analysis of and numerical schemes for a mouse population model in hantavirus epidemics

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

Post on 27-Mar-2017




0 download

Embed Size (px)


  • 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

    Journal of Difference Equations and ApplicationsPublication details, including instructions for authors and subscription information:

    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

    To link to this article:


    Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

    This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

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


    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]:


    dt bM 2 cMs 2MsM

    K2 aMsMI 1


    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


    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

    DOI: 10.1080/10236190600779791

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

    Journal of Difference Equations and Applications,

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




    by [



    ity o

    f C


    go L



    at 1





    r 20


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


    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




    by [



    ity o

    f C


    go L



    at 1





    r 20


  • 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


    dt b2 cV 2 V



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