analysis of and numerical schemes for a mouse population model in hantavirus epidemics
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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
To link to this article: http://dx.doi.org/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
The following basic model of mouse populations has been used in the study of Hantavirus
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
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
*Corresponding author. Email: email@example.comEmail: firstname.lastname@example.org
Journal of Difference Equations and Applications,
Vol. 12, No. 9, September 2006, 887899
non-standard finite difference (NSFD)  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,  and ).
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
. 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
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