STABILITY ANALYSIS OF A FRACTIONAL-ORDER MODEL FOR HIV INFECTION OF CD4+T CELLS WITH TREATMENT

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1 STABILITY ANALYSIS OF A FRACTIONAL-ORDER MODEL FOR HIV INFECTION OF CD4+T CELLS WITH TREATMENT Alberto Ferrari, Eduardo Santillan Marcus Summer School on Fractional and Other Nonlocal Models Bilbao, May 28-31, 2018 Facultad de Ciencias Exactas, Ingeniería y Agrimensura (UNR) CONICET, Rosario, Argentina

2 OBJECTIVES To consider a model that represents how HIV acts in a human body, with the incorporation of the presence of a drug, allowing to estimate the number of healthy and infected cells and the viral load over time. To verify the existence and uniqueness of the problem, and the nonnegativity of the solution. To study if the model is positively invariant, that is, if the solution exhibits strictly positive values given similar initial conditions. To determine the equilibrium points of the model and to study sufficient conditions for the non-infection status to be asymptotically stable.

3 The dynamics of HIV WHAT IS HIV? The Human Immunodeficiency Virus (HIV) is a microorganism that, in order to replicate itself, it must penetrate certain types of cells, progressively destroying them if left untreated. The preferred target of HIV is the T-lymphocytes CD4 +, which are the most abundant white blood cells in the immune system, thereby weakening the body s defenses. The antiretroviral therapy (ART) consists of the use of drugs (generally, three or more) to effectively prevent the reproduction of the virus.

4 The dynamics of HIV MODEL VARIABLES At first, three variables could be considered in the model: Uninfected CD4 + T-cells (or healthy) Infected CD4 + T-cells Viral load

5 The dynamics of HIV MODEL VARIABLES At first, three variables could be considered in the model. Uninfected CD4 + T-cells (or healthy) T Infected CD4 + T-cells Infected CD4 + T-cells before reverse transcription (and therefore do not produce virus) I Infected CD4 + T-cells after reverse transcription (and therefore they are capable of producing virus) V Viral load L

6 The dynamics of HIV MATHEMATICAL MODEL The mathematical model considered for the dynamics of HIV with the presence of a reverse transcriptase (RT) inhibitor is [Arafa, Rida, Khalil (2014)]: D α c (T) = s klt µt+(ηε+ b)i D α c (I) = klt (µ 1 + ε+ b)i D α c (V) = (1 η)εi δv D α c (L) = NδV cl (1) VARIABLES T: density of susceptible CD4 + T-cells I: density of infected CD4 + T-cells (pre-rt class) V: density of infected CD4 + T-cells (post-rt class) L: viral load PARAMETERS s: inflows rate of CD4 + T-cells k: interaction infection rate of CD4 + T-cells µ: natural death rate of CD4 + T-cells η: efficacy of RT inhibitor ε: transition rate of pre-rt class b: reverting rate of infected cells µ 1 : death rate of infected CD4 + T-cells δ: death rate of actively infected CD4 + T cells N: total number of viral particles produced by an infected CD4 + T-cell c: clearance rate of the virus

7 Fractional calculus FRACTIONAL CALCULUS The fractional calculation is a generalization of ordinary differentiation and integration to real orders. It goes back to the times when Leibniz and Newton invented differential calculus. APPLICATIONS Models admitting backgrounds of heat transfer Viscoelasticity Electrical circuits Electro-chemistry Economics Polymer physics Biology Etc.

8 Fractional calculus FRACTIONAL CALCULUS Definition: The Riemann-Liouville fractional integral of order α > 0 of a function f :R + R is given by [Diethelm (2010)]: where J 0 f(x)=f(x), x>0. J α f(x)= 1 x (x t) α 1 f(t)dt, Γ(α) 0 Definition: The Caputo fractional derivative of order α > 0 of a continous function f :R + R is given by [Diethelm (2010)]: where m 1<α m, m N. D α c f(x)=j m α (D m f(x)),

9 Previous Results PREVIOUS RESULTS THEOREM 1 (EXISTENCE AND UNIQUENEES OF THE SOLUTION) There exists an unique solution x(t)=(t(t),i(t),v(t),l(t)) of system (1) with t 0. THEOREM 2 (NON-NEGATIVITY OF THE SOLUTION) R 4 + is a non-negative invariant domain for the solution of system (1). LEMMA 1 (EQUILIBRIUM POINTS) There are two equilibrium points of (1): E 1 =( s µ,0,0,0) and E 2 =( T,Ī, V, L) where T = (µ 1+ ε+ b)c NKε(1 η), Ī = s Tµ, ε(1 η)+ µ 1 V = (1 η)εi δ, L= NVδ. c

10 Positive Invariance POSITIVE INVARIANCE Now we will prove that system (1) is positively invariant, that is, it only exhibits strictly positive values given similar initial conditions. LEMMA 2 (BIOLOGICAL CONSISTENCE) Let B ={(T,I,V,L), T > 0, I 0, V 0, L 0} subset of R + 4. If the initial condition (T 0,I 0,V 0,L 0 ) is in B, then T(t)>0 for all time where T,I,V,L are defined.

11 Positive Invariance POSITIVE INVARIANCE Now we will prove that system (1) is positively invariant, that is, it only exhibits strictly positive values given similar initial conditions. LEMMA 2 (BIOLOGICAL CONSISTENCE) Let B ={(T,I,V,L), T > 0, I 0, V 0, L 0} subset of R + 4. If the initial condition (T 0,I 0,V 0,L 0 ) is in B, then T(t)>0 for all time where T,I,V,L are defined. Proof: Suposse that t/t(t)=0, let t the least of them. Then the first equation of (1) in t writes: D α c T(t )=s+(ηε+ b)i(t )>0. Therefore, for t smaller and close enough to t, T( t)<0. This contradicts Theorerm 2 and completes the proof.

12 Positive Invariance POSITIVE INVARIANCE THEOREM 3 (POSITIVE INVARIANCE OF THE SOLUTION) Let B ={(T,I,V,L), T > 0, I > 0, V > 0, L>0}. Then B is a positively invariant domain for the solution of system (1).

13 Positive Invariance POSITIVE INVARIANCE THEOREM 3 (POSITIVE INVARIANCE OF THE SOLUTION) Let B ={(T,I,V,L), T > 0, I > 0, V > 0, L>0}. Then B is a positively invariant domain for the solution of system (1). Proof: By the previous Lemma we already know that T does not vanish. Suppose that I vanishes first and alone at the time t. Then the second equation of system (1) writes: D α c I(t )=kl(t )T(t )>0. This implies that there exists a time t<t such that I( t)<0, contradicting Theorem 2.

14 Positive Invariance POSITIVE INVARIANCE Similar arguments allow to prove that: V does not vanish first and alone. L does not vanish first and alone. I and V do not vanish simultaneously and alone. I and L do not vanish simultaneously and alone. V and L do not vanish simultaneously and alone. It only rests to prove that I, V and L can not vanish simultaneously.

15 Positive Invariance POSITIVE INVARIANCE Similar arguments allow to prove that: V does not vanish first and alone. L does not vanish first and alone. I and V do not vanish simultaneously and alone. I and L do not vanish simultaneously and alone. V and L do not vanish simultaneously and alone. It only rests to prove that I, V and L can not vanish simultaneously. Let us consider the following IVP: { D α c T(t) = s µt(t) T(0) = T 0 Applying the Laplace transform and using Theorema 7.1 of [Diethelm (2010)] we obtain: r α L T(r) r α 1 T 0 = s µl T(r) r

16 Positive Invariance POSITIVE INVARIANCE Therefore: L T(r)= rα 1 r α + µ T s 0+ r α+1 + µr

17 Positive Invariance POSITIVE INVARIANCE Therefore: L T(r)= rα 1 r α + µ T s 0+ r α+1 + µr By Theorem 4.5 of [Diethelm (2010)] we know that: THEOREM ([DIETHELM (2010)]) Let z(t)=e α ( µt α z ) where E α (z)= j is the Mittag-Leffler function j=0 Γ(jα+ 1) of order α > 0. Then L z(r)= rα 1 r α + µ.

18 Positive Invariance POSITIVE INVARIANCE Therefore: L T(r)= rα 1 r α + µ T s 0+ r α+1 + µr By Theorem 4.5 of [Diethelm (2010)] we know that: THEOREM ([DIETHELM (2010)]) Let z(t)=e α ( µt α z ) where E α (z)= j is the Mittag-Leffler function j=0 Γ(jα+ 1) of order α > 0. Then L z(r)= rα 1 r α + µ. If we define w(t)= s, by the differentation theorem for the Laplace transform µ we have: L w(r)l z (r)= s s rα 1 (rl z(r) z(0))= (r µr µr r α + µ 1)= s r α+1 + µr

19 Positive Invariance POSITIVE INVARIANCE By the convolution Theorem for the Laplace transform: t t L w(r)l z (r)=l w(t r)z (r)dr=l s 0 0 µ E α( µr α )( µ)αr α 1 dr= t = sαl E α( µr α )r α 1 dr 0 t L T(r)=sαL E α( µr α )r α 1 dr+ T 0 L E α ( µt α ). 0 t T(t)=sα E α( µr α )r α 1 dr+ T 0 E α ( µt α ). 0

20 Positive Invariance POSITIVE INVARIANCE By the convolution Theorem for the Laplace transform: t t L w(r)l z (r)=l w(t r)z (r)dr=l s 0 0 µ E α( µr α )( µ)αr α 1 dr= t = sαl E α( µr α )r α 1 dr 0 t L T(r)=sαL E α( µr α )r α 1 dr+ T 0 L E α ( µt α ). 0 t T(t)=sα E α( µr α )r α 1 dr+ T 0 E α ( µt α ). 0 If we suppose that there is a time at which the other variables are identically zero and we define t as the lowest time where I, V and L vanish, applying Theorem 1 with reverse time, we conclude that the (unique) solution is such that: t T(t)=sα E α( µr α )r α 1 dr+ T 0 E α ( µt α ) 0 I(t)=V(t)=L(t)=0 for all time t<t. This contradicts the definition of t as the lowest time where I, V and L vanish.

21 Stability analysis STABILITY ANALYSIS THEOREM ([DIETHELM (2010)]) A sufficient condition for the local asymptotic stability of an equilibrium point is that the eigenvalues λ of the Jacobian matrix A=(a ij )= ( f i /x j ) evaluated at that point of equilibrium satisfy the following condition [Diethelm (2010)]: arg(λ) > απ 2. (2) In [Ahmed, Elgazzar (2007)] we have the following Theorem: THEOREM ([AHMED, ELGAZZAR (2007)]) For n=3, if the discriminant of the characteristic polynomial P(λ), D(P)=18a 1 a 2 a 3 +(a 1 a 2 ) 2 4a 3 a 3 1 4a3 2 27a2 3 is positive, then Routh-Hurwitz conditions are the necessary and sufficient conditions for (2) to be satisfied, i.e.: a 1 > 0, a 1 a 2 > a 3, a 3 > 0.

22 Stability analysis STABILITY ANALYSIS THEOREM 4 Let a 1,a 2,b 1,b 2 > 0, β (0,1]. If the following conditions are satisfied simultaneously: a 2 (a a2 2 ) < 1; 3(a 1 a 2 ) 2 + a6 1 9 b 1 ( 9 2 a 1a 2 a b 1 ) { ( ) } > mín 0; a 2 2 a 2 a b2 1 ; b 2 (a b a 1a 2 27 ) { ( ) 2 b 2(1 β) > mín 0; a 2 2 a 2 a ) +a 1 (a a 2 (b 1 + b 2 β)+ 27 ( ) 2 b b b 2β 27 } 4 b2 2 (1 β 2 ). then D(P)=18a 1 a 2 a 3 +(a 1 a 2 ) 2 4a 3 a 3 1 4a3 2 27a2 3 > 0, where a 3 = b 1 b 2 (1 η) for all η [β,1].

23 Stability analysis STABILITY ANALYSIS Now that we know sufficient conditions under which D(P)>0, we analyze the Routh-Hurwitz conditions for the non infectious state E 1, i.e.: a 1 > 0, a 1 a 2 > a 3, a 3 > 0. It is easy to conclude that a 1 > 0 and a 1 a 2 > a 3 are verified for all η [β,1]. Therefore, it should only be verified that a 3 > 0 so that the non-infection state E 1 is asymptotically stable. This condition translates to η > η crit where: η crit = 1 µc(µ 1+ ε+ b) Nεks THEOREM 5 Let system (1). If the conditions given by Theorem 4 and η > 1 µc(µ 1 + ε+ b)/nεks are satisfied, then the non-infectious status E 1 is asymptotically stable.

24 Stability Analysis INTRODUCING VALUES TO THE PARAMETERS According to the biological literature, we consider the following initial conditions and the following parameters: T(0)=300/mm 3 (initial density of susceptible CD4 + T-cells), I(0)=10/mm 3 (initial density of infected CD4 + T-cells (pre-rt class)), V(0)=10/mm 3 (initial density of infected CD4 + T-cells (post-rt class)), L(0)=10/mm 3 (initial viral load). s=10/mm 3 dia (inflows rate of CD4 + T-cells), k=0,000024mm 3 /dia (interaction infection rate of CD4 + T-cells), µ = 0,01/dia (natural death rate of CD4 + T-cells), ε = 0,4/dia (transition rate of pre-rt class), b=0,05/dia (reverting rate of infected cells), µ 1 = 0,015/dia (death rate of infected CD4 + T-cells), δ = 0,26/dia (death rate of actively infected CD4 + T cells), N = 1000 (total number of viral particles produced by an infected CD4 + T-cell), c=2,4/dia (clearance rate of the virus), α = 0, 99 (fractional derivative order)

25 Stability analysis CRITICAL EFFICACY Using the obtained stability results together with the stablished values, the equilibrium point of non-infectious E 1 turns out to be: ( ) s E 1 = µ,0,0,0 =(1000,0,0,0) It is easy to check that the conditions of Theorem 4 are satisfied for β = 0.6 (approximate current efficacy of a RT inhibitor). Finally, referring to Theorem 5, E 1 is asymptotically stable if η > η crit, being: η crit = 0,88375

26 Conclusions CONCLUSIONS A model for HIV infection has been analiyzed along with the immune system, the CD4 + T-lymphocites, where a control parameter has been introduced given by the treatment, where it was considered that a reverse transcriptase inhibitor was provided. A fractional order model was considered, in view of the advantages that this entails. Fundamental solutions of fractional equations exhibit useful properties of adjustment that are attractive to applications. We showed that the solution of this model is positively invariant. This has biological sense since it indicates that a person who lives with the virus will continue infected. After analyzing the stability of the solution, we concluded that if the efficacy of the drug is greater than %, then an undetectable level of virus density will be achieved.

27 Conclusions FUTURE WORK Consider a new model with other drugs different from the RT inhibitor, or combinations of them. Study the stability corresponding to the infectious state E 2. The recent apparition of fractional differential equations in Applied Mathematics makes necessary to investigate more analytical and numerical methods for these equations, in particular to determine (analitically) which is the fractional derivative order α that best suits to reallity. Deep the contacts made with doctors an biochemists to confirm that the values assigned to the parameters are correct and with patient data, determine (empirically) which is the fractional derivative order α that bests suits to reallity.

28 Conclusions Thank you very much for your attention!!

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