

∗

= n

then

∗

h) +

thus the BFGS algorithm becomes as

s2. While kε

(ψ

)k >

h) +

s3. ∆ψ

= −α

−1

∇ε

(ψ

);

s4. Compute ∇ε

(ψ

k+1

);

where ∇ε

(ψ

) represents the gradient descent as

ﬁrst-order iterative optimization tool. So,

s5. E

(ψ) = ∇ε

(ψ

k+1

) − ∇ε

(ψ

);

s6. β

= I − E

∆ψ

;

Next, from step (s3.) and (8), by using the pseu-

doinverse of σ(p,h), σ

(p,h) yields

s7.

= −



(ψ

− ψ

) − α

−1

∇ε

(ψ

)



;

s8. B

−1

k+1

=β

−1

+ ∆ψ

∆ψ

;

s9. ψ

k+1

= ψ

+ ∆ψ

;

s10. Compute kε

(ψ

)k and go to step (s2.);

It can be seen that the term −K

h is used to make

h converge to ρ while the term −

∗

h) is used to

compensate the reconstruction error kε

(ψ

)k of NN

I. The term

σ in (7) uses a dynamic learning rule

based on laws (8)-(9) and BFGS algorithm to adjust

the synaptic weights of NN I and to enhance the ro-

bustness of the un-modeled dynamics N(p).

4 SIMULATION

The simulation tests were done by using the nonlinear

NARMA-L2-based controller with a two-layer NN in

order to implement (7) (in Theorem 1). All this, when

the antiviral therapy is applied with initial uninfected

CD4+ T cells.

The NARMA-L2 model was set with n = 3 re-

gressors (in TDL blocks), 20 iterations at BFGS al-

gorithm and 30 hidden neurons at the middle layer

of NNI. In previous simulation tests, it was noted

that when the number of hidden neurons is less than

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

680

t (s)

(t)

Figure 7: Validation of the drug-sensitive infectious virus

based on the global feedback law (7), in Theorem 1,

by comparing the behavior of each variable with control u

(solid line) and without control u (dash line).

t (s)

(t)

Figure 8: Validation of the drug-resistant infectious virus

based on the global feedback law (7), in Theorem 1,

by comparing the behavior of each variable with control u

(solid line) and without control u (dash line).

30 the weights were adjusted to values that were not

enough robust. In order to train the NNI, it was con-

sidered that the antiviral therapy u represents 10% at

PIs efﬁcacy, i.e., n

= 0.1 and n

= 0.1 for ∀ t > 0.

Complementary, since the signals B

(t), for i = 1, 2,

have independent increments for ∀t > 0, the future

increments of the death rates are independent of the

past values. So, such signals were set as pseudo-

random binary signals (PRBS) with maxim ampli-

tude 1 and samples between 10 and 100 ms. The

parameters of model (1a)-(1e) were set as in (Wang

et al., 2014; Pinto and Carvalho, 2015b): λ = 100;

d = 0.1; m = 0.01; k

= 2.4× 10

−6

; k

= 2× 10

−6

;

s = 3× 10

−5

; δ = 1; N

= 4800; N

= 4000; c = 23;

τ = 1; n

= 0.4; n

= 0.2; σ

= σ

= 0.1; with

initial conditions T(0) = T

∗

= 1000, T

(0) = 1 and

t (s)

(t)

Figure 9: Validation of the drug-resistant infected CD4+ T

cells T

based on the global feedback law (7), in Theorem

1, by comparing the behavior of each variable with control

u (solid line) and without control u (dash line).

t (s)

T(t)

Figure 10: Validation of the uninfected CD4+ T cells T

based on the global feedback law (7), in Theorem 1, by

comparing the behavior of each variable with control u

(solid line) and without control u (dash line).

(0) = V

(0) = 0.01. In order to set the con-

troller, the parameters of nonlinear law (7) were set

as K

= γ

= k

= 1, δ

= δ

= 0.01 and

h) = tanh(

h).

In Fig. 5 and Fig. 6 can be seen the behavior of

variablesV

and T

modeled by NN II on NARMA-

L2 approach by considering B

(t), for i = 1,2, as

Brownian signals. Complementarily, in Fig. 3 and

Fig. 4 can be seen the adjustment error of ﬁve sig-

nals associated with the NN II. For each NN was used

three algorithms based on the error backpropagation

principle. It can be noted that the BFGS algorithm

has a better approximation than Levenberg-Marquardt

(LM) and Gradient Descent (GD) approaches, except

for the behavior of T. For this reason, it will be

used the BFGS approach to model the responses of

NARMA-L2-based Antiviral Therapy for Infected CD4+ T Cells in a Nonlinear Model for HIV Dynamics: Protease Inhibitors-based

Approach

681

the variables T

, T

, V

and V

, while the Levenberg-

Marquardt algorithm will be used to model the re-

sponse of T.

4.1 Validation of the Antiviral Therapy

Since the main objective is to reduce drug-resistant in-

fected CD4+ T cells in order to guarantee T close to

∗

, then the global feedback law (7) will use the abil-

ity of NNs to approximate nonlinear and un-modeled

behaviors (i.e., N(p)) in order to improve the perfor-

mance of the antiviral therapy. To this end, the signals

, for i = 1,2, will be set as exogenous inputs to the

closed-loop of (2) and the input u will be represented

by the outputs of the NARMA-L2-based controller.

By comparing the Fig. 7 and Fig. 8 can be seen

that the drug-sensitive infectious virus (V

) decreases

faster than the drug-resistant infectious virus (V

)

when the antiviral therapy u is applied during 100

days. It can be noted that the antiviral therapy guaran-

tees a decreasing of the contagion effect of the HIV,

see that T

decreases signiﬁcantly on time domain in-

stead of the case when the antiviral therapy is not used

(see dash lines in Fig. 8).

Although the behavior of T was model by using

the LM approach in NN II, Fig. 10 shows that the

control law (7) guarantees that the concentration of

uninfected CD4+T cells decreasing close to a neigh-

borhood of T

∗

(= T(0) = 1000). Without treatments

(without control u), the number of uninfected cells de-

creases drastically.

5 FINAL REMARKS

The controller in Theorem 1 differs from other con-

tributions mainly because the optimization approach

used to maximize any objective functional J(u) is

based on a nonlinear programming rule. Never-

theless, here was chosen the use of NARMA-L2

approach together with nonlinear programming for

training NN because it improves the performance of

the model and the controller used as antiviral ther-

apy, both reducing viral load, and minimizing the cost

treatment. Here, it was assumed that un-modeled in-

formation could be related to the learning ability of

the NNs as well as it was proposed an alternative

learning technique based on the dynamic of the back-

propagation approach. In this way, the robustness of

the NN-based controller (7) was improved by includ-

ing the nonlinear term −

∗

h) into the nonlinear

control law (see (7) in Theorem 1 and step (s2.)) such

that the convergence radius of the tracking error can

be regulated by known parameters. It is hoped that in

future works the NN-based controller can be designed

by using three or more layers to improve the perfor-

mance of the term

σ(q

∗

,h) and the compensation

of the un-modeled dynamics N(p).

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