Stability analysis of a novel Delay Differential Equation of HIV Infection of CD4^+ T-cells
SS TABILITY ANALYSIS OF A NOVEL DELAY DIFFERENTIALEQUATION MODEL OF
HIV
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CD4 + T- CELLS A PREPRINT
Hoang Anh NGO
LIX, École PolytechniqueInstitut Polytechnique de Paris91120 Palaiseau, France
Hung Dang NGUYEN
Vietnamese - German Faculty of Medicine (VGFM)Pham Ngoc Thach University of MedicineHo Chi Minh City 710000, Vietnam›
Mehmet Dik ∗ Department of Mathematics and Computer ScienceBeloit CollegeWI 53511, United StatesJanuary 5, 2021 A BSTRACT
In this paper, we investigate a novel 3-compartment model of HIV infection of CD4 + T-cells witha mass action term by including two versions: one baseline ODE model and one delay-differentialequation (DDE) model with a constant discrete time delay.Similar to various endemic models, the dynamics within the ODE model is fully determined by thebasic reproduction term R . If R < , the disease-free (zero) equilibrium will be asymptoticallystable and the disease gradually dies out. On the other hand, if R > , there exists a positiveequilibrium that is globally/orbitally asymptotically stable within the interior of a predefined region.To present the incubation time of the virus, a constant delay term τ is added, forming a DDE model.In this model, this time delay (of the transmission between virus and healthy cells) can destabilize thesystem, arising periodic solutions through Hopf bifurcation.Finally, numerical simulations are conducted to illustrate and verify the results. K eywords HIV · Globally asymptotical stability · Periodic solution · Delay term · Steady state
In the field of epidemiology, although our knowledge of viral dynamics and virus-specific immmune responses has notfully developed, numerous mathematical models have been developed an investigated to describe the immunologicalresponse to HIV infection (for example, [5, 14, 8, 28, 54, 43] and references therein). The models have been usedto explain different phenomena within the host body, and by directly applying the models to real clinical data, theycan also predicts estimates of many measures, including the death rate of productively infected cells, the rate of viralclearance or the viral production rate.These simple HIV models have played an essential role in providing a better understanding in the dynamics of thisinfectious diseases, while providing very important biological meanings for the (combined) drug therapies used againstit. For more references and detailed meta mathematical analysis on these models in general, we can refer to surveypapers written by Kirschner, 1996 [29] or Perelson and Nelson, 1999 [58] ∗ Corresponding author, Email address: [email protected] a r X i v : . [ q - b i o . P E ] J a n TABILITY ANALYSIS OF A NOVEL
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5, 2021The simplest HIV model, only considering the dynamics of the virus concentration, is dVdt = P − cV (1)where • P is an unknown function representing the rate of production of the virus,• V is the virus concentration.The dynamics of the population of target cells (CD4 + T-cells for HIV or hepatic cells for HBV and HCV) is still notfully understood. Nevertheless, a reasonable, simple model for this population of cells, which can be extended furtherin various models, is dTdt = s − dT + aT (cid:18) − TT max (cid:19) (2)with • s representing the rate at which new T-cells are created from sources within the body, such as the thymus, orfrom the proliferation of existing T-cells,• d being the death rate per T-cells,• a is the maximum proliferation rate of target T-cells, when the proliferation is represented by a logistic function,and• T max is the population density of T-cells at which proliferation shuts off.Human immunodeficiency virus, or HIV, is a virus belonging to the genus Lentivirus, part of the family Retroviridae[71]. It has an outer envelope of lipid and viral proteins, which encloses its core. The virion core contains twopositive-sense single-stranded RNA and the enzyme reverse transcriptase, an RNA-dependent DNA polymerase.HIV, like most viruses, cannot reproduce by itself. Therefore, they require a host cell and its materials to replicate.For HIV, it infects a variety of immune cells, including helper T cells, lymphocytes, monocytes, and dendritic cellsby attaching to a specific receptor called the CD4 receptor contained in the cell membrane. Along with a chemokinecoreceptor, the virus is granted entry into the cell. Inside the host cell, the viral RNA is transcribed into DNA by theenzyme reverse transcriptase. However, the enzyme has no proofreading capacity, so errors often occur during thisprocess, giving rise to 1 to 3 mutations per newly synthesized virus particle. The DNA provirus is then transported intothe nucleus and inserts itself into the host cell DNA with the aid of viral integrase. Thus, the viral genetic code becomesa stable part of the cell genome, which is then transcribed into a full-length mRNA by the host cell RNA polymerase.The full-length mRNA would be(a) the genomes of progeny virus, which would be transported to the cytoplasm for assembly,(b) translated to produce the viral proteins, including reverse transcriptase and integrase, and(c) spliced, creating new translatable sequencesThe nonstructural genes on the virus also encode regulatory proteins that have diverse effects on the host cell, includingdown-regulating host cell receptors like CD4 and major histocompatibility complex class I molecules, aiding insynthesizing full-length HIV RNAs and enabling transportation of the viral mRNAs out of the nucleus without beingspliced by the host cell. Altogether, these effects enable viral mRNAs to be correctly translated into polypeptides andpackaged into virions. These components are then transported to the plasma membrane and assembled into the maturevirion, exiting the cell.A person can contract the virus through one of four routes: sexual contact, either homo- or heterosexual; transfusionswith whole blood, plasma, clotting factors and cellular fractions of blood; contaminated needles; perinatal transmission.The virus causes tissue destruction, immunodeficiency and can progress to acquired immunodeficiency syndrome(AIDS), completely breaking down the human body’s defense mechanisms. These patients are now more susceptible toinfections that should be harmless to a normal person, such as P.jiroveci pneumonia or tuberculosis, and the conditionsare worse as well. So far, treatments for the disease mainly target reverse transcriptase, viral proteases, and viralintegration and fusion, dealing with the virus infection before it progresses to AIDS. Currently, one treatment for HIV is2 TABILITY ANALYSIS OF A NOVEL
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5, 2021highly active antiretroviral therapy (HAART), which includes a combination of drugs including nucleoside/nucleotideanalog reverse transcriptase inhibitors, nonnucleoside reverse transcriptase inhibitors, protease inhibitors, fusioninhibitors, integrase inhibitors, and coreceptor blockers. These drugs are administered based on individualized criteriasuch as tolerability, drug-drug interactions, convenience/adherence, and possible baseline resistance. Although HAARTcan lower the viral load, the virus reemerges if the treatment is stopped. Therefore, HIV infection is currently bothchronic and incurable. [20]Whenever the population reaches T max , it will decrease, allowing us to impose an upper constrain dT max < s . Withthis constrain, the equation (2) has a unique equilibrium at ˆ T = T max a (cid:20) a − d + (cid:114) ( a − d ) + 4 asT max (cid:21) (3)In 1989, Perelson [56] proposed a general model for the interaction between the human immune system and HIV; in thesame paper, he also simplified that general model into a simpler model with four compartments, whose dynamics aredescribed by a system of four ODEs:• Concentration of cells that are uninfected ( T ),• Concentration of cells that are latently infected ( T ∗ ),• Concentration of cells that are actively infected ( T ∗∗ ), and• Concentration of free infectious virus particles ( v ).Later, he extended his own model in Perelson et al. (1993) [55] by proving various mathematical properties of themodel, choosing parameter values from a restricted set that give rise to the long incubation period characteristic of HIVinfection, and presenting some numerical solutions. He also observed that his model exhibits many clinical symptomsof AIDS, including:• Long latency period,• Low levels of free virus in the environment, and• Depletion of CD4 + cells.Time delay, of one type or another, have been incoporated into biological models in various research papers (forexample, [56]); particularly, by the similar theoretical analysis to dynamical population system (in [50]), they alsoplay an important role in the dynamical properties of the HIV infection models. Generally speaking, systems ofdelay-differential equations (DDEs) have much more complicated dynamics than that of ordinary differential equations(ODEs), as the time delay can cause a stable equilibrium of the ODE system to become unstable, leading to thefluctuation of popolations. In studying the viral clearance rate, Perelson et al. (1996) [60] stated that there are twodifferent types of delay that can occur in an HIV infection model:• Pharmacological delay : This delay occurs between the ingestion of drug and its appearance within cells,•
Intracellular delay : This delay happens between the initial HIV infection of a cell and the release of virionswithin the environment.There has also been various attempts by different authors, trying to come up with the most realistic model byimplementing these delays, in one form or another (constant delay, discrete delay, continuous delay, etc.). For example,• Herz et al. (1996) [23], who implemented a discrete delay to represent the intracellular delay in the HIV model.He showed that the incorporation of the delay would significantly shorten the estimate for the half-life of freevirus particles.• Mittler et al. (1998) [47] stated that a γ - distribution delay would be more realistic to describe the intracellulardelay, then implemented it to the original model proposed by Perelson et al. (1996) [60].• Mittler et al. [46] et al. and Tam et al. (1999) [70] also derived an analytic expression for the rate of decline ofvirus following drug treatment by assuming the drug to be completely effcacious.• Song and Neumann (2007) [66] proposed a saturated mass-action term into the simplified model from Perelsonet al. (1996) [60], and later investigated the drug effectiveness under this saturation infection.3 TABILITY ANALYSIS OF A NOVEL
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5, 2021The paper will be organized as follows: First, we will investigate a simplified ODE model from Perelson et al. (1993)[55] by considering three main components: the uninfected CD4 + T-cells ( T ), the infected CD4 + T-cells ( I ), and thefree virus ( V ) with. This model is also assumed to have a saturation response of the infection rate. Next, the existenceand stability of he infected steady state are considered. Then, we incorporate a discrete constant delay into the modelto indicate the time range between the infection of a CD4 + T-cell and the emission of viral particles at the cellularlevel, resulting in a system of three delay-differential equations (DDEs). To understand the dynamics of this delaymodel and obtain sufficient conditions for local/global asymptotic stability of the equilibria of all time delay, we carriedout a complete analysis on the transcendental characteristic equations of the linearized system at both the viral-freeequilibrium and the infected (positive) equilibrium. Finally, numerical simulations are carried out, using
Julia , toconfirm the obtained results, before some remarks are included in the conclusion.
Simplifying the model proposed in Perelson et al. (1993) [55] by reducing the number of dimensions and assuming thatall of the infected cells have the ability of producing virus at an equal rate, we propose the following epidemic model ofHIV infection of CD4 + T-cells as follows: dTdt = s − dT + aT (cid:18) − TT max (cid:19) − βT V αV + ρIdIdt = βT V αV − ( δ + ρ ) IdVdt = qI − cV − k V T (4)where • T ( t ) is the concentration of healthy CD4 + T-cells at time t (target cells),• I ( t ) is the concentration of infected CD4 + T-cells at time t , and• V ( t ) is the viral load of the virions (concentration of free HIV at time t ).In infection modelling, it is very common to augment (4) with a "mass-action" term in which the rate of infectionis given by βT V . This type of term is sensible, since the virus must interact with T-cells in order to infect and theprobability of virus encountering a T-cell at a low concentration environment (where infected cells and viral load’smotions are regarded as independent) can be assumed to be proportional to the product of the density, which is calledlinear infection rate. As a result, it follows that the classical models can assume that T-cells are infected at rate − βT V and are generated at rate βT V .With that simple mass-action infection term, the rates of change of uninfected cells, T , productively infected cells I ,and free virus V , would be dTdt = s − dT + aT (cid:18) − TT max (cid:19) − βT VdIdt = βT V − δIdVdt = qI − cV (5)Moreover, although the rate of infection in most HIV models is bilinear for the virus V and the uninfected target cells T ,the actual incidence rates are probably not strictly linear for each variable in over the whole valid range. For example,a non-linear or less-than-linear response in V could occur due to the saturation at a high enough viral concentration,where the infectious fraction is significant for exposure to happen very likely. Thus, is it reasonable to assume that theinfection rate of HIV modelling in saturated mass action is βT V x αV y , x, y, α > (6)In this paper, we will investigate the viral model with saturation response of the infection rate where x = y = 1 , for thesake of simplicity. With that being said, we will proceed to explain the parameters within the model, with4 TABILITY ANALYSIS OF A NOVEL
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5, 2021• s is the rate at which new T-cells are created from source from precursors,• d is the natural death rate of the CD4 + T-cells,• a is the maximum proliferation rate (growth rate) of T-cells (this means that a > d in general),• T max is the T-cells population density at which proliferation shuts off (their carrying capacity),• β is the rate constant of infection of T-cells with free virus,• ρ is the "cure" rate, or the non-cytolytic loss of infected cells,• δ is the death rate of the infected cells,• q is the reproduction rate of the infected cells, and• c is the clearance rate constant (loss rate) of the virions.From the explanations above, we can say that• δ + ρ is the total rate of disappearance of infected cells from the environment,• /δ is the average lifespan of a productively infected cell• q/δ is the total number of virions produced by an actively infected cell during its lifespan, and• q is the average rate of virus released by each cell.Under the absence of virus (i.e I ( t ) = V ( t ) = 0 ∀ t > ), the T-cell population has a steady state value of T = T max a (cid:20) ( a − d ) + (cid:114) ( a − d ) + 4 aT max (cid:21) (7)The system (4) needs to be initialized with the following initial conditions T (0) > , I (0) > , V (0) > , (8)which leads us to denote that R = { ( T, I, V ) ∈ R (cid:107) T ≥ , I ≥ , V ≥ } (9) The system (4) has two steady states: the uninfected steady state E = ( T , , and the (positive) infected steady state ¯ E = (cid:0) ¯ T , ¯ I, ¯ V (cid:1) , where: ¯ T = T max a a − d − δ qβ − ( δ + ρ ) qα ( δ + ρ ) + (cid:115)(cid:18) a − d − δ qβ − ( δ + ρ ) qα ( δ + ρ ) (cid:19) − aT max (cid:18) δcqα − s (cid:19) ¯ I = [ qβ − ( δ + ρ ) k ] ¯ T − ( δ + ρ ) cqα ( δ + ρ )¯ V = 1 α (cid:20) qβ ¯ Tα ( δ + ρ )( c + k T − (cid:21) (10)Now, we will proceed to analyse the stability of the equilibria of system (4).Since T and ¯ T satisfy s − dT + aT (cid:18) − T T max (cid:19) = 0 s − d ¯ T + a ¯ T (cid:18) − ¯ TT max (cid:19) = δ ¯ I = δqα ( δ + ρ ) [( qβ − ( δ + ρ )) T − ( δ + ρ ) c ] (11)5 TABILITY ANALYSIS OF A NOVEL
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5, 2021we get that ¯ T > c ( δ + ρ ) qβ − ( δ + ρ ) k ⇒ s − d ¯ T + a ¯ T (cid:18) − ¯ TT max (cid:19) > ⇒ T > ¯ T (12)and ¯ T < c ( δ + ρ ) qβ − ( δ + ρ ) k ⇒ s − d ¯ T + a ¯ T (cid:18) − ¯ TT max (cid:19) < ⇒ T < ¯ T (13)Hence,• If ¯ T > c ( δ + ρ ) qβ − ( δ + ρ ) k , then T > ¯ T > c ( δ + ρ ) qβ − ( δ + ρ ) k , which means that E ( T , , is unstable, while the positiveequilibrium ¯ E ( ¯ T , ¯ I, ¯ V ) exists.• If ¯ T < c ( δ + ρ ) qβ − ( δ + ρ ) k , then T < ¯ T < c ( δ + ρ ) qβ − ( δ + ρ ) k , which means that E ( T , , is locally asymptoticallystable, while the positive equilibrium ¯ E ( ¯ T , ¯ I, ¯ V ) is not feasible, as ¯ I < , ¯ V < .Let R = (cid:18) qβ − ( δ + ρ ) k c ( δ + ρ ) (cid:19) ¯ T (14)We can see that R is the bifurcation parameter. When R < , the uninfected steady state E is stable and the infectedsteady state ¯ E does not exist (unphysical). When R > , E becomes unstable and ¯ E exists.For system (5), it is known that the basic reproductive ratio is given by: R = (cid:18) qβ − ( δ + ρ ) k c ( δ + ρ ) (cid:19) T (15)Once again, we emphasize the large difference of the basic reproduction ratio between the linear infection rate and thesaturation infection rate.• If α → , then ¯ T → c ( δ + ρ ) qβ − ( δ + ρ ) , R → ;• If α → + ∞ , then ¯ T → T , R → R .The Jacobian matrix of system (4) is: ( a − d ) − aTT max − βV αV ρ − βT (1+ αV ) βV αV − ( δ + ρ ) βT (1+ αV ) − k V q − c − k T (16)Let E ∗ ( T ∗ , I ∗ , V ∗ ) be any arbitrary equilibrium. Then, the characteristic equation about E ∗ is: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ + (cid:16) ( d − a ) + aT ∗ T max + βV ∗ αV ∗ (cid:17) − ρ βT ∗ (1+ αV ∗ ) − βV ∗ αV ∗ λ + ( δ + ρ ) − βT ∗ (1+ αV ∗ ) k V ∗ − q λ + ( c + k T ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (17)For equilibrium E = ( T , , , (17) reduces to (cid:18) λ − a + d + 2 aT T max (cid:19) (cid:2) λ + ( c + δ + ρ ) λ + c ( δ + ρ ) − qβT (cid:3) = 0 (18)Hence, we can see that E = ( T , , is locally asymptotically stable if R < , and it is a saddle point if dim W s ( E ) = 2 , or if dim W s ( E ) = 1 while R > . As a result, we have the following theorems6 TABILITY ANALYSIS OF A NOVEL
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Theorem 1. If R < , E = ( T , , is locally asymptotically stable; else, if R > , E = E = ( T , , isunstable. Theorem 2.
There exists
M > , M ∈ R such that for any positive solution ( T ( t ) , I ( t ) , V ( t )) of system (4) , T ( t ) ≤ M, I ( t ) ≤ M, V ( t ) ≤ M (19) for all large enough t .Proof. Let L ( t ) = T ( t ) + I ( t ) and assume that L (0) = T (0) + I (0) = const = c . Calculating the derivative of L ( t ) using the equations in system (4), we have: dL ( t ) dt = dT ( t ) dt + dI ( t ) dt = s − dT + aT (cid:18) − TT max (cid:19) − δI = − dt − δI − aT max (cid:18) T − T max a (cid:19) + 4 s + aT max ≤ − ( T + I ) min ( d, δ ) − aT max (cid:18) T − T max a (cid:19) + 4 s + aT max − hL ( t ) − M (cid:18) h = min ( d, δ ) , M = 4 s + aT max (cid:19) (20)Let U ( t ) = L ( t ) − M h . This means that U (0) = L (0) − M h = c − M hdU ( t ) dt = dL ( t ) dt (21)The inequality (20) can be rewritten as dU ( t ) dt ≤ ( − h ) U ( t ) (22)which yields, according to Gronwall’s inequality, U ( t ) ≤ U (0) exp (cid:18)(cid:90) t ( − h ) ds (cid:19) = (cid:18) c − M h (cid:19) exp (cid:16) [ − hs ] t (cid:17) = (cid:18) c − M h (cid:19) exp( − ht ) ≤ c − M h (23)or T ( t ) + I ( t ) = L ( t ) = U ( t ) + M h = c − M h + M h = c (24)As T ( t ) > , I ( t ) > ∀ i ∈ Z + , we can say that V ( t ) ≤ c, I ( t ) ≤ c (25)7 TABILITY ANALYSIS OF A NOVEL
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5, 2021Moreover, we also know that dVdt = qI − cV − k V T ≤ qI − cV ≤ qc − cV = − c ( V − q ) (26)Setting V (0) = const = c V , using the exact same procedure with Gronwall’s inequality, we obtain V ( t ) ≤ c V ∀ t ∈ Z + (27)With M = max ( c, c V ) , we would then conclude that T ( t ) ≤ M, I ( t ) ≤ M, V ( t ) ≤ M ∀ t ∈ Z + (28)We can easily see that this set is convex. As a consequence, the system (4) is dissipative.The proof is complete.From this theorem, we define D = (cid:8) ( T, I, V ) ∈ R , ≤ T, I, V ≤ M (cid:9) . (29)Denote M = d − a + 2 a ¯ TT max , N = β ¯ V α ¯ V , P = β ¯ T (1 + α ¯ V ) . (30)Then, the characteristic equation of the system around the equilibrium ¯ E ( ¯ T , ¯ I, ¯ V ) reduces to: λ + a λ + ( a + a ) λ + ( a + a ) = 0 (31)where a = M + ( δ + ρ + c + k ¯ T ) a = ( δ + ρ )( c + k T ) + M ( δ + ρ + c + k ¯ T ) + ( − k ¯ V P ) a = ρ (cid:2) − N ( c + k ¯ T ) + P k ¯ V (cid:3) + P N qa = − N Pa = M ( δ + ρ )( c + k ¯ T ) − P ( δ + ρ ) k ¯ V (32)By the Routh-Hurwitz criterion [34], it follows that all eigenvalues of equation (31) have negative real parts if and onlyif a > , a + a > , a ( a + a ) − ( a + a ) > (33)This leads us to the following theorem Theorem 3.
Suppose that1. R > ,2. a > , a + a > , a ( a + a ) − ( a + a ) > .Then, the positive equilibrium ¯ E ( ¯ T , ¯ I, ¯ V ) is asymptotically stable. Theorem 4. If R < , then E ( T , , is globally asymptotically stable. TABILITY ANALYSIS OF A NOVEL
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Proof.
First of all, as R < , we would have T < ¯ T < c ( δ + ρ ) qβ − ( δ + ρ ) (34)which means that p < ( c + k T )( δ + ρ ) βT (35)From the system (4), we would have dIdt ≤ βT V − ( δ + ρ ) I,dVdt = qI − cV − k V T. (36)Now, we would consider the following comparative system dz dt = βT z − ( δ + ρ ) z dz dt = pz − cz − k z T (37)We will consider the following form of Lyapunov function: L ( X ) = V ( z , z ) = δ + ρ ( βT ) z + 1 c + k T z (38)The derivative of the function can be calculated as follows dLdt = ∂L∂z dz dt + ∂L∂z dz dt = 2 δ + ρ ( βT ) z ( βT z − ( δ + ρ ) z ) + 2 1 c + k T z ( qz − cz − k T z )= − (cid:34)(cid:18) δ + ρβT z (cid:19) + z − (cid:18) δ + ρβT z z + qc + k T (cid:19) z z (cid:35) ≤ − (cid:34)(cid:18) δ + ρβT z (cid:19) + z − (cid:18) δ + ρβT + β + ρβT (cid:19) z z (cid:35) = − (cid:20) δ + ρβT z − z (cid:21) ≤ ∀ z , z (39)We can see that the derivative is negative definite everywhere except at (0 , . This means that ( z , z ) = (0 , isglobally asymptotically stable.As we can also see that ≤ I (0) ≤ z (0) , ≤ V (0) ≤ z (0) (40)which means that, if the system (37) admits the initial values ( z (0) , z (0)) , we have that I ( t ) ≤ z ( t ) , V ( t ) ≤ z ( t ) ∀ t > t (41)9 TABILITY ANALYSIS OF A NOVEL
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5, 2021or, in other words, lim t → + ∞ I ( t ) = lim t → + ∞ V ( t ) = 0 (42)From this, using the first equation of the system (4), for an (cid:15) in (0 , infinitesimal, s + ( a − d − δ(cid:15) ) T − aT T max ≤ dT ( t ) dt ≤ s + ( a − d ) T − aT T max ∀ t > t (43)which shows that lim t → + ∞ T ( t ) = T . (44)From (42) and (44), we conclude that the system is globally asymptotically stable. The proof is complete. Theorem 5. If R > , then the system (4) is permanent.Proof. If R > , we would have ( qβ − ( δ + ρ ) k ) T > ( qβ − ( δ + ρ ) k ) ¯ T > c ( δ + ρ ) (45)We will proceed to prove the weak permanence of this system using contradiction.Assume that the system is not weakly permanent, from Theorem 4, there exists a positive orbit ( T ( t ) , I ( t ) , V ( t )) suchthat lim t → + ∞ T ( t ) = T , lim t → + ∞ I ( t ) = lim t → + ∞ V ( t ) = 0 (46)Since T > c ( δ + ρ ) qβ − ( δ + ρ ) , combining with (46), we choose an arbitrary infinitesimal (cid:15) > such that there exists a t > ,for all t > t , T − (cid:15) α(cid:15) > c ( δ + ρ ) qβ − ( δ + ρ ) T ( t ) > T − (cid:15),V ( t ) < (cid:15) (47)Under these conditions, the system (4) becomes dIdt = βT V αV − ( δ + ρ ) I ≥ β ( T − (cid:15) )1 + α(cid:15) V − ( δ + ρ ) I ( t ) dVdt = qI − ( c + k T ) ≈ qI − cV − k T (48)Consider the following Jacobian matrix J (cid:15) = (cid:18) − ( δ + ρ ) β ( T − (cid:15) )1+ α(cid:15) q − ( c + k T ) (cid:19) (49)Since J (cid:15) has positive off-diagonal element, according to the Perron - Frobenius theorem, for the maximum positiveeigenvalue j of J (cid:15) , there is an associated positive eigenvector v = (cid:18) v v (cid:19) .Next, we consider a system associated with the Jacobian matrix J (cid:15) dz dt = β ( T − (cid:15) )1 + α(cid:15) z − ( δ + ρ ) z dz dt = qz − ( c + k T ) z (50)10 TABILITY ANALYSIS OF A NOVEL
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5, 2021Let z ( t ) = ( z ( t ) , z ( t )) be a solution of (50) through ( lv , lv ) at t = t , where l > satisfies that lv < I ( t ) , lv < V ( t ) (51)As we know that the semi-flow of (50) is monotone and J (cid:15) v = v > , z i ( t )( t = 1 , is strictly increasing, meaning lim t to + ∞ z i ( t ) = + ∞ . This contradicts the Theorem (2), saying that the positive solution of (4) is bounded fromabove. This contradiction says that there exists no positive orbit of (4) tends to ( T , , and t → + ∞ . Combining thisand a result provided in [11], we conclude that the system (4) is permanent.The proof is complete. Theorem 6.
Assume that D is convex and bounded. Suppose that the system dXdt = F ( X ) , X ∈ D (52) is competitive, permanent and has the property of stability of periodic orbits. If ¯ X is the only equilibrium point inint D and if it is locally asymptotically stable, then it is globally asymptotically stable in int D .Proof. This matrix can easily be proven by considering the Jacobian matrix and choose the matrix H as H = (cid:32) − (cid:33) (53)By simple calculation, we obtain that H ∂f∂x H = ( a − d ) − aTT max − βV αV − ρ − βT (1+ αV ) − βV αV − ( δ + ρ ) − βT (1+ αV ) − k V − q − c − k T (54)This means that the system (4) is competitive in D , with respect to the partial order defined by the orthant K = (cid:8) ( T, I, V ) ∈ R (cid:107) T ≤ , I ≥ , V ≥ (cid:9) (55) Remark 1. As D is convex and the system (4) is competitive in D . we can say that the system (4) satisfies the Poincare- Bendixson property . This has been proven by Hirsch (1990) [24], Zhu and Smith (1994) [76] and Smith and Thieme(1991) [64] that any three-dimensional competitive system that lie in convex sets would have the Poincaré - Bendixsonproperty; in other words, any non-empty compact omega limit set that contains no equilibria must be a closed orbit.
Theorem 7.
Let c = I (0) + T (0) and suppose that1. R > ,2. a > , a + a > , a ( a + a ) − ( a + a ) > .Then, the positive equilibrium ¯ E ( ¯ T , ¯ I, ¯ V ) of system (4) is globally asymptotically stable provided that one of thefollowing two assumptions hold1. T max a − d + k c a < m < T < T max a − d + δ + k c a ,2. m > T max a − d + δ + k c a . As we have already known that the system (4) is competitive and permanent (from Theorem 5 and Theorem 6),while ¯ E ( ¯ T , ¯ I, ¯ V ) is locally asymptotically stable if the two properties (i) and (ii) of Theorem 7 holds. As a result, inaccordance with Theorem 6 (choosing D = Ω ), Theorem 7 if we can prove that the system (4) has the stability ofperiodic orbits. We will proceed to prove this under the following proposition.11 TABILITY ANALYSIS OF A NOVEL
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Proposition 1.
Assume condition (iii) or (iv) of 7 hold true. Then, system (4) has the property of stability of periodicorbits.Proof.
Let P ( t ) = (( T ( t ) , I ( t ) , V ( t )) be a periodic solution whose orbit Γ is contained in int Ω . In accordance with thecriterion given by Muldowney in [48], for the asymptotic orbital stability of a periodic orbit of a general autonomoussystem, it is sufficient to prove that the linear non-autonomous system dW ( t ) dt = (cid:16) DF [2] ( P ( t )) (cid:17) W ( t ) (56)is asymptotically stable, where DF [2] is the second additive compound matrix of the Jacobian DF [66].The Jacobian matrix of the system (4) is given by J = ( a − d ) − aTT max − βV αV ρ − βT (1+ αV ) βV αV − ( δ + ρ ) βT (1+ αV ) − k V q − ( c + k T ) (57)For the solution P ( t ) , the equation (56) becomes dW dt = − (cid:18) δ + ρ − ( a − d ) + 2 aTT max + βV αV (cid:19) W + βT (1 + αV ) ( W + W ) ,dW dt = qW + (cid:18) a − d − aTT max − βV αV − ( c + k T ) (cid:19) W + ρW ,dW dt = k V W + βV αV W − ( δ + ρ + c + k T ) W . (58)To prove that the system (58) is asymptotically stable, we shall use the following Lyapunov function, which is similarto the one found in [38] for the SEIR model: L ( W ( t ) , W ( t ) , W ( t ) , T ( t ) , I ( t ) , V ( t )) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) W ( t ) , I ( t ) V ( t ) W ( t ) , I ( t ) V ( t ) W ( t ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (59)where (cid:107)·(cid:107) is the norm in R defined by (cid:107) ( W , W , W ) (cid:107) = sup {| W | , | W + W |} (60)From Theorem 5, we obtain that the orbit of P ( t ) remains at a positive distance from the boundary of Ω . Therefore, I ( t ) ≥ η, V ( t ) ≥ η, η = min { I , V } ∀ t → + ∞ (61)Hence, the function L ( t ) is well defined along P ( t ) and L ( W , W , W ; T, I, V ) ≥ ηM (cid:107) ( W , W , W ) (cid:107) (62)Along a solution ( W , W , W ) of the system (58), L ( t ) becomes L ( t ) = sup (cid:26) | W ( t ) | , I ( t ) V ( t ) ( | W ( t ) | + | W ( t ) | ) (cid:27) (63)12 TABILITY ANALYSIS OF A NOVEL
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5, 2021Then, we would have the following inequalities D + | W ( t ) | ≤ − (cid:18) δ + ρ − ( a − d ) + 2 aTT max + βV αV (cid:19) | W | + βT (1 + αV ) ( | W ( t ) | + | W ( t ) | ) D + | W ( t ) | ≤ q | W ( t ) | + (cid:18) a − d − aTT max − βV αV − ( c + k T ) (cid:19) | W ( t ) | + ρ | W ( t ) | D + | W ( t ) | ≤ k V | W ( t ) | + βV αV | W ( t ) | − ( δ + ρ + c + k T ) | W ( t ) | (64)From this, we get D + IV ( | W | + | W | ) = (cid:18) dI/dtV − IdV /dtV (cid:19) ( | W | + | W | ) + IV D + ( | W | + | W | ) ≤ (cid:18) dI/dtI − dV /dtV (cid:19) IV ( | W | + | W | ) + (cid:18) qIV + k I (cid:19) | W |− (cid:18) − a + d + 2 aTT max + ( c + k T ) (cid:19) IV | W ( t ) | − ( δ + c + k T ) IV | W ( t ) | (65)Thus, we can obtain D + L ( t ) ≤ sup { g ( t ) , g ( t ) } L ( t ) , (66)where g ( t ) = − δ − ρ + a − d − aTT max − βV αV + βT VI (1 + αV ) g ( t ) = qIV + k I + dI/dtI − dV /dtV − G G = min (cid:26) − a + d + 2 aTT max + ( c + k T ) , δ + c + k T (cid:27) (67)From the second equation of the system (4), we obtain g ( t ) = − δ − ρ + a − d − aTT max − βV αV + βT VI (1 + αV ) ≤ − δ − ρ + a − d − aTT max − βV αV + βtVI (1 + αV )= a − d − aTT max − βT αV + dI/dtI (68)Here, we consider two different cases.• Case 1:
If Point 3 of Theorem 7 holds, then − δ < a − d − aTT max < , (69)that is G = − a + d + 2 aTT m ax + ( c + k T ) (70)Then, we would obtain g ( t ) = a − d − aTT max + k I + dI/dtI = g ( t ) + k I + βV αV > g ( t ) (71)13 TABILITY ANALYSIS OF A NOVEL
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5, 2021Hence, sup { g ( t ) , g ( t ) } ≤ a − d − aTT max + k I + dI/dtI ≤ − µ + dI/dtI (72)where µ > , a − d − aTT max + k I ≤ − µ < (73)with the assumption that k I is negligible compare to the term a − aTT max . This assumption would be verifiedin the examples of the simulation part below.• Case 2:
If Point 4 of Theorem 7 holds, then − a + d + 2 aTT max ≤ δ, (74)which means that G = δ + c + k T . Then, we obtain that µ < , g ( t ) < g ( t ) = k T − δ + dI/dtI ≤ − µ + dI/dtI (75)with the same assumption that k T < σ in reasonably practical scenarios. Hence, sup { g ( t ) , g ( t ) } ≤ − µ + dI/dtI (76)Let µ = min { µ , µ } . Then, form (72) and (75), we have sup { g ( t ) , g ( t ) } ≤ − µ + dI/dtI , (77)or D + L ( t ) ≤ (cid:18) − µ + dI/dtI (cid:19) L ( t ) . (78)According to Gronwall’s inequality, we would have L ( t ) ≤ L (0) exp (cid:18)(cid:90) t (cid:20) − µ + dI/dtI (cid:21) ds (cid:19) = L (0) exp (cid:16) [ − µs + ln( I ( s ))] t (cid:17) = L (0) exp( − µt ) exp (ln( I ( t )) − ln( I (0)))= L (0) exp( − µt ) I ( t ) I (0) ≤ M L (0) I (0) exp( − µt ) → as t → + ∞ (79)From (62), we conclude that ( W ( t ) , W ( t ) , W ( t )) → as t → + ∞ (80)This implies that the linear system equation (58) is asymptotically stable, and therefore the periodic solution isasymptotically orbitally stable. This proves proposition 114 TABILITY ANALYSIS OF A NOVEL
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Theorem 8.
Suppose that1. R > ,2. a > , a + a > , a ( a + a ) − ( a + a ) > .Then, system (4) has an orbitally asymptotically stable periodic solution.Proof. First, we perform a change of variables as follows: z ( t ) = − T ( t ) , z ( t ) = I ( t ) , z ( t ) = − V ( t ) (81)Applying this transformation to the system (4), we obtain dz ( t ) dt = − s − dz + az (cid:18) z T max (cid:19) + βz z − αz + ρz dz ( t ) dt = βz z − αz − ( δ + ρ ) z dz ( t ) dt = − qz − cz + k z z (82)The Jacobian matrix of the system (82) is then given by J ( z ) = a − d + az T max + βz − αz ρ βz (1+ αz ) βz − αz − ( δ + ρ ) βz (1+ αz ) k z − q − c + k z (83)Similar to the definition of the set D at 29, we define set E as: E = { ( z , z , z ) : z ≤ , z ≥ , z ≤ } (84)Since J ( z ) has non-positive off diagonal elements at each point of E , (82) is competitive at E . Set z ∗ = ( − T ∗ , I ∗ , V ∗ ) .It is easy to see that z ∗ is unstable and det J ( z ∗ ) < . Furthermore, it follows from Theorem 5 that there exists acompact set B in the interior of E such that for any z ∈ int E , there exists T ( z ) > such that z ( t, z ) ∈ B for all t > T ( z ) . Consequently, by Theorem 1.2 in Zhu and Smith (1994) [76] for the class of three-dimensional competitivesystems, it has an orbitally asymptotically stable periodic solution.The proof is complete. In this section, we introduce a time delay into the model (5) to represent the incubation time that the vectors need tobecome infectious. The model is rewritten as follows dTdt = s − dT + aT (cid:18) − TT max (cid:19) − βT V αV + ρIdIdt = βT ( t − τ ) V ( t − τ )1 + αV ( t − τ ) − ( δ + ρ ) IdVdt = qI − cV − k V T (85)under the initial values T ( θ ) = T , I ( θ ) = I , V ( θ ) = V ∀ θ ∈ [ − τ, (86)15 TABILITY ANALYSIS OF A NOVEL
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5, 2021All parameters of this delay model are the same as those of the system (4), except that the additional positive constant τ represents the length of the delay, in days.This time delay parameter can be explained as follows: At time t , only healthy cells that have been infected by thevirus τ days ago (i.e at time t − τ are infectious, provided that they have survived the incubation period of τ days andwere alive at the time t − τ when they infect the healthy cells. As a result, the incidence term of healthy cells in thederivative of infected cells with respect to time is modified from βT ( t ) V ( t ) to βT ( t − τ ) V ( t − τ ) .The reproduction of this delay differential equation can be given the same as the original ODE model, which is R = qβ − ( δ + ρ ) k c ( δ + ρ ) T (87)Its biological meaning is that, if one virus is introduced in the population of uninfected cells, the total number ofsecondary infected cells during the infectious period would be qβ − ( δ + ρ ) k c ( δ + ρ ) . Within this section, we will study the local and global stability of the disease-free equilibrium E of the delay model intwo cases: when R > and when R < . Theorem 9.
The disease-free equilibrium of the system (85) is locally asymptotically stable if R < , and is unstableif R > .Proof. Linearizing the system (85) around E = ( T , , , we obtain one negative characteristic root λ = a − d − aT T max (88)and the following transcendental characteristic equation whose roots are the remaining eigenvalues λ + ( δ + ρ + c + k T ) + ( c + k T )( δ + ρ ) − qβT e − λτ = 0 (89)For τ = 0 , we obtain the exact same quadratic equation as the original ODE system. In this case, we have provenpreviously that all eigenvalues of the characteristic equation (89) have negative real parts. According to the Routh -Hurwitz criterion, the disease free equilibirum E will be locally asymptotically stable when R < and is unstablewhen R > .As a result, we now only need to prove that the statement holds true for all τ (cid:54) = 0 .• Case 1: R > . In this case, we expect that (89) has one positive root and the disease-free equilibrium isunstable. Indeed, we arrange the characteristic equation into the form of λ + ( δ + ρ + c + k T ) λ = qβT e − λτ − ( c + k T )( δ + ρ ) (90)Now, suppose that δ ∈ R and denote F ( λ ) = λ + ( δ + ρ + c + k T ) λG ( λ ) = qβT e − λτ − ( c + k T )( δ + ρ ) (91)We would then have that F (0) = 0 , lim λ → + ∞ F ( λ ) = + ∞ (92)while G (0) = qβT − ( c + k T )( δ + ρ ) = c ( δ + ρ )( R − > ,G (cid:48) ( λ ) < ∀ λ > (93)As a result, the two functions must intersect at a point λ > , which means that the equation (89) admits apositive real root, which means that the disease-free equilibrium is unstable.16 TABILITY ANALYSIS OF A NOVEL
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Case 2: R < . First, we can notice that (89) can not have any non-negative roots since , while F (0) = 0 , F (cid:48) ( λ ) > ∀ λ ≥ G ( λ ) < , G (cid:48) ( λ ) < ∀ λ > (94)As a result, if (89) has roots with non-negative real parts, they must be complex and should be obtained froma pair of complex conjugate with cross the imaginary axis. This means that (89) must have a pair of purelyimaginary roots for τ > .As a result, we assume that λ = iω , and without loss of generality, we assume that ω > is a root of (89),meaning that − ω + iω ( δ + ρ + c + k T ) + ( c + k T )( δ + ρ ) − qβT (cos( ωτ ) + i sin( ωτ )) = 0 (95)Separating the real and imaginary part, we would have − ω + ( c + k T )( δ + ρ ) = qβT cos( ωτ )( δ + ρ + c + k T ) ω = − qβT sin( ωτ ) (96)Squaring and adding up both sides of the two equations above, we obtain the following fourth-order equationfor ω as ω + ω (cid:2) ( δ + ρ + c + k T ) − c + k T )( δ + ρ ) (cid:3) + [( c + k T )( δ + ρ )] − ( qβT ) = 0 (97)To reduce this fourth-order equation into a quadratic equation, let z = ω and denote the coefficients as a = ( δ + ρ + c + k T ) − c + k T )( δ + ρ ) a = ( c + k T ) ( δ + ρ ) − ( qβT ) (98)The equation (97) can be rewritten as z + a z + a = 0 (99)Since R < , we would have that a = ( δ + ρ ) + ( c + k T ) > a = [( c + k T )( δ + ρ ) + qβT ] c ( δ + ρ )(1 − R ) > (100)As a result, this means that the two roots of (99) have positive product, which means that they have the samesign, regardless of being real or complex. As these two roots also have negative real products, they wouldbe either negative real numbers, or complex conjugate with negative real parts. As a result, the equation (99)can not have any positive real roots, leading to the fact that there would be no ω such that iω is a root of (89).Using Rouche’s theorem, we conclude that the real parts of all eigenvalues of the characteristic equation of thedisease-free equilibrium (89) are all negative for all delay values τ > .In conclusion, if R < , the disease-free equilibrium E is locally asymptotically stable.The proof is complete. To study the stability of the steady states ¯ E , we define x ( t ) = T ( t ) − ¯ T , y ( t ) = I ( t ) − ¯ I, z ( t ) = V ( t ) − ¯ V . (101)17
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5, 2021Then, the linearized system of (85) at ¯ E is given by dx ( t ) dt = (cid:20) − d + a − a ¯ TT max − β ¯ V (1 + α ¯ V ) (cid:21) x ( t ) + ρy ( t ) − β ¯ T (1 + α ¯ V ) z ( t ) dy ( t ) dt = β ¯ V α ¯ V x ( t − τ ) − ( δ + ρ ) y ( t ) + β ¯ V (1 + α ¯ V ) z ( t − τ ) dz ( t ) dt = − k ¯ V x ( t ) + qy ( t ) − ( c + k ¯ T ) z ( t ) (102)The system (102) can be expressed in matrix form as follows ddt (cid:32) x ( t ) y ( t ) z ( t ) (cid:33) = A (cid:32) x ( t ) y ( t ) z ( t ) (cid:33) + A (cid:32) x ( t − τ ) y ( t − τ ) z ( t − τ ) (cid:33) , (103)where A and A are × matrices given by A = a − d − a ¯ TT max ρ − β ¯ T (1+ α ¯ V ) − ( δ + ρ ) 0 − k ¯ V q − ( c + k ¯ T ) , A = β ¯ V α ¯ V β ¯ T (1+ α ¯ V ) . (104)The characteristic equation of system (102) is given by ∆( λ ) = (cid:12)(cid:12) λI − A − e − λτ A (cid:12)(cid:12) = 0 , (105)that is, λ + a λ + a λ + a = − e − λτ ( a + a λ ) , (106)with a i ( i = 1 , ..., previously defined in (32).Next, we shall study the distribution of the roots of the transcendental equation (106) with respect to analytically.Based on the point (or assumption) that the positive steady state of the original ODE model (4) is stable, we will derivefurther conditions on the parameters to ensure that the steady state of the delay model is still stable.First, we will consider the base case when τ = 0 . Then, the characeristic equation (106) will become (89). Now, wewill assume that all roots of his equation, in this case, has all negative real parts, which is equivalent to the fact thatthe conditions in Theorem 8 are satisfied. As the delay term τ is considered to be continuous on R , from Rouché’sTheorem [16], the transcendental equaion (106) can only have roots with negative real parts if and only if it has purelyimaginary roots. We will investigate whether (106) can admit any purely imaginary roots; from which, we will be ableto determine the conditions under which all eigenvalues would have negative real parts.We assume that λ = η ( τ ) + iω ( τ ) ( ω > is the eigenvalue of the characteristic equation (106), where η ( τ ) and ω ( τ ) are functions depending on the delay term τ . As the positive equilibrium ¯ E of the model (4) is stable, we can saythat η (0) < at τ = 0 .If η ( τ ) = 0 for some certain values of of τ > (which means that λ = iω ( τ ) are purely imaginary roots of thecharacteristic equation (106), the steady state ¯ E would lose is stability and become unstable whenever η ( τ ) is greaterhan . In other words, if there exists no ω ( τ ) such that the condition above happens, or, if the characteristic equation(106) does not have any purely imaginary roots for all values of τ , the positive equilibrium ¯ E is always stable. In thefollowing part, we will prove that this statement is indeed correct for equation (106).Clearly iω ( ω > is a root of equation (106) if and only if − iω − a ω + ia ω + a = − a (cos( ωτ ) − i sin( ωτ )) − a ω (sin( ωτ ) + icos ( ωτ )) (107)18 TABILITY ANALYSIS OF A NOVEL
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5, 2021Separating the real and imaginary parts, we would have a ω − a = a cos( ωτ ) + a ω sin( ωτ ) ω − a ω = − a sin( ωτ ) + a ω cos( ωτ ) (108)Squaring both sides of each equation above and adding up, we obtain the following sixth-degree equation for ω : ω + ( a − a ) ω + ( a − a a − a ) ω + ( a − a ) = 0 (109)Since this equation contains only even powers of ω , we can reduce the order by letting once again z = ω and m = a − a ,m = a − a a − a ,m = a − a , (110)the equation (109) becomes h ( z ) = z + m z + m z + m = 0 (111)In order to show that the positive equilibrium ¯ E is locally stable, we have to prove that the equation (111) does not haveany positive real root which associates to the square of ω ; that is, (106) can not have any purely imaginary roots. TheTheorem below provides us with necessary conditions satisfying the result. Theorem 10. If m ≥ and m > , the equation (109) has no positive real roots.Proof. We will proceed to prove the lemma above using contradiction.Assume that there exists at least one positive real roots for the equation h ( z ) = 0 .Notice that h (0) = m ≥ . This means that in order for the equation (109) to have a positive real roots, there exists z ≥ such that dh ( z ) dz ≤ . (112)This is equivalent to dh ( z ) dz = 3 z + 2 m z + m ≤ (113)or − m − (cid:112) m − m ≤ z ≤ − m + (cid:112) m − m < (114)This contradicts our original assumption that z ≥ , which means that there does not exist any z ≥ such that dh ( z ) dz > , or the equation does not have any positive real roots.The proof is complete.This theorem has implied that there exists no ω such that iω is an eigenvalue of the characteristic function (106). As aresult, from Rouche’s theorem [16], the real parts of the eigenvalues of (106) are negative for all τ ≥ . Summarizingall the above analysis, we have the following theorem Theorem 11.
Suppose that1. a > , a + a > , a ( a + a ) − ( a + a ) > ;2. m ≥ and m > .Then, the infected steady state ¯ E of the delay model (85) is absolutely stable; that is, ¯ E is asymptotically stable for all τ ≥ . TABILITY ANALYSIS OF A NOVEL
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Remark 2.
The Theorem 11 indicates that if the parameters satisfy both of the conditions, the equilibrium ¯ E of (85) isasymptotically stable regardless of the value of the delay (independent asymptotic stability). However, we also need tonote that if any of the conditions in Theorem 11 is violated (particularly the inequalities in Point 2), the stability of theequilibrium will then depend on the delay value; and when the delay value varies, the equilibrium can lose stability,leading to oscillations For example, if1. If m < : From equation (111), we would have that h (0) < , lim z → + ∞ = + ∞ , (115)which means that (111) has at least one positive real root, denoted by ω .2. If m < , we would have that − m + (cid:112) m − m − m m + (cid:112) m + 3 m > (116)which means that the equation (111) has one positive real root ω .These two cases implies that the characteristic equation (106) has a pair of purely imaginary roots ± iω .Next, we would focus on the bifurcation analysis, using the delay term τ as the bifurcation parameter, in light that thesolutions of (106) as function of this parameter.Let λ ( τ ) = µ ( τ ) + iω ( τ ) be the eigenvalue of (108) such that for some initial values of the bifurcation parameter τ ,we would have µ ( τ ) = 0 , ω ( τ ) = ω . From the system (108), we would have: τ j = 1 ω arccos (cid:18) a ω + ( a a − a a ) ω − a a a + a ω (cid:19) + 2 jπω (117)Moreover, we can verify the following transversal condition: ddτ (cid:60) ( λ ( τ )) (cid:107) τ = τ = ddτ µ ( τ ) (cid:107) τ = τ > (118)holds. By continuity, the real part of λ ( τ ) becomes positive when τ > τ and the steady state becomes unstable.Moreover, a Hopf bifurcation occurs when τ passes through the critical value τ (see [21]).To apply the Hopf bifurcation theorem stated in Marsden and McCracken [41], we state the following theorem Theorem 12.
Suppose that ω is the largest positive simple root of (109) . Then, iω ( τ ) = iω is a simple root of (109) ,and η ( τ ) + iω ( τ ) is differentiable with respect to τ in a neighborhood of τ = τ After previous reasoning, we admit that iω is a simple root of (109), which is an analytic equation; as a result, using theanalytic version of the implicit function theorem mentioned in Chow and Hale (1982) [12], we have that η ( τ ) + iω ( τ ) is well-defined and analytic in neighborhood of τ = τ To establish the Hopf bifurcation at τ = τ , we need to show that d (cid:60) ( λ ( τ )) dτ (cid:107) τ = τ > . (119)In order to prove this inequality, we first start with a lemma and its respective proof. Lemma 1.
Suppose that z , z , z are the roots of h ( z ) = z + m z + m z + m = 0 ( m < , and z ∈ R + isthe largest positive simple root, then dh ( z ) dz (cid:107) z = z > . (120) Proof.
We will proceed to prove the lemma above with contradiction.Assume that the largest positive simple root z of the equation h ( z ) = 0 and dh ( z ) dz (cid:107) z = z ≤ . (121)20 TABILITY ANALYSIS OF A NOVEL
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5, 2021As a result, there exists a ¯ z ∈ R + , ¯ z > z such that h (¯ z ) < .According to the Intermediate Value Theorem, for any h ∈ R , h ∈ [ h (¯ z, + ∞ ) , there always exists z ∈ [¯ z, + ∞ ) .Taking h = 0 , we would have (cid:26) h ( z ) = 0 z > ¯ z > z , (122)which means that z is the highest positive simple root of h ( z ) = 0 , not z . This contradicts our original assumption.In conclusion, if z is the largest positive simple root, dh ( z ) dz (cid:107) z = z > . (123)The proof is complete.From the equation (106), derivating both sides with respect to τ , we obtain (cid:0) λ + 2 a λ + a (cid:1) dλdτ = [ − τ exp( − λτ )( − a − a λ ) + exp( − λτ )( − a )] dλdτ − λ exp( − λτ )( − a − a λ ) . (124)This gives us (cid:18) dλdτ (cid:19) − = 3 λ + 2 a λ + a + τ exp( − λτ )( − a − a λ ) − exp( − λτ )( − a ) − λ exp( − λτ )( − a − a λ )= 3 λ + 2 a λ + a − λ exp( − λτ )( − a − a λ ) + a λ ( a + a λ ) − τλ = 2 λ + a λ − a − λ ( λ + a λ + a λ + a ) + − a λ ( a + a λ ) − τλ (125)Thus, Sign (cid:26) d ( (cid:60) ( λ )) dτ (cid:27) = Sign (cid:40) (cid:60) (cid:18) dλdτ (cid:19) − (cid:41) = Sign (cid:40) (cid:60) (cid:20) λ + a λ − a − λ ( λ + a λ + a λ + a ) (cid:21) λ = iω + (cid:60) (cid:20) − a λ ( a + a λ ) (cid:21) λ = iω (cid:41) = Sign (cid:26) (cid:60) (cid:20) − ω i − a ω − a ω ( − ω i − a ω + a ω i + a ) (cid:21) + (cid:60) (cid:20) − a − ω ( a + a ω i ) (cid:21)(cid:27) = Sign (cid:26) ω + ( a − a ) ω − a ω [( a ω − ω ) + ( a − a ω ) ] + a ω ( a ω + a ) (cid:27) = Sign (cid:26) ω + 2( a − a ) ω + ( a − a a − a )( a ω − ω ) + ( a − a ω ) (cid:27) (126)Since h ( z ) = z + m z + m z + m , (127)we would have dh ( z ) dz = 3 z + 2 m z + m = 3 z + 2( a − a ) z + ( a − a a − a ) (128)21 TABILITY ANALYSIS OF A NOVEL
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5, 2021As we have assumed that ω is the largest positive simple root of the equation (109), from Lemma 1, we get dh ( z ) dz (cid:107) z = ω > . (129)Hence, ω + 2( a − a ) ω + ( a − a a − a )( a ω − ω ) + ( a − a ω ) = dh ( z ) dz (cid:107) z = ω ( a ω − ω ) + ( a − a ω ) > (130)or Sign (cid:26) d ( (cid:60) ( λ )) dτ (cid:27) = Sign (cid:26) ω + 2( a − a ) ω + ( a − a a − a )( a ω − ω ) + ( a − a ω ) (cid:27) = 1 (131)i.e d ( (cid:60) ( λ )) dτ > (132)The Hopf bifurcation analysis above can be summarized in the following theorem. Theorem 13.
Suppose that a > , a + a > , a ( a + a ) − ( a + a ) > (133) and R > (134) If m < ∨ m ≥ , m < , (135) the infected steady state ¯ E of the delay model (85) is asymptotically stable when τ < τ and unstable when τ > τ ,where τ = 1 ω arccos (cid:18) a ω + ( a a − a a ) ω − a a a + a ω (cid:19) (136) When τ = τ , a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcates from ¯ E as τ passes throughthe critical value τ . After providing all the analytical tools and qualitatively analysing the system for patterns on its dynamics, in this section,we will perform some numerical analysis on the model to verify the previous results.
The numerical simulation is conducted on the programming language
Julia through the package
DifferentialEquation.jl , A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Juliaby Rackauckas and Nie (2017) [61].In order to avoid any stiffness in the ODE/DDE models, the algorithm for the Method of Steps in
Julia is set to
Rosenbrock23 , which is the same as the classic ODE solver ode23s in MATLAB .For the complete version of the
Julia notebooks for simulation, please refer to the
Github repository at https://github.com/hoanganhngo610/DDE-HIV-NGO2020etal. TABILITY ANALYSIS OF A NOVEL
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Parameters and Variables Values
Dependent variables T Uninfected CD4 + T-cell population size mm − I Infected CD4 + T-cell density mm − V Initial density of HIV RNA mm − Parameters and Constants s Source term for uninfected CD4 + T-cells day − mm − d Natural death rate of CD4 + T-cells 0.01 day -1 a Growth rate of CD4 + T-cell population . day − T max Maximal population level of CD4 + T-cells mm − β Rate CD4 + T-cells became infected with virus . × − mm − α Saturated mass-action term . ρ Rate of cure . day − δ Blanket death rate of infected CD4 + T-cells . day − q Reproduction rate of the infected CD4 + T-cells mm − day − c Death rate of free virus day − Table 1: Preliminary values of variables and parameters for viral spread.Parameters Original scenario Scenario s − − − d . − − − a . − − T max − − − β . × − − . . α .
001 0 . . . ρ .
01 0 . − − δ . − − q − . . c . . Table 2: Values of parameters for viral spread in different scenarios.Within the range of parameters that are proven to be realistic in medical research, we investigate the behavior of themodel within 4 different scenarios.•
The original scenario:
In this scenario, the condition , and in Theorem 7 are satisfied. This means that,the positive equilibrium of the system (4) is globally asymptotically stable.23 TABILITY ANALYSIS OF A NOVEL
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5, 2021Figure 1: The ODE model is locally asymptotically stable with parameters in the original scenario • Scenario
In this scenario, the conditions , and in Theorem 7 are satisfied. This means that, thepositive equilibrium of the system (4) is also globally asymptotically stable.Figure 2: The ODE model is locally asymptotically stable with parameters in Scenario • Scenario
In this scenario, the conditions and of Theorem 3 is satisfied. This means that, the positiveequilibrium of the system (4) is locally asymptotically stable.24 TABILITY ANALYSIS OF A NOVEL
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5, 2021Figure 3: The ODE model is locally asymptotically stable with parameters in
Scenario • Scenario
In this scenario, the conditions and of Theorem 8 is satisfied. This means that, the positiveequilibrium of the system (4) is orbitally asymptotically stable.Figure 4: The ODE model is orbitally asymptotically stable with parameters in Scenario TABILITY ANALYSIS OF A NOVEL
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Parameters (DDE) Original scenario (ODE) Scenario s − − − − d . − − − − a . − T max − − − − β . × − − − − − α .
001 0 . . . . ρ . − − − . δ . − − − − q − − − − c − − − − τ N/A . Table 3: Values of parameters for viral spread in different scenarios.First of all, instead of keeping α = 0 . , we modify this parameter into α = 0 . so that we can observe differentbehaviors while modifying other parameters.• Scenario
When τ = 0 . and all other variables are kept the same as the original scenario in theODE setting (apart from α ), T ( t ) , I ( t ) and V ( t ) all converges to their positive equilibrium. We say that, inthis setting, the positive equilibrium ¯ E is globally asymptotically stable.Figure 5: The DDE model is globally asymptotically stable with parameters in Scenario , with the delay term τ = 0 . .• Scenario
When we modify a = 5 and τ = 10 , the parameters, in this setting, satisfy the conditionsof Theorem 13. This means that, the positive equilibrium ¯ E is orbitally asymptotically stable, or in otherwords, there exists a positive periodic solution for all the components of the system.26 TABILITY ANALYSIS OF A NOVEL
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5, 2021Figure 6: The DDE model is globally asymptotically stable with parameters in
Scenario , with a = 5 and the delayterm τ = 10 .• From the last two scenarios, Scenario and
Scenario , we can draw a conclusion that thesolution of the system would return to stability when the cure rate ρ is increased. For example, if we select ρ = 0 . instead of ρ = 0 . with all other parameters kept identical, the system would admit a lower globalasymptotic stability. We can conclude that ρ is an important parameter in the sense that increasing it helps uscontrol the disease.Figure 7: The DDE model is globally asymptotically stable with parameters in Scenario and
Scenario , with ρ = 0 . and ρ = 0 . , respectively. These graphs show that the cure rate is an important parameter in controlling thedisease. 27 TABILITY ANALYSIS OF A NOVEL
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A List of macros for formatting text, figures and tables
Theorem 14 (Gronwall, 1919) . Let I denote an interval of the real line of the form [ a, inf) or [ a, b ] lr [ a, b ) with a < b Let β and u be real-valued continuous functions defined on I . If u is a differentiable in the interior I of I (the interval I without the end points a and possibly b ) and satisfies the differential inequality u (cid:48) ( t ) ≤ β ( t ) u ( t ) , t ∈ I (137) then u is bounded by the solution of the corresponding differential equation ν (cid:48) ( t ) = β ( t ) ν ( t ) : u ( t ) ≤ u ( a ) exp (cid:18)(cid:90) ta β ( s ) ds (cid:19) (138) Theorem 15 (Lyapunov’s stability) . Let a function V ( X ) be continuously differentiable in a neighbourhood U of theorigin. The function V ( X ) is called the Lyapunov function for an autonomous system X (cid:48) = f ( X ) (139) if the following conditions are met:1. V ( X ) > for all X ∈ U \ { } ;2. V(0) = 0;3. dVdt ≤ for all X ∈ U .Then, if in a neighborhood U of the zero solution X = 0 of an autonomous system there is a Lyapunov function V ( X ) with a negative definite derivative dVdt for all X ∈ U \ { } , then the equilibrium point X = 0 of the system isasymptotically stable. Theorem 16 (Perron - Frobenius theorem) . [17] Let A be a irreducible Metzler matrix (A Metzler matrix is a matrixwhose all of its off-diagonal elements are non-negative). Then, λ M , the eigenvalue of A of largest real part is real, andthe elements of its associated eigenvector v M are positive. Moreover, any eigenvector of A with non-negative elementsbelongs the the span of v M . Theorem 17 (Implicit Function Theorem (Chow and Hale, 1982)) . Suppose that • X, Y, Z are Banach spaces, • F : U × V → Z is continuously differentiable, • F ( x , y ) = 0 and D x F ( x , y ) has a bounded inverse.Then, there exists a neighborhood U × V ∈ U × V of ( x , y ) and a function f : V → U , f ( y ) = x such that F ( x, y ) = 0 for ( x, y ) ∈ U × V iff x = f ( y ) (140) If F ∈ C k ( U × V, Z ) , k ≥ or analytic in a neighborhood ( x , y ) , then f ∈ C k ( V , X ) or is analytic in aneighborhood of y . Theorem 18 (Poincaré - Bendixson theorem) . [73]Given a differentiable real dynamical system defined on an open subset of the plane, every non-empty compact ω -limitset of an orbit, which contains only finitely many fixed points, is either • a fixed point, • a periodic orbit, or • a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbitsconnecting these.Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could becountably many homoclinic orbits connecting one fixed point. TABILITY ANALYSIS OF A NOVEL
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5, 2021Next, we will give the definition of an additive compound matrix and consider the particular case when it’s a squarematrix [66]. A survey of properties of additive compound matrices, along with their connections to differential equationshave been investigated in [48, 38]We will start with the definition of the k -th exterior power (or multiplicative compound) of an n × m matrix. Definition 1 (Multiplicative compound of a matrix) . Let A be an n × m matrix of real or complex numbers. Let a i ,i ,...,i k ,j ,j ,...,j k be the minor of A determined by the rows ( i , ..., i k ) and the columns ( j , ..., j k ) , ≤ i < i <... < i k ≤ n, ≤ j < j < ... < j k ≤ m . The k -th multiplicative compound matrix A ( k ) of A is the (cid:0) nk (cid:1) × (cid:0) mk (cid:1) matrix whose entries, written in lexicographic order, are a i ,...,i k ,j ,...,j k . In particular, when A is an n × k matrix with columns a , a , ..., a k , A ( k ) is the exterior product a ∨ a ∨ ... ∨ a k .In the case m = n , the additive compound matrices are defined as follows. Definition 2.
Let A be an n × n matrix. The k -th additive compound A [ k ] of A is the (cid:0) nk (cid:1) × (cid:0) nk (cid:1) matrix given by A [ k ] = D ( I + hA ) (cid:107) h =0 (141) If B = A [ k ] , the following formula for b i,j can be deduced from the equation (141) , For any integer i = 1 , ..., (cid:0) nk (cid:1) , let ( i ) = ( i , i , ..., i k ) be the i -th member in the lexicographic ordering of all k -tuples of integers such that ≤ i < i <... < i k ≤ n . Then, b i,j = a i ,i + ... + a i k ,i k if ( i ) = ( j )( − r + s a i s ,j r if exactly one entry i s in ( i ) does not occur in ( j ) and j r does not occur in ( i ) , if ( i ) differs from ( j ) in two or more entries. (142)In the extreme cases when k = 1 and k = n , we would have that A [1] = A and A [ n ] = tr ( A ) . For n = 3 ,which is thecase that we are considering in this paper, we would have the matrices A [ k ] , k = 0 , , as follows: A [1] = A, A [2] = (cid:32) a + a a − a a a + a a − a a a + a , (cid:33) , A [3] = a + a + a (143) Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgement
The authors would like to thank Nguyen Tran Hai Yen, undergraduate student at the Faculty of Biology and Biotechnol-ogy, Ho Chi Minh University of Science, VNU - HCM, Class of 2022 for providing valuable biological insights andideas to support this research.
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