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  • Appendix A: Basic Concept of Liao’s Homotopy Analysis Method (HAM)

    Consider the following nonlinear differential equation

    (A1)

    where N is a nonlinear operator, t denotes an independent variable, is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the conventional Homotopy method, Liao constructed the so-called zero-order deformation equation as:

    (A2)

    where is the embedding parameter, is a nonzero auxiliary parameter, is an auxiliary function, L is an auxiliary linear operator, is an initial guess of, is an unknown function. It is important, that one has great freedom to choose auxiliary unknowns in HAM. Obviously, when and, it holds:

    and       (A3)

    respectively. Thus, as p increases from 0 to 1, the solution varies from the initial guess to the solution. Expanding in Taylor series with respect to p, we have:

    (A4)

    where

    (A5)

    If the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the auxiliary function are so properly chosen, the series (A4) converges at p = 1 then we have:

    . (A6)

    Define the vector

    (A7)

    Differentiating Equation (A2) for m times with respect to the embedding parameter p, and then setting and finally dividing them by m!, we will have the socalled mth-order deformation equation as:

    (A8)

    where

    (A9)

    and

    (A10)

    Applying on both side of Equation (A8), we get

    (A11)

    In this way, it is easily to obtain for at order, we have

    (A12)

    When, we get an accurate approximation of the original Equation (A1). For the convergence of the above method we refer the reader to Liao. If Equation (A1) admits unique solution, then this method will produce the unique solution. If Equation (A1) does not possess unique solution, the HAM will give a solution among many other (possible) solutions.

    Appendix B: Steady State Solution

    For the case of steady-state condition, the Equations (6) and (7) becomes as follows:

    (B1)

    (B2)

    Solving the above Equations (B1) and (B2), we get

    (B3)

    and

    . (B4)

    Thus we can obtain and as in the text (Equations (10) and (13)).

    Appendix C: Non-Steady State Solution of the Equations Using the HAM

    The given differential equations for the non-steady state condition are of the form as:

    (C1)

    (C2)

    For the transient part, the initial conditions are redefined as

    (C3)

    (C4)

    In order to solve the Equations (C1) and (C2) by means of the HAM, we first construct the Zeroth-order deformation equation by taking,

    (C5)

    (C6)

    We have the solution series as

    (C7)

    and

    (C8)

    where

    (C9)

    Substituting Equations (C7) and (C8) into Equations (C5) and (C6) and comparing the coefficient of like powers of p, we get

    (C10)

    (C11)

    (C12)

    (C13)

    and so on.

    The initial conditions are redefined as

    (C14)

    (C15)

    and

    for     (C16)

    Solving the Equations (C10) and (C10) by using the boundary conditions given in Equations (C14) and (C15), we get

    (C17)

    and

    (C18)

    Substituting the Equations (C17) and (C18) in Equations (C12) and (C12) and by using the boundary conditions given in Equation (C16), we get

    (C19)

    and

    (C20)

    Adding the Equations (C17) and (C19) and the Equations (C18) and (C20), we get the Equations (9) and (12) as in the text.

    Appendix D: MATLAB Program to Find the Numerical Solution of Non-Linear Equations (6) and (7)

    function main1 options= odeset('RelTol',1e-6,'Stats','on');

    Xo = [10; 2];

    tspan = [0,10];

    tic

    [t,X] = ode45(@TestFunction,tspan,Xo,options);

    toc figure hold on plot(t, X(:,1),'blue')

    plot(t, X(:,2),'green')

    return function [dx_dt]= TestFunction(t,x)

    a=16;

    b=3;

    r=10;

    dx_dt(1)  =x(1)*(1-(x(1)+x(2)))-(b*x(1)*x(2))/(x(1)+x(2));

    dx_dt(2)=r*x(2)*(1-(x(1)+x(2)))+(b*x(1)*x(2))/(x(1)+x(2))-a*x(2);

    dx_dt = dx_dt';

    Appendix E: Nomenclature

    Symbol     Meaning

            Size of the uninfected cell population

             Size of the infected cell population

              Rate of infected cell killing by the virus

              Transmission coefficient

              Maximum per capita growth rates of uninfected cells

              Maximum per capita growth rates of infected cells

             Carrying capacity

                Time

            Size of the dimensionless uninfected cell population

            Size of the dimensionless infected cell population

               Dimensionless time

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