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Analysis of a multiscale HIV-1 model coupling within-host viral dynamics and between-host transmission dynamics

  • There are many challenges to constitute the linkage from the macroscale to the microscale and analyze the multiscale model. We proposed a bidirectional coupling model with standard incidence which includes the interaction of between-host transmission dynamics and within-host viral dynamics, and investigated the dynamic behaviors of the multiscale system on two time-scales. We found that the multiscale system exhibits more complex dynamics including backward bifurcation, which means that the usual thresholds for infection control or virus elimination obtained from the epidemiological model or virus dynamic model may not act as threshold parameter under a certain condition. There may be multiple epidemic equilibriums, one of which is stable, although the basic reproduction number is less than 1. We numerically examine the synergistic impact between the macro and micro dynamics. In particular, increasing the drug efficacy can decrease the prevalence of disease. The contact rate may affect the number and size of equilibria of viral dynamics model by inducing the occurrence of backward bifurcation. The finding suggests that the effective control measures may include both the reduction in contact rate or transmission rate at the population level and the increase in drug efficacy at the individual level, and using these control measures together can effectively control the diseases.

    Citation: Yuyi Xue, Yanni Xiao. Analysis of a multiscale HIV-1 model coupling within-host viral dynamics and between-host transmission dynamics[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6720-6736. doi: 10.3934/mbe.2020350

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  • There are many challenges to constitute the linkage from the macroscale to the microscale and analyze the multiscale model. We proposed a bidirectional coupling model with standard incidence which includes the interaction of between-host transmission dynamics and within-host viral dynamics, and investigated the dynamic behaviors of the multiscale system on two time-scales. We found that the multiscale system exhibits more complex dynamics including backward bifurcation, which means that the usual thresholds for infection control or virus elimination obtained from the epidemiological model or virus dynamic model may not act as threshold parameter under a certain condition. There may be multiple epidemic equilibriums, one of which is stable, although the basic reproduction number is less than 1. We numerically examine the synergistic impact between the macro and micro dynamics. In particular, increasing the drug efficacy can decrease the prevalence of disease. The contact rate may affect the number and size of equilibria of viral dynamics model by inducing the occurrence of backward bifurcation. The finding suggests that the effective control measures may include both the reduction in contact rate or transmission rate at the population level and the increase in drug efficacy at the individual level, and using these control measures together can effectively control the diseases.


    Fractional calculus is a main branch of mathematics that can be considered as the generalisation of integration and differentiation to arbitrary orders. This hypothesis begins with the assumptions of L. Euler (1730) and G. W. Leibniz (1695). Fractional differential equations (FDEs) have lately gained attention and publicity due to their realistic and accurate computations [1,2,3,4,5,6,7]. There are various types of fractional derivatives, including Riemann–Liouville, Caputo, Grü nwald–Letnikov, Weyl, Marchaud, and Atangana. This topic's history can be found in [8,9,10,11]. Undoubtedly, fractional calculus applies to mathematical models of different phenomena, sometimes more effectively than ordinary calculus [12,13]. As a result, it can illustrate a wide range of dynamical and engineering models with greater precision. Applications have been developed and investigated in a variety of scientific and engineering fields over the last few decades, including bioengineering [14], mechanics [15], optics [16], physics [17], mathematical biology, electrical power systems [18,19,20] and signal processing [21,22,23].

    One of the definitions of fractional derivatives is Caputo-Fabrizo, which adds a new dimension in the study of FDEs. The new derivative's feature is that it has a nonsingular kernel, which is made from a combination of an ordinary derivative with an exponential function, but it has the same supplementary motivating properties with various scales as in the Riemann-Liouville fractional derivatives and Caputo. The Caputo-Fabrizio fractional derivative has been used to solve real-world problems in numerous areas of mathematical modelling for example, numerical solutions for groundwater pollution, the movement of waves on the surface of shallow water modelling [24], RLC circuit modelling [25], and heat transfer modelling [26,27] were discussed.

    Rach (1987), Bellomo and Sarafyan (1987) first compared the Adomian Decomposition method (ADM) [28,29,30,31,32] to the Picard method on a variety of examples. These methods have many benefits: they effectively work with various types of linear and nonlinear equations and also provide an analytic solution for all of these equations with no linearization or discretization. These methods are more realistic compared with other numerical methods as each technique is used to solve a specific type of equations, on the other hand ADM and Picard are useful for many types of equations. In the numerical examples provided, we compare ADM and Picard solutions of multidimentional fractional order equations with Caputo-Fabrizio.

    The fractional derivative of Caputo-Fabrizio for the function x(t) is defined as [33]

    CFDα0x(t)=B(α)1αt0dds(x(s)) eα1α(ts)ds, (1.1)

    and its corresponding fractional integral is

    CFIαx(t)=1αB(α)x(t)+αB(α)t0x (s)ds,    0<α<1, (1.2)

    where x(t) be continuous and differentiable on [0, T]. Also, in the above definition, the function B(α)>0 is a normalized function which satisfy the condition B(0)=B(1)=0. The relation between the Caputo–Fabrizio fractional derivate and its corresponding integral is given by

    (CFIα0)(CFDα0f(t))=f(t)f(a). (1.3)

    In this section, we will introduce a multidimentional FDE subject to the initial condition. Let α(0,1], 0<α1<α2<...,αm<1, and m is integer real number,

    CFDx=f(t,x,CFDα1x,CFDα2x,...,CFDαmx,) ,x(0)=c0, (2.1)

    where x=x(t),tJ=[0,T],TR+,xC(J).

    To facilitate the equation and make it easy for the calculation, we let x(t)=c0+X(t) so Eq (2.1) can be witten as

    CFDαX=f(t,c0+X,CFDα1X,CFDα2X,...,CFDαmX), X(0)=0. (2.2)

    the algorithm depends on converting the initial condition from a constant c0 to 0.

    Let CFDαX=y(t) then X=CFIαy, so we have

    CFDαiX= CFIααi CFDαX= CFIααiy,  i=1,2,...,m. (2.3)

    Substituting in Eq (2.2) we obtain

    y=f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy). (2.4)

    Assume f satisfies Lipschtiz condition with Lipschtiz constant L given by,

    |f(t,y0,y1,...,ym)||f(t,z0,z1,...,zm)|Lmi=0|yizi|, (2.5)

    which implies

    |f(t,c0+CFIαy,CFIαα1y,..,CFIααmy)f(t,c0+CFIαz,CFIαα1z,..,CFIααmz)|Lmi=0| CFIααiy CFIααiz|. (2.6)

    The solution algorithm of Eq (2.4) using ADM is,

    y0(t)=a(t)yn+1(t)=An(t), j0. (2.7)

    where a(t) pocesses all free terms in Eq (2.4) and An are the Adomian polynomials of the nonlinear term which takes the form [34]

    An=f(Sn)n1i=0Ai, (2.8)

    where f(Sn)=ni=0Ai. Later, this accelerated formula of Adomian polynomial will be used in convergence analysis and error estimation. The solution of Eq (2.4) can be written in the form,

    y(t)=i=0yi(t). (2.9)

    lastly, the solution of the Eq (2.4) takes the form

    x(t)=c0+X(t)=c0+ CFIαy(t). (2.10)

    At which we convert the parameter to the initial form y to x in Eq (2.10), so we have the solution of the original Eq (2.1).

    Define a mapping F:EE where E=(C[J],) is a Banach space of all continuous functions on J with the norm x= maxtϵJx(t).

    Theorem 3.1. Equation (2.4) has a unique solution whenever 0<ϕ<1 where ϕ=L(mi=0[(ααi)(T1)]+1B(ααi)).

    Proof. First, we define the mapping F:EE as

    Fy=f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy).

    Let y and zE are two different solutions of Eq (2.4). Then

    FyFz=f(t,c0+CFIαy,CFIαα1y,..,CFIααmy)f(t,c0+CFIαz,CFIαα1z,...,CFIααmz)

    which implies that

    |FyFz|=|f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy)f(t,c0+ CFIαz, CFIαα1z,..., CFIααmz)|Lmi=0| CFIααiy CFIααiz|Lmi=0|1(ααi)B(ααi)(yz)+ααiB(ααi)t0(yz)ds|FyFzLmi=01(ααi)B(ααi)maxtϵJ|yz|+ααiB(ααi)maxtϵJ|yz|t0dsLmi=01(ααi)B(ααi)yz+ααiB(ααi)yzTLyz(mi=01(ααi)B(ααi)+ααiB(ααi)T)Lyz(mi=0[(ααi)(T1)]+1B(ααi))ϕyz.

    under the condition 0<ϕ<1, the mapping F is contraction and hence there exists a unique solution yC[J] for the problem Eq (2.4) and this completes the proof.

    Theorem 3.2. The series solution of the problem Eq (2.4)converges if |y1(t)|<c and c isfinite.

    Proof. Define a sequence {Sp} such that Sp=pi=0yi(t) is the sequence of partial sums from the series solution i=0yi(t), we have

    f(t,c0+ CFIαy, CFIαα1y,..., CFIααmy)=i=0Ai,

    So

    f(t,c0+ CFIαSp, CFIαα1Sp,..., CFIααmSp)=pi=0Ai,

    From Eq (2.7) we have

    i=0yi(t)=a(t)+i=0Ai1

    let Sp,Sq be two arbitrary sums with pq. Now, we are going to prove that {Sp} is a Caushy sequence in this Banach space. We have

    Sp=pi=0yi(t)=a(t)+pi=0Ai1,Sq=qi=0yi(t)=a(t)+qi=0Ai1.
    SpSq=pi=0Ai1qi=0Ai1=pi=q+1Ai1=p1i=qAi1=f(t,c0+ CFIαSp1, CFIαα1Sp1,..., CFIααmSp1)f(t,c0+ CFIαSq1, CFIαα1Sq1,..., CFIααmSq1)
    |SpSq|=|f(t,c0+ CFIαSp1, CFIαα1Sp1,..., CFIααmSp1)f(t,c0+ CFIαSq1, CFIαα1Sq1,..., CFIααmSq1)|Lmi=0| CFIααiSp1 CFIααiSq1|Lmi=0|1(ααi)B(ααi)(Sp1Sq1)+ααiB(ααi)t0(Sp1Sq1)ds|Lmi=01(ααi)B(ααi)|Sp1Sq1|+ααiB(ααi)t0|Sp1Sq1|ds
    SpSqLmi=01(ααi)B(ααi)maxtϵJ|Sp1Sq1|+ααiB(ααi)maxtϵJ|Sp1Sq1|t0dsLSpSqmi=0(1(ααi)B(ααi)+ααiB(ααi)T)LSpSq(mi=0[(ααi)(T1)]+1B(ααi))ϕSpSq

    let p=q+1 then,

    Sq+1SqϕSqSq1ϕ2Sq1Sq2...ϕqS1S0

    From the triangle inequality we have

    SpSqSq+1Sq+Sq+2Sq+1+...SpSp1[ϕq+ϕq+1+...+ϕp1]S1S0ϕq[1+ϕ+...+ϕpq+1]S1S0ϕq[1ϕpq1ϕ]y1(t)

    Since 0<ϕ<1,pq then (1ϕpq)1. Consequently

    SpSqϕq1ϕy1(t)ϕq1ϕmaxtϵJ|y1(t)| (3.1)

    but |y1(t)|< and as q then, SpSq0 and hence, {Sp} is a Caushy sequence in this Banach space then the proof is complete.

    Theorem 3.3. The maximum absolute truncated error Eq (2.4)is estimated to be maxtϵJ|y(t)qi=0yi(t)|ϕq1ϕmaxtϵJ|y1(t)|

    Proof. From the convergence theorm inequality (Eq 3.1) we have

    SpSqϕq1ϕmaxtϵJ|y1(t)|

    but, Sp=pi=0yi(t) as p then, Spy(t) so,

    y(t)Sqϕq1ϕmaxtϵJ|y1(t)|

    so, the maximum absolute truncated error in the interval J is,

    maxtϵJ|y(t)qi=0yi(t)|ϕq1ϕmaxtϵJ|y1(t)| (3.2)

    and this completes the proof.

    In this part, we introduce several numerical examples with unkown exact solution and we will use inequality (Eq 3.2) to estimate the maximum absolute truncated error.

    Example 4.1. Application of linear FDE

    CFDx(t)+2aCFD1/2x(t)+bx(t)=0,       x(0)=1. (4.1)

    A Basset problem in fluid dynamics is a classical problem which is used to study the unsteady movement of an accelerating particle in a viscous fluid under the action of the gravity [36]

    Set

    X(t)=x(t)1

    Equation (4.1) will be

    CFDX(t)+2aCFD1/2X(t)+bX(t)=0,       X(0)=0. (4.2)

    Appling Eq (2.3) to Eq (4.2), and using initial condition, also we take a = 1, b = 1/2,

    y=122I1/2y12I y (4.3)

    Appling ADM to Eq (4.3), we find the solution algorithm become

    y0(t)=12,yi(t)=2 CFI1/2yi112 CFI yi1,     i1. (4.4)

    Appling Picard solution to Eq (4.2), we find the solution algorithm become

    y0(t)=12,yi(t)=122 CFI1/2yi112 CFI yi1,     i1. (4.5)

    From Eq (4.4), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.5), the solution using Picard technique is given by y(t)=Limiyi(t). Lately, the solution of the original problem Eq (4.2), is

    x(t)=1+ CFI y(t).

    One the same processor (q = 20), the time consumed using ADM is 0.037 seconds, while the time consumed using Picard is 7.955 seconds.

    Figure 1 gives a comparison between ADM and Picard solution of Ex. 4.1.

    Figure 1.  ADM and Picard solution of Ex. 4.1.

    Example 4.2. Consider the following nonlinear FDE [35]

    CFD1/2x=8t3/23πt7/44Γ(114)t44+18 CFD1/4x+14x2, x(0)=0. (4.6)

    Appling Eq (2.3) to Eq (4.6), and using initial condition,

    y=8t3/23πt7/44Γ(114)t44+18 CFI1/4y+14(CFI1/2y)2. (4.7)

    Appling ADM to Eq (4.7), we find the solution algorithm will be become

    y0(t)=8t3/23πt7/44Γ(114)t44,yi(t)=18 CFI1/4yi1+14(Ai1),     i1. (4.8)

    at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.

    Appling Picard solution to Eq (4.7), we find the the solution algorithm become

    y0(t)=8t3/23πt7/44Γ(114)t44,yi(t)=y0(t)+18 CFI1/4yi1+14(CFI1/2yi1)2,     i1. (4.9)

    From Eq (4.8), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.9), the solution using Picard technique is given by y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.7), is.

    x(t)= CFI1/2y.

    One the same processor (q = 2), the time consumed using ADM is 65.13 seconds, while the time consumed using Picard is 544.787 seconds.

    Table 1 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 2):

    Table 1.  Max. absolute error.
    q max. absolute error
    2 0.114548
    5 0.099186
    10 0.004363

     | Show Table
    DownLoad: CSV

    Figure 2 gives a comparison between ADM and Picard solution of Ex. 4.2.

    Figure 2.  ADM and Picard solution of Ex. 4.2.

    Example 4.3. Consider the following nonlinear FDE [35]

    CFDαx=3t2128125πt5+110(CFD1/2x)2,x(0)=0. (4.10)

    Appling Eq (2.3) to Eq (4.10), and using initial condition,

    y=3t2128125πt5+110(CFI1/2y)2 (4.11)

    Appling ADM to Eq (4.11), we find the solution algorithm become

    y0(t)=3t2128125πt5,yi(t)=110(Ai1),     i1 (4.12)

    at which Ai are Adomian polynomial of the nonliner term (CFI1/2y)2.

    Then appling Picard solution to Eq (4.11), we find the solution algorithm become

    y0(t)=3t2128125πt5,yi(t)=y0(t)+110(CFI1/2yi1)2,     i1. (4.13)

    From Eq (4.12), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.13), the solution is y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.11), is

    x(t)=CFIy(t).

    One the same processor (q = 4), the time consumed using ADM is 2.09 seconds, while the time consumed using Picard is 44.725 seconds.

    Table 2 showed the maximum absolute truncated error of of ADM solution (using Theorem 3.3) at different values of m (when t = 0:5; N = 4):

    Table 2.  Max. absolute error.
    q max. absolute error
    2 0.00222433
    5 0.0000326908
    10 2.88273*108

     | Show Table
    DownLoad: CSV

    Figure 3 gives a comparison between ADM and Picard solution of Ex. 4.3 with α=1.

    Figure 3.  ADM and Picard solution where of Ex. 4.3.

    Example 4.4. Consider the following nonlinear FDE [35]

    CFDαx=t2+12 CFDα1x+14 CFDα2x+16 CFDα3x+18x4,x(0)=0. (4.14)

    Appling Eq (2.3) to Eq (4.10), and using initial condition,

    y=t2+12(CFIαα1y)+14(CFIαα2y)+16(CFIαα3y)+18(CFIαy)4, (4.15)

    Appling ADM to Eq (4.15), we find the solution algorithm become

    y0(t)=t2,yi(t)=12(CFIαα1y)+14(CFIαα2y)+16(CFIαα3y)+18Ai1,  i1 (4.16)

    where Ai are Adomian polynomial of the nonliner term (CFIαy)4.

    Then appling Picard solution to Eq (4.15), we find the solution algorithm become

    y0(t)=t2,yi(t)=t2+12(CFIαα1yi1)+14(CFIαα2yi1)+16(CFIαα3yi1)+18(CFIαyi1)4     i1. (4.17)

    From Eq (4.16), the solution using ADM is given by y(t)=Limqqi=0yi(t) while from Eq (4.17), the solution using Picard technique is y(t)=Limiyi(t). Finally, the solution of the original problem Eq (4.14), is

    x(t)=CFIαy(t).

    One the same processor (q = 3), the time consumed using ADM is 0.437 seconds, while the time consumed using Picard is (16.816) seconds. Figure 4 shows a comparison between ADM and Picard solution of Ex. 4.4 atα=0.7,α1=0.1,α2=0.3,α3=0.5.

    Figure 4.  ADM and Picard solution where of Ex. 4.4.

    The Caputo-Fabrizo fractional deivative has a nonsingular kernel, and consequently, this definition is appropriate in solving nonlinear multidimensional FDE [37,38]. Since the selected numerical problems have an unkown exact solution, the formula (3.2) can be used to estimate the maximum absolute truncated error. By comparing the time taken on the same processor (i7-2670QM), it was found that the time consumed by ADM is much smaller compared with the Picard technique. Furthermore Picard gives a more accurate solution than ADM at the same interval with the same number of terms.

    The authors declare there is no conflict of interest.



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