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Threshold dynamics of an HIV infection model with two distinct cell subsets
Journal Pre-proof Threshold dynamics of an HIV infection model with two distinct cell subsets Xia Wang, Qing Ge, Yuming Chen
PII: DOI: Reference:
S0893-9659(20)30035-5 https://doi.org/10.1016/j.aml.2020.106242 AML 106242
To appear in:
Applied Mathematics Letters
Received date : 23 October 2019 Revised date : 14 January 2020 Accepted date : 14 January 2020 Please cite this article as: X. Wang, Q. Ge and Y. Chen, Threshold dynamics of an HIV infection model with two distinct cell subsets, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106242. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Journal Pre-proof *Manuscript Click here to view linked References
Threshold dynamics of an HIV infection model with two distinct cell subsets✩ Xia Wanga,∗, Qing Gea , Yuming Chenb a College b
of Mathematics and Statistics, Xinyang Normal University, 464000, Xinyang, P. R. China Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada
Abstract
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In this paper, we study an HIV infection model with two distinct cell subsets proposed by Hill et al. (Insight into treatment of HIV infection from viral dynamics models, Immunological Reviews, 285 (2018) 9-25). With the approach of Lyapunov functions, we establish a threshold dynamics with the basic reproduction number R0 being the threshold parameter. When R0 < 1 the infection-free equilibrium is globally asymptotically stable while when R0 > 1 the infectious equilibrium is globally asymptotically stable. Keywords: HIV infection, basic reproduction number, Lyapunov function, global stability 2000 MSC: 34D23, 92D30
1. Introduction
Jo
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HIV (Human Immunodeficiency Virus) is a virus that damages the immune system. It continues to be a major global public health issue. In 2018, around 770 000 people died from HIV-related causes in the world [8]. Without treatment, a person with HIV is likely to develop a serious condition called AIDS (Acquired Immune Deficiency Syndrome). The main treatment for HIV is antiretroviral therapy, a combination of daily medications that stops the virus from reproducing. Yet, despite the intensive studies done so far, we still do not have therapies that can permanently cure the infection. Mathematical modeling has played an important role in interpreting and predicting the time-course of viral levels during infection and how they are altered by treatment. This new field of research is called viral dynamics. In a recent survey, Hill et al. [3] summarized the contributions that viral dynamics models have made to understand the pathophysiology of HIV infection and to design effective therapies. Based on the work by Cardozo et al. [1], which suggests that “HIV infects two distinct cell subsets, one with a fast rate of integration and another with a slow rate of integration”, they proposed the following HIV infection model with two distinct cell subsets, T˙1 (t ) = λ1 − d T1 T1 (t ) − β1T1 (t )V (t ), I˙1 (t ) = β1 T1 (t )V (t ) − d I1 I1 (t ) − m1 I1 (t ), T˙2 (t ) = λ2 − d T2 T2 (t ) − β2T2 (t )V (t ), I˙2 (t ) = β2 T2 (t )V (t ) − d I2 I2 (t ) − m2 I2 (t ), (1.1) L˙ (t ) = f m2 I2 (t ) − a L (t ) − d L L (t ), ˙ (t ) = m1 I1 (t ) + (1 − f )m2 I2 (t ) − d M M (t ) + a L (t ), M ˙ V (t ) = k M (t ) − c V (t ). Here T1 and T2 are the densities of the short-and long-lived target cells, respectively; I1 and I2 denote the densities of corresponding immature infected cells; L and M stand for the densities of latently infected cells and mature infected cells; V represents the density of free viruses. The infection diagram is shown in Figure 3(A) of [3] and from which we can easily see the biological meanings of the parameters. Though they provided some decay ✩ This work is supported partially by the NSFC (No. 11771374), the Nanhu Scholar Program for Young Scholars of Xinyang Normal University, and the NSERC of Canada. ∗ Corresponding author at: College of Mathematics and Statistics, Xinyang Normal University, 464000, Xinyang, P. R. China. Email address: [email protected] (Xia Wang)
Preprint submitted to Elsevier
January 14, 2020
Journal Pre-proof
curves that are produced under different treatment regimes (which change some parameter values in (1.1) from the view of modeling), the dynamics of (1.1) has not been studied yet. The goal of this paper is to establish a threshold dynamics for (1.1) by employing the approach of Lyapunov functions in Section 2. The threshold dynamics is determined by the basic reproduction number. The paper concludes with a brief discussion. 2. A threshold dynamics
2
0 0 F = 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
β1 T10 β2 T20 0 0 0
and
d I1 + m 1 0 0 V= −m 1 0
0 d I2 + m 2 − f m2 −(1 − f )m2 0
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λ1 d T1
We start with the existence of equilibria of (1.1). Obviously, system (1.1) always has the unique infection-free equilibrium E 0 = (T10 , 0, T20 , 0, 0, 0, 0), where T10 = λ and T20 = d T2 . Let 0 0 a + dL −a 0
0 0 0 dM −k
0 0 0 . 0 c
Then by applying the next generation matrix method developed by van den Driessche and Watmough [7], the basic reproduction number R0 is the spectral radius of the next generation operator FV−1 . Thus R0 = ρ(FV−1 ) =
k m1 k m2 af β1 λ1 β2 λ2 + +1− f . d M (d I1 + m1 ) c d T1 d M (d I2 + m2 ) c d T2 a + d L
From the equilibrium equations, we can easily see that an equilibrium of (1.1) other than E 0 must be infectious. In fact, all components of such an equilibrium are positive. Let E ∗ = (T1∗ , I1∗ , T2∗ , I2∗ , L ∗ , M ∗ , V ∗ ) be an infectious equilibrium. Then after a simple calculation, we see that T1∗ = I2∗ =
λ1 d T1 +β1 V ∗ , β2 λ2 V ∗ (d I2 +m2 )(d T2 +β2 V ∗ ) ,
T2∗ = L∗ =
λ2 d T2 +β2 V ∗ , m2 f ∗ a +d L I 2 ,
β1 λ1 V ∗ (d I1 +m1 )(d T1 +β1 V ∗ ) , M ∗ = kc V ∗ ,
I1∗ =
(2.1)
where V ∗ is a positive zero of F (V ) with
m1 β1 λ1 m2 β2 λ2 dM c af F (V ) = + +1− f − . (d I1 + m1 )(d T1 + β1 V ) (d I2 + m2 )(d T2 + β2 V ) a + d L k m β 2λ
(2.2) m β 2λ
af
1 2 2 2 Note that F (0) = d Mk c (R0 − 1), limV →∞ F (V ) = − d Mk c < 0, and F ′ (V ) = − (d I +m11)(d1T +β 2 − (d +m )(d +β V )2 ( a +d + 1V ) I2 2 T2 2 L 1 1 1 − f ) < 0. By the Intermediate Value Theorem, F has no positive zeros if R0 ≤ 1 and it only has a unique positive zero if R0 > 1. Therefore, we have shown the following result on the equilibria of (1.1).
(i) If R0 ≤ 1 then (1.1) only has the infection-free equilibrium E 0 .
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Proposition 2.1.
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(ii) If R0 > 1 then besides E 0 , (1.1) also has a unique infectious equilibrium E ∗ = (T1∗ , I1∗ , T2∗ , I2∗ , L ∗ , M ∗ , V ∗ ), where V ∗ is the unique positive zero of F defined by (2.2) and the other components are given by (2.1). Though the theory of van den Driessche and Watmough [7] implies the stability of E 0 , in the following we use linearization to obtain the stability of equilibria of (1.1). Theorem 2.2. (i) The infection-free equilibrium E 0 of system (1.1) is locally asymptotically stable when R0 < 1 and unstable when R0 > 1. (ii) When R0 > 1, the infectious equilibrium E ∗ of system (1.1) is locally asymptotically stable. Proof. (i) The characteristic equation at E 0 is ∆1 (λ) ¬ (λ + d I1 + m1 )(λ + d I2 + m2 )(λ + a + d L )(λ + d M )(λ + c )q1(λ) = 0, 2
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β2 T20 m2 k a f +(1−f )(λ+a +d L )
β T 0m k
1 1 where q1 (λ) = 1 − (λ+d I +m2 )(λ+a +d L )(λ+d M )(λ+c ) − (λ+d I +m1 1 )(λ+d . Clearly, the stability of E 0 is determined by the M )(λ+c ) 2 1 roots of q1 (λ) = 0. If R0 > 1, then q1 (0) = 1 − R0 < 0. As limλ→∞ q1 (λ) = 1 > 0, we see that q1 (λ) = 0 has a positive root and hence E 0 is unstable if R0 > 1. Now assume that R0 < 1. We claim that all roots of q1 (λ) = 0 have negative real parts. Otherwise, let λ0 with Re(λ0 ) ≥ 0 satisfy q1 (λ0 ) = 0. Then from q1 (λ0 ) = 0 we get β2 T20 m2 k a f + (1 − f )(λ0 + a + d L ) β1 T10 m1 k + 1 = (λ0 + d I2 + m2 )(λ0 + a + d L )(λ0 + d M )(λ0 + c ) (λ0 + d I1 + m1 )(λ0 + d M )(λ0 + c ) β2 T20 m2 k β1 T10 m1 k af ≤ +1− f + (d I2 + m2 )d M c a + d L (d I1 + m1 )d M c = R0 ,
a contradiction. This proves the claim and it follows that E 0 is locally asymptotically stable when R0 < 1. (ii) The characteristic equation at E ∗ is ¬
=
(λ + d T1 + β1 V ∗ )(λ + d I1 + m1 )(λ + d T2 + β2 V ∗ )(λ + d I2 + m2 )(λ + a + d L )(λ + d M )(λ + c ) −(λ + d T1 + β1 V ∗ )(λ + d I1 + m1 )(λ + d T2 )β2 T2∗ k m2 [a f + (1 − f )(λ + a + d L )] −(λ + d T1 )β1 T1∗ k m1 (λ + d T2 + β2 V ∗ )(λ + d I2 + m2 )(λ + a + d L ) 0.
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∆2 (λ)
(2.3)
We use by way of contradiction to show that all roots of (2.3) have negative real parts. Otherwise, suppose that λ0 with Re(λ0 ) ≥ 0 is a root of ∆2 (λ) = 0. Then we get a contradiction as follows, β2 T2∗ m2 k (λ0 + d T2 )[a f + (1 − f )(λ0 + a + d L )] 1 = (λ0 + d T2 + β2 V ∗ )(λ0 + d I2 + m2 )(λ0 + a + d L )(λ0 + d M )(λ0 + c ) β1 T1∗ m1 k (λ0 + d T1 ) + 0 ∗ 0 0 0 (λ + d T1 + β1 V )(λ + d I1 + m1 )(λ + d M )(λ + c ) β2 T2∗ m2 k β2 T2∗ m2 k a f 1 − f + < 0 0 0 0 0 0 0 (λ + d I2 + m2 ) (λ + a + d L )(λ + d M )(λ + c ) (λ + d I2 + m2 ) (λ + d M )(λ + c ) β1 T1∗ m1 k + (λ0 + d I1 + m1 )(λ0 + d M )(λ0 + c ) β2 T2∗ m2 k β1 T1∗ m1 k af < +1− f + (d I2 + m2 )d M c a + d L (d I1 + m1 )d M c = 1. Therefore, E ∗ is locally asymptotically stable when R0 > 1.
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Now we establish the main result of this paper, a threshold dynamics of (1.1), by employing the approach of Lyapunov functions. During the discussion, we need the Volterra-type function g : (0, ∞) ∋ x 7→ x −1−ln x . Note that g (x ) ≥ 0 for x > 0 and g (x ) = 0 if and only if x = 1. We start with the global stability of E 0 .
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Theorem 2.3. Suppose that R0 < 1. Then the infection-free equilibrium E 0 of system (1.1) is globally asymptotically stable in R7+ . Proof. As E 0 is locally asymptotically stable when R0 < 1 by Theorem 2.2, it suffices to show that E 0 is globally attractive in R7+ . It is not difficult to see that T1 (t ) > 0 and T2 (t ) > 0 for t > 0 and every solution of (1.1). Thus the Lyapunov function U (t ) = U1 (t ) + U2(t ) + U3 (t ) is well-defined (without loss of generality), where T2 (t ) k m1 T10 k m2 T20 a f T1 (t ) g + 1 − f g U1 (t ) = + , d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L T10 T20 af k m2 k m1 I1 (t ) + + 1 − f I2 (t ), U2 (t ) = d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L 3
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U3 (t )
k ka L (t ) + V (t ) + M (t ). d M (a + d L ) dM
=
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Clearly, U (·) is non-negative and U (x ) = 0 if and only if x = E 0 . Along solutions of (1.1), we have T10 k m1 d U1 (t ) λ1 − d T1 T1 − β1 T1 V = 1− dt d M (d I1 + m1 ) T1 T20 k m2 af + +1− f 1− λ2 − d T2 T2 − β2 T2 V d M (d I2 + m2) a + d L T2 T10 T1 k m1 d T1 T10 k m1 − − β1 T1 V 2− = d M (d I1 + m1 ) T1 T10 d M (d I1 + m1 ) k m1 af k m2 0 + + 1 − f β2 T20 V β1 T1 V + d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L k m2 d T2 T20 T 0 T2 af +1− f 2− 2 − 0 + d M (d I2 + m2 ) a + d L T2 T2 af k m2 + 1 − f β2 T2 V − d M (d I2 + m2 ) a + d L
(2.4)
(here we have used λi = d Ti Ti 0 for i = 1, 2),
d U2 (t ) k m1 k m2 af = (β1 T1 V − d I1 I1 − m1 I1 ) + + 1 − f (β2 T2 V − d I2 I2 − m2 I2 ), (2.5) dt d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L
and
d U3 (t ) ka k = ( f m I2 − a L − d L L ) + k M − c V + m1 I1 + (1 − f )m2 I2 − d M M + a L . dt d M (a + d L ) dM Recall that R0 =
k m1 β1 T10 d M (d I1 +m1 )c
km β T 0
(2.6)
af
2 2 + d M (d I2 +m ( + 1 − f ). It follows from (2.4)–(2.6) that 2 )c a +d L 2
k m2 d T2 T20 k m1 d T1 T10 T10 T1 T20 T2 af d U (t ) + + (R0 − 1)c V . − +1− f 2− − = 2− dt d M (d I1 + m1 ) T1 T10 d M (d I2 + m2 ) a + d L T2 T20 Thus dUd t(t ) ≤ 0 when R0 < 1. Moreover if dUd t(t ) = 0 then T1 (t ) = T10 and T2 (t ) = T20 . This, combined with (1.1), easily implies that the largest invariant set of (1.1) in { dUd t(t ) = 0} is the singleton {E 0 }. Then the LaSalle’s Invariance Principle (see, for example, [2]) implies that E 0 is attractive in R7+ . This completes the proof. We mention that E 0 is globally attractive in R7+ when R0 = 1 by the proof of Theorem 2.3. Let
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Γ0 = {(T1 , I1 , T2 , I2 , L , M , V ) ∈ R7+ : I1 + I2 + L + M + V > 0}.
Then Γ0 is positively invariant. In fact, one can show that every solution of (1.1) is positive for t > 0 (i.e., every component is positive for t > 0). Moreover, R7+ \ Γ0 is positively invariant and every solution of (1.1) with initial conditions in it tends to E 0 as t → ∞.
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Theorem 2.4. Suppose that R0 > 1. Then the infectious equilibrium E ∗ of (1.1) is globally asymptotically stable in Γ0 . Proof. As for Theorem 2.3, it suffices to show the global attractivity of E ∗ in Γ0 . By the discussion before the theorem, we can define the Lyapunov function, W (t ) = W1 (t ) + W2(t ) + W3(t ), where W1 (t ) =
k m1 T1∗ k m2 T2∗ k m1 I1∗ k m2 I2∗ af af T2 (t ) T1 (t ) I1 (t ) d M (d I1 +m1 ) g ( T1∗ ) + d M (d I2 +m2 ) ( a +d L + 1 − f )g ( T2∗ ), W2 (t ) = d M (d I1 +m1 ) g ( I1∗ ) + d M (d I2 +m2 ) ( a +d L + 1 − ∗ ∗ W3 (t ) = d Mk(aa L+d L ) g ( LL(t∗ ) ) + kdMM g ( MM(t∗ ) ) + V ∗ g ( VV(t∗ ) ). Obviously, the Lyapunov function W (·) is non∗
f )g ( I2I(t∗ ) ), and 2 negative and W (E ) = 0.
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In the following, we calculate the time derivatives of W1 (t ), W2 (t ), and W3 (t ) along solutions of (1.1) one by one. Firstly, T∗ d W1 (t ) k m1 1 − 1 (λ1 − d T1 T1 − β1 T1 V ) = dt d M (d I1 + m1 ) T1 T∗ af k m2 + 1 − f 1 − 2 (λ2 − d T2 T2 − β2 T2 V ). + d M (d I2 + m2 ) a + d L T2
Similarly, d W2 (t ) dt
=
(2.8)
ka k a f m2 I2 L ∗ k a L ∗ k m1 I1 k k M∗ f m2 I2 − + + + (1 − f )m2 I2 − m1 I1 d M (a + d L ) d M (a + d L )L dM dM dM dM M ∗ ∗ k M∗ k M V − (1 − f )m2 I2 +kM ∗ − aL − cV −kM + c V ∗. dM M dM M V
(2.9)
=
Note that
β1 T1∗ V ∗ = (d I1 + m1 )I1∗ , m β T ∗V ∗ m β T ∗V ∗ af 2 2 2 1 + + 1 − f , V ∗ = kc M ∗ = c dkM d1 I 1+m d +m a +d 1 I 2 L ∗
∗
cV = kM =
k dM
1
m1 I1∗ + dkM
2
(1 − f
)m2 I2∗ + dkM
∗
aL ,
β2 T2∗ V ∗ = (d I2 + m2 )I2∗ = af ∗ a +d L m 2 I 2 , ∗ ∗ β k m 1 T1 V k ∗ 1 d M m 1 I 1 = d M d I1 +m1 .
a L∗ =
(d I2 +m2 )(a +d L )L ∗ , m2 f
It follows from (2.7)–(2.9) that =
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=
d W1 (t ) d W2 (t ) d W3 (t ) + + dt dt dt k m2 d T2 T2∗ k m1 d T1 T1∗ T1∗ T1 T2∗ T2 af − +1− f 2− − + 2− d M (d I1 + m1 ) T1 T1∗ d M (d I2 + m2 ) a + d L T2 T2∗ ∗ k m2 β2 T2∗ V ∗ T∗ T∗ k m1 β1 T1∗ V ∗ T1∗ T2 af − 1 − ln 1 − +1− f − 1 − ln 2 − d M (d I1 + m1 ) T1 T1 d M (d I2 + m2 ) a + d L T2 T2 ∗ ∗ ∗ ∗ k m1 β1 T1 V T1 V I T1 V I1 − − 1 − ln ∗ 1 d M (d I1 + m1 ) T1∗ V ∗ I1 T1 V ∗ I1 k m2 β2 T2∗ V ∗ T2 V I2∗ T2 V I2∗ af − +1− f − 1 − ln ∗ d M (d I2 + m2 ) a + d L T2∗ V ∗ I2 T2 V ∗ I2 ∗ ka f I2 L ∗ I1 M ∗ I2 L ∗ k ∗ I1 M − m I m2 I2∗ − 1 − ln − 1 − ln − 1 1 d M (a + d L ) I2∗ L I2∗ L dM I1∗ M I1∗ M I2 M ∗ I2 M ∗ LM ∗ k a L∗ LM ∗ k (1 − f )m2 I2∗ − 1 − ln − 1 − ln − − dM I2∗ M I2∗ M dM L ∗M L ∗M
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d W (t ) dt
(2.7)
af k m2 k m1 β1 T1 V k m1 β1 T1 V I1∗ k m1 − (I1 − I1∗ ) + + 1 − f β2 T2 V − d M (d I1 + m1 ) d M (d I1 + m1 ) I1 dM d M (d I2 + m2 ) a + d L I2∗ k m2 k m2 af af − + 1 − f β2 T2 V − + 1 − f (I2 − I2∗ ) d M (d I2 + m2 ) a + d L I2 dM a + dL
and d W3 (t ) dt
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This, combined with λ1 = d T1 T1∗ + β1 T1∗ V ∗ and λ2 = d T2 T2∗ + β2 T2∗ V ∗ , gives us k m1 d T1 T1∗ T ∗ T1 d W1 (t ) k m1 k m1 = β1 T1 V + β1 T1∗ V 2− 1 − ∗ − dt d M (d I1 + m1 ) T1 T1 d M (d I1 + m1 ) d M (d I1 + m1 ) T1∗ k m2 d T2 T2∗ T2∗ T2 af k m1 ∗ ∗ + β1 T1 V 1 − +1− f 2− − + d M (d I1 + m1 ) T1 d M (d I2 + m2 ) a + d L T2 T2∗ T∗ af k m2 + 1 − f β2 T2∗ V ∗ 1 − 2 − (β2 T2 V − β2 T2∗ V ) . + d M (d I2 + m2 ) a + d L T2
5
(2.10)
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=
Therefore,
d W (t ) dt
MV ∗ MV ∗ −k M − 1 − ln M ∗V M ∗V k m2 d T2 T2∗ T ∗ T2 T1∗ T1 k m1 d T1 T1∗ af − ∗ + +1− f 2− 2 − ∗ 2− d M (d I1 + m1 ) T1 T1 d M (d I2 + m2 ) a + d L T2 T2 ∗ ∗ ∗ T1 V I1 T T k m1 k m2 k m1 af − g 1 − +1− f g 2 − g dM T1 dM a + dL T2 dM T1∗ V ∗ I1 ∗ ∗ T2 V I2 k m1 I1∗ k a f m2 I2 k m2 I1 M ∗ I2 L ∗ af − +1− f g g g − − dM a + dL T2∗ V ∗ I2 d M (a + d L ) I2∗ L dM I1∗ M I2 M ∗ MV ∗ LM ∗ k a L∗ k − k M ∗g . (1 − f )m2 I2∗ g g − − ∗ dM I2 M dM L ∗M M ∗V ∗
≤ 0. Moreover, if
d W (t ) dt
= 0 then T1 (t ) = T1∗ , T2 (t ) = T2∗ , and
(T1 (t ), I1(t ), T2 (t ), I2(t ), L (t ), M (t ), V (t )) be a solution in
kM ∗ − c V (t ) = 0 V∗
= 0}. Then M (t ) =
= ∗
M V∗
I1 (t ) I1∗
=
I2 (t ) I2∗
=
L (t ) L∗
=
M (t ) M∗ .
Let
V (t ) and it follows from
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V˙ (t ) = k M (t ) − c V (t ) =
(t ) { dW dt
V (t ) V∗
that V is a positve constant and hence (T1 (t ), I1(t ), T2 (t ), I2(t ), L (t ), M (t ), V (t )) is a positive constant solution of (1.1), namely, an infectious equilibrium of it. By the unqiueness of infectious equilibria, we know that the (t ) maximal compact invariant set in { d W = 0} is the singleton {E ∗ }. Then an application of the LaSalle’s invaridt ance principle completes the proof. 3. Conclusion
References References
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In this paper, we considered an HIV infection model proposed by Hill et al. [3], where the target cells are divided into two subsets (one with a fast rate of integration and another with a slow rate of integration). With the approach of Lyapunov functions, we established a threshold dynamics determined by the basic reproduction number. When the basic reproduction number is less than unity, the virus will be cleared and otherwise the infection will persist. The theoretical result can help us to determine whether the therapy would be successful or not. On the one hand, the incidences in (1.1) are bilinear, β1 T1 (t )V and β2 T2 (t )V . However, it is well-known that nonlinear incidence rates are frequently used to describe the viral infection process based on experiment data and reasonable assumptions [4]. On the other hand, there is another transmission mode for HIV infection, the cell-to-cell transmission [5, 6]. We believe that the method here can be modified to deal with extensions of (1.1) by incorporating nonlinear incidences and/or the cell-to-cell transmission mode.
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[1] E. F. Cardozo, et al., Treatment with integrase inhibitor suggests a new interpretation of HIV RNA decay curves that reveals a subset of cells with slow integration, PLoS pathogens, 13(7) (2017) e1006478. [2] J. K. Hale, S. M. V. Lunel. Introduction to Functional Differential Equations, Applied Mathematical Sciences, vol 99. Springer-Verlag, New York, 1993. [3] A. L. Hill, D. I. S. Rosenbloom, M. A. Nowak, et al., Insight into treatment of HIV infection from viral dynamics models, Immunological Reviews, 285(1) (2018) 9-25. [4] A.R. Mclean, C.J. Bostock, Scrapie infections initiated at varying does: an analysis of 117 titration experiments, Philos. Trans. R. Soc. Lond. Ser. B, 355(1400) (2000) 1043-1050. [5] B. Monel, E. Beaumont, D. Vendrame, et al., HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, Journal of Virology, 86(7) (2012) 3924-3933. [6] X. Wang, L. Rong, HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission, Applied Mathematics Letters, 94 (2019) 44-51. [7] P. van den Driessche, J. Watmough., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1) (2002) 29-48. [8] The World Health Organization, HIV/AIDS., Key Facts, https://www.who.int/news-room/fact-sheets/detail/hiv-aids.
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*Credit Author Statement
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Credit author statement Xia Wang: Model analysis, Writing - Original draft preparation. Qing Ge: Model analysis,
Writing- Original draft preparation.
Jo
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na lP repr oo f
Yuming Chen: Writing- Reviewing and Editing
PII: DOI: Reference:
S0893-9659(20)30035-5 https://doi.org/10.1016/j.aml.2020.106242 AML 106242
To appear in:
Applied Mathematics Letters
Received date : 23 October 2019 Revised date : 14 January 2020 Accepted date : 14 January 2020 Please cite this article as: X. Wang, Q. Ge and Y. Chen, Threshold dynamics of an HIV infection model with two distinct cell subsets, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106242. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
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Threshold dynamics of an HIV infection model with two distinct cell subsets✩ Xia Wanga,∗, Qing Gea , Yuming Chenb a College b
of Mathematics and Statistics, Xinyang Normal University, 464000, Xinyang, P. R. China Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada
Abstract
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In this paper, we study an HIV infection model with two distinct cell subsets proposed by Hill et al. (Insight into treatment of HIV infection from viral dynamics models, Immunological Reviews, 285 (2018) 9-25). With the approach of Lyapunov functions, we establish a threshold dynamics with the basic reproduction number R0 being the threshold parameter. When R0 < 1 the infection-free equilibrium is globally asymptotically stable while when R0 > 1 the infectious equilibrium is globally asymptotically stable. Keywords: HIV infection, basic reproduction number, Lyapunov function, global stability 2000 MSC: 34D23, 92D30
1. Introduction
Jo
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HIV (Human Immunodeficiency Virus) is a virus that damages the immune system. It continues to be a major global public health issue. In 2018, around 770 000 people died from HIV-related causes in the world [8]. Without treatment, a person with HIV is likely to develop a serious condition called AIDS (Acquired Immune Deficiency Syndrome). The main treatment for HIV is antiretroviral therapy, a combination of daily medications that stops the virus from reproducing. Yet, despite the intensive studies done so far, we still do not have therapies that can permanently cure the infection. Mathematical modeling has played an important role in interpreting and predicting the time-course of viral levels during infection and how they are altered by treatment. This new field of research is called viral dynamics. In a recent survey, Hill et al. [3] summarized the contributions that viral dynamics models have made to understand the pathophysiology of HIV infection and to design effective therapies. Based on the work by Cardozo et al. [1], which suggests that “HIV infects two distinct cell subsets, one with a fast rate of integration and another with a slow rate of integration”, they proposed the following HIV infection model with two distinct cell subsets, T˙1 (t ) = λ1 − d T1 T1 (t ) − β1T1 (t )V (t ), I˙1 (t ) = β1 T1 (t )V (t ) − d I1 I1 (t ) − m1 I1 (t ), T˙2 (t ) = λ2 − d T2 T2 (t ) − β2T2 (t )V (t ), I˙2 (t ) = β2 T2 (t )V (t ) − d I2 I2 (t ) − m2 I2 (t ), (1.1) L˙ (t ) = f m2 I2 (t ) − a L (t ) − d L L (t ), ˙ (t ) = m1 I1 (t ) + (1 − f )m2 I2 (t ) − d M M (t ) + a L (t ), M ˙ V (t ) = k M (t ) − c V (t ). Here T1 and T2 are the densities of the short-and long-lived target cells, respectively; I1 and I2 denote the densities of corresponding immature infected cells; L and M stand for the densities of latently infected cells and mature infected cells; V represents the density of free viruses. The infection diagram is shown in Figure 3(A) of [3] and from which we can easily see the biological meanings of the parameters. Though they provided some decay ✩ This work is supported partially by the NSFC (No. 11771374), the Nanhu Scholar Program for Young Scholars of Xinyang Normal University, and the NSERC of Canada. ∗ Corresponding author at: College of Mathematics and Statistics, Xinyang Normal University, 464000, Xinyang, P. R. China. Email address: [email protected] (Xia Wang)
Preprint submitted to Elsevier
January 14, 2020
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curves that are produced under different treatment regimes (which change some parameter values in (1.1) from the view of modeling), the dynamics of (1.1) has not been studied yet. The goal of this paper is to establish a threshold dynamics for (1.1) by employing the approach of Lyapunov functions in Section 2. The threshold dynamics is determined by the basic reproduction number. The paper concludes with a brief discussion. 2. A threshold dynamics
2
0 0 F = 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
β1 T10 β2 T20 0 0 0
and
d I1 + m 1 0 0 V= −m 1 0
0 d I2 + m 2 − f m2 −(1 − f )m2 0
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λ1 d T1
We start with the existence of equilibria of (1.1). Obviously, system (1.1) always has the unique infection-free equilibrium E 0 = (T10 , 0, T20 , 0, 0, 0, 0), where T10 = λ and T20 = d T2 . Let 0 0 a + dL −a 0
0 0 0 dM −k
0 0 0 . 0 c
Then by applying the next generation matrix method developed by van den Driessche and Watmough [7], the basic reproduction number R0 is the spectral radius of the next generation operator FV−1 . Thus R0 = ρ(FV−1 ) =
k m1 k m2 af β1 λ1 β2 λ2 + +1− f . d M (d I1 + m1 ) c d T1 d M (d I2 + m2 ) c d T2 a + d L
From the equilibrium equations, we can easily see that an equilibrium of (1.1) other than E 0 must be infectious. In fact, all components of such an equilibrium are positive. Let E ∗ = (T1∗ , I1∗ , T2∗ , I2∗ , L ∗ , M ∗ , V ∗ ) be an infectious equilibrium. Then after a simple calculation, we see that T1∗ = I2∗ =
λ1 d T1 +β1 V ∗ , β2 λ2 V ∗ (d I2 +m2 )(d T2 +β2 V ∗ ) ,
T2∗ = L∗ =
λ2 d T2 +β2 V ∗ , m2 f ∗ a +d L I 2 ,
β1 λ1 V ∗ (d I1 +m1 )(d T1 +β1 V ∗ ) , M ∗ = kc V ∗ ,
I1∗ =
(2.1)
where V ∗ is a positive zero of F (V ) with
m1 β1 λ1 m2 β2 λ2 dM c af F (V ) = + +1− f − . (d I1 + m1 )(d T1 + β1 V ) (d I2 + m2 )(d T2 + β2 V ) a + d L k m β 2λ
(2.2) m β 2λ
af
1 2 2 2 Note that F (0) = d Mk c (R0 − 1), limV →∞ F (V ) = − d Mk c < 0, and F ′ (V ) = − (d I +m11)(d1T +β 2 − (d +m )(d +β V )2 ( a +d + 1V ) I2 2 T2 2 L 1 1 1 − f ) < 0. By the Intermediate Value Theorem, F has no positive zeros if R0 ≤ 1 and it only has a unique positive zero if R0 > 1. Therefore, we have shown the following result on the equilibria of (1.1).
(i) If R0 ≤ 1 then (1.1) only has the infection-free equilibrium E 0 .
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Proposition 2.1.
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(ii) If R0 > 1 then besides E 0 , (1.1) also has a unique infectious equilibrium E ∗ = (T1∗ , I1∗ , T2∗ , I2∗ , L ∗ , M ∗ , V ∗ ), where V ∗ is the unique positive zero of F defined by (2.2) and the other components are given by (2.1). Though the theory of van den Driessche and Watmough [7] implies the stability of E 0 , in the following we use linearization to obtain the stability of equilibria of (1.1). Theorem 2.2. (i) The infection-free equilibrium E 0 of system (1.1) is locally asymptotically stable when R0 < 1 and unstable when R0 > 1. (ii) When R0 > 1, the infectious equilibrium E ∗ of system (1.1) is locally asymptotically stable. Proof. (i) The characteristic equation at E 0 is ∆1 (λ) ¬ (λ + d I1 + m1 )(λ + d I2 + m2 )(λ + a + d L )(λ + d M )(λ + c )q1(λ) = 0, 2
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β2 T20 m2 k a f +(1−f )(λ+a +d L )
β T 0m k
1 1 where q1 (λ) = 1 − (λ+d I +m2 )(λ+a +d L )(λ+d M )(λ+c ) − (λ+d I +m1 1 )(λ+d . Clearly, the stability of E 0 is determined by the M )(λ+c ) 2 1 roots of q1 (λ) = 0. If R0 > 1, then q1 (0) = 1 − R0 < 0. As limλ→∞ q1 (λ) = 1 > 0, we see that q1 (λ) = 0 has a positive root and hence E 0 is unstable if R0 > 1. Now assume that R0 < 1. We claim that all roots of q1 (λ) = 0 have negative real parts. Otherwise, let λ0 with Re(λ0 ) ≥ 0 satisfy q1 (λ0 ) = 0. Then from q1 (λ0 ) = 0 we get β2 T20 m2 k a f + (1 − f )(λ0 + a + d L ) β1 T10 m1 k + 1 = (λ0 + d I2 + m2 )(λ0 + a + d L )(λ0 + d M )(λ0 + c ) (λ0 + d I1 + m1 )(λ0 + d M )(λ0 + c ) β2 T20 m2 k β1 T10 m1 k af ≤ +1− f + (d I2 + m2 )d M c a + d L (d I1 + m1 )d M c = R0 ,
a contradiction. This proves the claim and it follows that E 0 is locally asymptotically stable when R0 < 1. (ii) The characteristic equation at E ∗ is ¬
=
(λ + d T1 + β1 V ∗ )(λ + d I1 + m1 )(λ + d T2 + β2 V ∗ )(λ + d I2 + m2 )(λ + a + d L )(λ + d M )(λ + c ) −(λ + d T1 + β1 V ∗ )(λ + d I1 + m1 )(λ + d T2 )β2 T2∗ k m2 [a f + (1 − f )(λ + a + d L )] −(λ + d T1 )β1 T1∗ k m1 (λ + d T2 + β2 V ∗ )(λ + d I2 + m2 )(λ + a + d L ) 0.
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∆2 (λ)
(2.3)
We use by way of contradiction to show that all roots of (2.3) have negative real parts. Otherwise, suppose that λ0 with Re(λ0 ) ≥ 0 is a root of ∆2 (λ) = 0. Then we get a contradiction as follows, β2 T2∗ m2 k (λ0 + d T2 )[a f + (1 − f )(λ0 + a + d L )] 1 = (λ0 + d T2 + β2 V ∗ )(λ0 + d I2 + m2 )(λ0 + a + d L )(λ0 + d M )(λ0 + c ) β1 T1∗ m1 k (λ0 + d T1 ) + 0 ∗ 0 0 0 (λ + d T1 + β1 V )(λ + d I1 + m1 )(λ + d M )(λ + c ) β2 T2∗ m2 k β2 T2∗ m2 k a f 1 − f + < 0 0 0 0 0 0 0 (λ + d I2 + m2 ) (λ + a + d L )(λ + d M )(λ + c ) (λ + d I2 + m2 ) (λ + d M )(λ + c ) β1 T1∗ m1 k + (λ0 + d I1 + m1 )(λ0 + d M )(λ0 + c ) β2 T2∗ m2 k β1 T1∗ m1 k af < +1− f + (d I2 + m2 )d M c a + d L (d I1 + m1 )d M c = 1. Therefore, E ∗ is locally asymptotically stable when R0 > 1.
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Now we establish the main result of this paper, a threshold dynamics of (1.1), by employing the approach of Lyapunov functions. During the discussion, we need the Volterra-type function g : (0, ∞) ∋ x 7→ x −1−ln x . Note that g (x ) ≥ 0 for x > 0 and g (x ) = 0 if and only if x = 1. We start with the global stability of E 0 .
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Theorem 2.3. Suppose that R0 < 1. Then the infection-free equilibrium E 0 of system (1.1) is globally asymptotically stable in R7+ . Proof. As E 0 is locally asymptotically stable when R0 < 1 by Theorem 2.2, it suffices to show that E 0 is globally attractive in R7+ . It is not difficult to see that T1 (t ) > 0 and T2 (t ) > 0 for t > 0 and every solution of (1.1). Thus the Lyapunov function U (t ) = U1 (t ) + U2(t ) + U3 (t ) is well-defined (without loss of generality), where T2 (t ) k m1 T10 k m2 T20 a f T1 (t ) g + 1 − f g U1 (t ) = + , d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L T10 T20 af k m2 k m1 I1 (t ) + + 1 − f I2 (t ), U2 (t ) = d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L 3
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U3 (t )
k ka L (t ) + V (t ) + M (t ). d M (a + d L ) dM
=
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Clearly, U (·) is non-negative and U (x ) = 0 if and only if x = E 0 . Along solutions of (1.1), we have T10 k m1 d U1 (t ) λ1 − d T1 T1 − β1 T1 V = 1− dt d M (d I1 + m1 ) T1 T20 k m2 af + +1− f 1− λ2 − d T2 T2 − β2 T2 V d M (d I2 + m2) a + d L T2 T10 T1 k m1 d T1 T10 k m1 − − β1 T1 V 2− = d M (d I1 + m1 ) T1 T10 d M (d I1 + m1 ) k m1 af k m2 0 + + 1 − f β2 T20 V β1 T1 V + d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L k m2 d T2 T20 T 0 T2 af +1− f 2− 2 − 0 + d M (d I2 + m2 ) a + d L T2 T2 af k m2 + 1 − f β2 T2 V − d M (d I2 + m2 ) a + d L
(2.4)
(here we have used λi = d Ti Ti 0 for i = 1, 2),
d U2 (t ) k m1 k m2 af = (β1 T1 V − d I1 I1 − m1 I1 ) + + 1 − f (β2 T2 V − d I2 I2 − m2 I2 ), (2.5) dt d M (d I1 + m1 ) d M (d I2 + m2 ) a + d L
and
d U3 (t ) ka k = ( f m I2 − a L − d L L ) + k M − c V + m1 I1 + (1 − f )m2 I2 − d M M + a L . dt d M (a + d L ) dM Recall that R0 =
k m1 β1 T10 d M (d I1 +m1 )c
km β T 0
(2.6)
af
2 2 + d M (d I2 +m ( + 1 − f ). It follows from (2.4)–(2.6) that 2 )c a +d L 2
k m2 d T2 T20 k m1 d T1 T10 T10 T1 T20 T2 af d U (t ) + + (R0 − 1)c V . − +1− f 2− − = 2− dt d M (d I1 + m1 ) T1 T10 d M (d I2 + m2 ) a + d L T2 T20 Thus dUd t(t ) ≤ 0 when R0 < 1. Moreover if dUd t(t ) = 0 then T1 (t ) = T10 and T2 (t ) = T20 . This, combined with (1.1), easily implies that the largest invariant set of (1.1) in { dUd t(t ) = 0} is the singleton {E 0 }. Then the LaSalle’s Invariance Principle (see, for example, [2]) implies that E 0 is attractive in R7+ . This completes the proof. We mention that E 0 is globally attractive in R7+ when R0 = 1 by the proof of Theorem 2.3. Let
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Γ0 = {(T1 , I1 , T2 , I2 , L , M , V ) ∈ R7+ : I1 + I2 + L + M + V > 0}.
Then Γ0 is positively invariant. In fact, one can show that every solution of (1.1) is positive for t > 0 (i.e., every component is positive for t > 0). Moreover, R7+ \ Γ0 is positively invariant and every solution of (1.1) with initial conditions in it tends to E 0 as t → ∞.
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Theorem 2.4. Suppose that R0 > 1. Then the infectious equilibrium E ∗ of (1.1) is globally asymptotically stable in Γ0 . Proof. As for Theorem 2.3, it suffices to show the global attractivity of E ∗ in Γ0 . By the discussion before the theorem, we can define the Lyapunov function, W (t ) = W1 (t ) + W2(t ) + W3(t ), where W1 (t ) =
k m1 T1∗ k m2 T2∗ k m1 I1∗ k m2 I2∗ af af T2 (t ) T1 (t ) I1 (t ) d M (d I1 +m1 ) g ( T1∗ ) + d M (d I2 +m2 ) ( a +d L + 1 − f )g ( T2∗ ), W2 (t ) = d M (d I1 +m1 ) g ( I1∗ ) + d M (d I2 +m2 ) ( a +d L + 1 − ∗ ∗ W3 (t ) = d Mk(aa L+d L ) g ( LL(t∗ ) ) + kdMM g ( MM(t∗ ) ) + V ∗ g ( VV(t∗ ) ). Obviously, the Lyapunov function W (·) is non∗
f )g ( I2I(t∗ ) ), and 2 negative and W (E ) = 0.
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In the following, we calculate the time derivatives of W1 (t ), W2 (t ), and W3 (t ) along solutions of (1.1) one by one. Firstly, T∗ d W1 (t ) k m1 1 − 1 (λ1 − d T1 T1 − β1 T1 V ) = dt d M (d I1 + m1 ) T1 T∗ af k m2 + 1 − f 1 − 2 (λ2 − d T2 T2 − β2 T2 V ). + d M (d I2 + m2 ) a + d L T2
Similarly, d W2 (t ) dt
=
(2.8)
ka k a f m2 I2 L ∗ k a L ∗ k m1 I1 k k M∗ f m2 I2 − + + + (1 − f )m2 I2 − m1 I1 d M (a + d L ) d M (a + d L )L dM dM dM dM M ∗ ∗ k M∗ k M V − (1 − f )m2 I2 +kM ∗ − aL − cV −kM + c V ∗. dM M dM M V
(2.9)
=
Note that
β1 T1∗ V ∗ = (d I1 + m1 )I1∗ , m β T ∗V ∗ m β T ∗V ∗ af 2 2 2 1 + + 1 − f , V ∗ = kc M ∗ = c dkM d1 I 1+m d +m a +d 1 I 2 L ∗
∗
cV = kM =
k dM
1
m1 I1∗ + dkM
2
(1 − f
)m2 I2∗ + dkM
∗
aL ,
β2 T2∗ V ∗ = (d I2 + m2 )I2∗ = af ∗ a +d L m 2 I 2 , ∗ ∗ β k m 1 T1 V k ∗ 1 d M m 1 I 1 = d M d I1 +m1 .
a L∗ =
(d I2 +m2 )(a +d L )L ∗ , m2 f
It follows from (2.7)–(2.9) that =
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=
d W1 (t ) d W2 (t ) d W3 (t ) + + dt dt dt k m2 d T2 T2∗ k m1 d T1 T1∗ T1∗ T1 T2∗ T2 af − +1− f 2− − + 2− d M (d I1 + m1 ) T1 T1∗ d M (d I2 + m2 ) a + d L T2 T2∗ ∗ k m2 β2 T2∗ V ∗ T∗ T∗ k m1 β1 T1∗ V ∗ T1∗ T2 af − 1 − ln 1 − +1− f − 1 − ln 2 − d M (d I1 + m1 ) T1 T1 d M (d I2 + m2 ) a + d L T2 T2 ∗ ∗ ∗ ∗ k m1 β1 T1 V T1 V I T1 V I1 − − 1 − ln ∗ 1 d M (d I1 + m1 ) T1∗ V ∗ I1 T1 V ∗ I1 k m2 β2 T2∗ V ∗ T2 V I2∗ T2 V I2∗ af − +1− f − 1 − ln ∗ d M (d I2 + m2 ) a + d L T2∗ V ∗ I2 T2 V ∗ I2 ∗ ka f I2 L ∗ I1 M ∗ I2 L ∗ k ∗ I1 M − m I m2 I2∗ − 1 − ln − 1 − ln − 1 1 d M (a + d L ) I2∗ L I2∗ L dM I1∗ M I1∗ M I2 M ∗ I2 M ∗ LM ∗ k a L∗ LM ∗ k (1 − f )m2 I2∗ − 1 − ln − 1 − ln − − dM I2∗ M I2∗ M dM L ∗M L ∗M
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d W (t ) dt
(2.7)
af k m2 k m1 β1 T1 V k m1 β1 T1 V I1∗ k m1 − (I1 − I1∗ ) + + 1 − f β2 T2 V − d M (d I1 + m1 ) d M (d I1 + m1 ) I1 dM d M (d I2 + m2 ) a + d L I2∗ k m2 k m2 af af − + 1 − f β2 T2 V − + 1 − f (I2 − I2∗ ) d M (d I2 + m2 ) a + d L I2 dM a + dL
and d W3 (t ) dt
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This, combined with λ1 = d T1 T1∗ + β1 T1∗ V ∗ and λ2 = d T2 T2∗ + β2 T2∗ V ∗ , gives us k m1 d T1 T1∗ T ∗ T1 d W1 (t ) k m1 k m1 = β1 T1 V + β1 T1∗ V 2− 1 − ∗ − dt d M (d I1 + m1 ) T1 T1 d M (d I1 + m1 ) d M (d I1 + m1 ) T1∗ k m2 d T2 T2∗ T2∗ T2 af k m1 ∗ ∗ + β1 T1 V 1 − +1− f 2− − + d M (d I1 + m1 ) T1 d M (d I2 + m2 ) a + d L T2 T2∗ T∗ af k m2 + 1 − f β2 T2∗ V ∗ 1 − 2 − (β2 T2 V − β2 T2∗ V ) . + d M (d I2 + m2 ) a + d L T2
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(2.10)
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=
Therefore,
d W (t ) dt
MV ∗ MV ∗ −k M − 1 − ln M ∗V M ∗V k m2 d T2 T2∗ T ∗ T2 T1∗ T1 k m1 d T1 T1∗ af − ∗ + +1− f 2− 2 − ∗ 2− d M (d I1 + m1 ) T1 T1 d M (d I2 + m2 ) a + d L T2 T2 ∗ ∗ ∗ T1 V I1 T T k m1 k m2 k m1 af − g 1 − +1− f g 2 − g dM T1 dM a + dL T2 dM T1∗ V ∗ I1 ∗ ∗ T2 V I2 k m1 I1∗ k a f m2 I2 k m2 I1 M ∗ I2 L ∗ af − +1− f g g g − − dM a + dL T2∗ V ∗ I2 d M (a + d L ) I2∗ L dM I1∗ M I2 M ∗ MV ∗ LM ∗ k a L∗ k − k M ∗g . (1 − f )m2 I2∗ g g − − ∗ dM I2 M dM L ∗M M ∗V ∗
≤ 0. Moreover, if
d W (t ) dt
= 0 then T1 (t ) = T1∗ , T2 (t ) = T2∗ , and
(T1 (t ), I1(t ), T2 (t ), I2(t ), L (t ), M (t ), V (t )) be a solution in
kM ∗ − c V (t ) = 0 V∗
= 0}. Then M (t ) =
= ∗
M V∗
I1 (t ) I1∗
=
I2 (t ) I2∗
=
L (t ) L∗
=
M (t ) M∗ .
Let
V (t ) and it follows from
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V˙ (t ) = k M (t ) − c V (t ) =
(t ) { dW dt
V (t ) V∗
that V is a positve constant and hence (T1 (t ), I1(t ), T2 (t ), I2(t ), L (t ), M (t ), V (t )) is a positive constant solution of (1.1), namely, an infectious equilibrium of it. By the unqiueness of infectious equilibria, we know that the (t ) maximal compact invariant set in { d W = 0} is the singleton {E ∗ }. Then an application of the LaSalle’s invaridt ance principle completes the proof. 3. Conclusion
References References
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In this paper, we considered an HIV infection model proposed by Hill et al. [3], where the target cells are divided into two subsets (one with a fast rate of integration and another with a slow rate of integration). With the approach of Lyapunov functions, we established a threshold dynamics determined by the basic reproduction number. When the basic reproduction number is less than unity, the virus will be cleared and otherwise the infection will persist. The theoretical result can help us to determine whether the therapy would be successful or not. On the one hand, the incidences in (1.1) are bilinear, β1 T1 (t )V and β2 T2 (t )V . However, it is well-known that nonlinear incidence rates are frequently used to describe the viral infection process based on experiment data and reasonable assumptions [4]. On the other hand, there is another transmission mode for HIV infection, the cell-to-cell transmission [5, 6]. We believe that the method here can be modified to deal with extensions of (1.1) by incorporating nonlinear incidences and/or the cell-to-cell transmission mode.
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[1] E. F. Cardozo, et al., Treatment with integrase inhibitor suggests a new interpretation of HIV RNA decay curves that reveals a subset of cells with slow integration, PLoS pathogens, 13(7) (2017) e1006478. [2] J. K. Hale, S. M. V. Lunel. Introduction to Functional Differential Equations, Applied Mathematical Sciences, vol 99. Springer-Verlag, New York, 1993. [3] A. L. Hill, D. I. S. Rosenbloom, M. A. Nowak, et al., Insight into treatment of HIV infection from viral dynamics models, Immunological Reviews, 285(1) (2018) 9-25. [4] A.R. Mclean, C.J. Bostock, Scrapie infections initiated at varying does: an analysis of 117 titration experiments, Philos. Trans. R. Soc. Lond. Ser. B, 355(1400) (2000) 1043-1050. [5] B. Monel, E. Beaumont, D. Vendrame, et al., HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, Journal of Virology, 86(7) (2012) 3924-3933. [6] X. Wang, L. Rong, HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission, Applied Mathematics Letters, 94 (2019) 44-51. [7] P. van den Driessche, J. Watmough., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1) (2002) 29-48. [8] The World Health Organization, HIV/AIDS., Key Facts, https://www.who.int/news-room/fact-sheets/detail/hiv-aids.
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*Credit Author Statement
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Credit author statement Xia Wang: Model analysis, Writing - Original draft preparation. Qing Ge: Model analysis,
Writing- Original draft preparation.
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Yuming Chen: Writing- Reviewing and Editing