Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

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  • This article was downloaded by: [University of Chicago Library]On: 20 November 2014, At: 11:24Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

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    Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemicsMingxiang Chen a & Dominic P. Clemence aa Department of Mathematics , North Carolina A&T State University , Greenboro, NC, 27411,USAPublished online: 25 Jan 2007.

    To cite this article: Mingxiang Chen & Dominic P. Clemence (2006) Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemics, Journal of Difference Equations and Applications, 12:9, 887-899, DOI:10.1080/10236190600779791

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  • Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemics

    MINGXIANG CHEN* and DOMINIC P. CLEMENCE

    Department of Mathematics, North Carolina A&T State University, Greenboro, NC 27411, USA

    (Received 20 February 2006; revised 14 April 2006; in final form 27 April 2006)

    This paper considers a non-linear system of ordinary differential equations, which arises in the study ofhantavirus epidemics. The system has the property that the total population obeys the logistic equation.For this system, we use linearization and the dynamical properties of the logistic equation to analyze thedynamics of the subpopulation system. In view of the usual numerical instabilities associated withstandard finite difference methods used for simulating such systems, we construct non-standard finitedifference (NSFD) schemes, which preserve the dynamic properties of the system, and may therefore beused for its simulation.

    Keywords: Nonstandard finite difference schemes; Hantavirus epidemics; Finite difference methods;Equilibrium points

    1. Introduction

    The following basic model of mouse populations has been used in the study of Hantavirus

    epidemics [1]:

    dMs

    dt bM 2 cMs 2MsM

    K2 aMsMI 1

    dMI

    dt 2cMI 2MIM

    K aMsMI 2

    where Ms $ 0 and MI $ 0 are, respectively, the populations of susceptible and infected

    mice, b is the birth rate, c is the death rate, a is the infection rate, and K is related to the

    carrying capacity of the environment. Note that for system (1) and (2), the conservation law

    describing the dynamics of the total population, M Ms MI, is the logistic equationdM

    dt b2 cM 2M

    2

    K: 3

    Therefore, the carrying capacity for the total population is M* (b 2 c)K.Since the closed-form general solutions of the non-linear system (1) and (2), are

    not possible, any useful information from the system must be deduced from dynamical

    analysis and numerical simulation. The purpose of this note is two-fold: (i) to describe

    the dynamical properties of the system (1) and (2), and (ii) to propose two

    Journal of Difference Equations and Applications

    ISSN 1023-6198 print/ISSN 1563-5120 online q 2006 Taylor & Francis

    http://www.tandf.co.uk/journals

    DOI: 10.1080/10236190600779791

    *Corresponding author. Email: chen@ncat.eduEmail: clemence@ncat.edu

    Journal of Difference Equations and Applications,

    Vol. 12, No. 9, September 2006, 887899

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  • non-standard finite difference (NSFD) [9] schemes which may be used for the simulation of

    (1)(3).

    It is reasonable to expect that, in order to give reliable simulation results, any finite

    difference scheme used for equations (1) and (2) should preserve their dynamic properties.

    Standard finite difference numerical schemes, such as the Euler and RungeKutta methods,

    are known to yield unstable, or simply incorrect, results even for equation (3) [3,7]. In fact,

    numerical simulations show that, even with small step sizes and for some positive initial data,

    it takes only a few iterations for Euler and RungeKutta schemes to achieve negative values

    and then blow up (see table 1 in section 4). However, a novel class of finite difference

    schemes, commonly known as NSFD schemes [6,7], have been shown to remove such

    method-dependent instabilities (see, for example, [5] and [7]).

    In this note, the proposed NSFD schemes are compared to the standard Euler, RK-2, and

    RK-4 methods on their performance in preserving the following properties of the system

    (1)(3) for various time-step sizes h and initial conditions:

    (P1), Positivity: The scheme does not admit negative solutions with non-negative initial data,

    i.e. Ms(0) $ 0, MI(0) $ 0 implies that Ms(t) $ 0, MI(t) $ 0 for all t . 0.; (P2),

    Convergence: The scheme produce data that converge to correct equilibria.; (P3), Non-

    periodicity: The scheme does not possess solutions which are periodic.

    The NSFD schemes are shown to remain faithful to the dynamics of equations (1) and (2) for

    much larger h than the standard methods, with one in particular retaining system dynamics

    for all h . 0.

    In the next section, we describe the details of the global dynamics of the system (1), (2) and

    (3), and give the local stability analysis of its equilibria. The biological significance of the

    results presented is also discussed. In section 3, we construct two NSFD schemes, which may

    be used to simulate equations (1) and (2). Finally, in section 4, the results of the numerical

    simulation are given and discussed.

    2. Analysis of equilibria

    The dynamics of equation (3) with various parameter values are well-known and have the

    following features:

    . If b c, then equation (3) becomes dM=dt 2M 2=K, which has one stable equilibriumM 0.

    . If b c, then equation (3) can be written as dM/dt (b 2 c)M[1 2 (M/K(b 2 c))]. Theequation then has one stable equilibrium M 0 if b , c; and the equation has twoequilibria if b . c: M 0, which is unstable, and M K(b 2 c), which is stable.

    Table 1. Comparison of RK-4, RK-2, Euler and NSFD.

    Method RK-4 RK-2 Euler NSFD

    Result When n 6,(Ms, MI) ! (1, 21)

    When n 4,(Ms, MI) ! (1, 21)

    When n 7,(Ms, MI) ! (1, 21)

    (Ms, MI) ! (10, 10)

    K 40, a 0.1, b 1, c 0.5, h 0.1 M0s 200, M0I 230.

    M. Chen and D. P. Clemence888

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  • Using the equilibria M 0 and M K(b 2 c) of equation (3), we observe that the system(1) and (2) has the following non-negative equilibria:

    (1) If b # c, then equations (1) and (2) has only a single equilibrium (0, 0);

    (2) If b . c, and

    if K(b 2 c) 2 (b/a) # 0, then equations (1) and (2) has equilibria (0, 0) and(K(b 2 c), 0);

    if K(b 2 c) 2 (b/a) . 0, then equations (1) and (2) has equilibria (0, 0), (K(b 2 c),0) and ((b/a), K(b 2 c) 2 (b/a)).

    2.1 Global analysis

    To examine the global behavior of the system (1) and (2), we first examine its dynamical flow

    on the boundary of the first quadrant.

    On the positive MI-axis, Ms 0, MI . 0, so by equation (1), dMs/dt bMI . 0 and theflow is into the first quadrant. On the positive Ms-axis, MI 0, Ms . 0, so by equations (1)and (2), dMs=dt b2 cMs 2 M2s=K, and (dMI/dt) 0, so the flow stays on the Ms-axisand the dynamics is the same as that of the logistic equation (3). By continuity and

    uniqueness, the flow is thus invariant in the first quadrant.

    We then use the Liapunov function V(Ms, MI) Ms MI $ 0 in the first quadrant. Notethat the derivative of V through equations (1) and (2) is

    dV

    dt b2 cV 2 V

    2

    K:

    If b # c, then (dV/dt) , 0 for all V . 0, from which we conclude that (0, 0) is globally

    asymptotically stable.

    For the case b . c, in the region where V . K(b 2 c), (dV/dt) , 0, and where

    V . K(b 2 c), (dV/dt) , 0, so the limit set of any trajectory is on the line segment

    Ms MI K(b 2 c) with 0 , Ms # K(b 2 c). By uniqueness of solutions, the limit setmust be an equilibrium point on this line segment.

    Since for K # b/(a(b 2 c)) there is only one equilibrium (K(b 2 c), 0) on this line

    segment, the limit set must be (K(b 2 c), 0) (figures 2(b), 3(b)). For the case K . b/a(b 2 c),

    there are two equilibria on the line segment: (K(b 2 c), 0) and ((b/a), K(b 2 c) 2 (b/a)).

    Next, consider the flow of the system on the line segment. Using Ms MI K(b 2 c) inequation (2), we have

    dMI

    dt 2bMI aMsMI MIaMs 2 b:

    So (dMI/dt) , 0 if Ms , (b/a) and (dMI/dt) . 0 if Ms . (b/a). Thus ((b/a),K(b 2 c) 2 (b/a))

    is attracting and it is a global attractor of the the system (1) and (2) (figure 1(b)).

    2.2 Local analysis

    To reveal more detail about the local behavior near the equilibria, we find the Jacobian of

    equation (1) and (2) at each equilibrium point:

    J0; 0 b2 c b

    0 2c

    !; JKb2 c; 0

    2b2 c c2 aKb2 c0 2b aKb2 c

    !;

    Stability and NSFD schemes for a Hantavirus epidemics model 889

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  • and

    Jb

    a;Kb2 c2 b

    a

    2aKb2 c b 1 2 1aK

    2 b

    aK

    aKb2 c baK

    2 2b c 2b 1 2 1aK

    c0@

    1A:

    The eigenvalues of J(0, 0) are (b 2 c) and 2c. If b , c, then (0, 0) is a stable node. If

    b c, then (0, 0) is a degenerate node, and the global analysis above show that it is stillattracting in the first quadrant. If b . c, then (0, 0) is a saddle, and it is repelling in the first

    quadrant.

    Now consider the case b . c and the equilibrium (K(b 2 c), 0). It is easy to see that

    J(K(b 2 c), 0) has eigenvalues 2 (b 2 c), which is negative, and 2b aK(b 2 c). IfK , b/(a(b 2 c)), then both eigenvalues are negative and (K(b 2 c), 0) is a stable node.

    If K b/(a(b 2 c)), then (K(b 2 c), 0), is a degenerate node, and the global analysis showsthat it is still attracting in the first quadrant (figure 2(b)). If K . b/(a(b 2 c)) . 0, then

    (K(b 2 c), 0) is a saddle with the positive Ms-axis being the stable manifold, and the

    equilibrium is repelling for all other trajectories.

    Finally consider the case b . c and K . b/(a(b 2 c)), and the equilibrium ((b/a),

    K(b 2 c) 2 (b/a)). As J((b/a), K(b 2 c) 2 (b/a)) has trace 2aK(b 2 c) c , c 2 b , 0,and determinant (b 2 c[aK(b 2 c) 2 b ] . 0, it has two eigenvalues with negative real

    parts: in fact, the eigenvalues are 2 (b 2 c) , 0 and 2aK(b 2 c) b , 0. Therefore,((b/a), (K(b 2 c) 2 (b/a)) is a stable node, and it is a global attractor by global analysis

    (figure 1(b)).

    We summarize the above analysis and discussion in the following theorem.

    Theorem 1 The model given by equations (1) and (2) possesses the following stability

    properties:

    (i) If b # c, then the system has the unique equilibrium (0, 0), and it is globally

    asymptotically stable.

    (ii) If b . c and K # b/(a(b 2 c)), then the system has two equilibria: (0, 0), which is

    unstable, and (K(b 2 c), 0), which is globally asymptotically stable.

    Figure 1. Densitytime profile for the case K . Kc: K 40, with a 0.1, b 1, c 0.5, h 0.1.

    M. Chen and D. P. Clemence890

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  • (iii) If b . c and K . (b/a(b 2 c)), then the system has three equilibria: (0, 0) and

    (K(b 2 c), 0), which are unstable; and ((b/a), K(b 2 c) 2 (b/a)), which is globally

    asymptotically stable.

    2.3 Biological significance

    It is worthy to note that the mathematically natural conditions in the theorem

    have an equally natural biological interpretation in terms of the Ross threshold

    theorem. In particular, the epidemic is known to persist if the basic reproduction number

    R0 . 1, and to die out if R0 , 1. For the model studied in this note, direct

    calculation using the next generation approach [2] shows that R0 (Ka(b 2 c)/b).This dimensionless parameter R0 has interpretation as the number of infected

    mice resulting from each infected mouse during its infected lifetime. It is also well-

    known [4] that R0 , 1 is equivalent to l* , 0, where l* is the largest eigenvalue of

    the Jacobian at the disease-free equilibrium. When b . c, system (1) and (2) has a

    non-trivial disease-free equilibrium (K(b 2 c), 0) with l* max{2 (b 2 c),2b aK(b 2 c)}. If K . b/(a(b 2 c)), then l* . 0 and the epidemic will persist.If K , (b/a(b 2 c)), then l* , 0 and the epidemic will die out. Thus, Kc b/(a(b 2 c))is the threshold capacity.

    In the study of populations, Theorem 1 therefore has the following interpretation.

    (i) When the birth rate is not higher than the death rate, all the mice will always

    eventually die.

    (ii) At low carrying capacity (under or at the threshold value K b/(a(b 2 c)), any mousepopulation, infected or uninfected, will eventually converge to K(b 2 c) uninfected

    mice.

    (iii) When the carrying capacity is above the threshold value, and if there are any initially

    infected mice, the mouse subpopulations will coexist, and eventually tend to (b/a)

    uninfected and K(b 2 c) 2 (b/a) infected mice; if there are initially no infected

    mice, then the whole mouse population will be uninfected and eventually tend to

    K(b 2 c) mice.

    Figure 2. Phase portrait for the case K . Kc: K 40, with a 0.1, b 1, c 0.5, h 0.1.

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  • 3. Construction of non-standard numerical schemes

    The purpose of this section is to construct numerical schemes for the system (1) and (2). The

    proposed schemes should faithfully preserve the dynamical properties (P1)(P3) which are

    not preserved by Euler and RK methods.

    Our purpose is achieved by implementing the following non-standard...

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