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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
dt¼ bM 2 cMs 2
MsM
K2 aMsMI ð1Þ
dMI
dt¼ 2cMI 2
MIM
Kþ aMsMI ð2Þ
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 ¼ Ms þ MI, is the logistic equation
dM
dt¼ ðb2 cÞM 2
M 2
K: ð3Þ
Therefore, the carrying capacity for the total population is M* ¼ (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: [email protected]†Email: [email protected]
Journal of Difference Equations and Applications,
Vol. 12, No. 9, September 2006, 887–899
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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 Runge–Kutta 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 Runge–Kutta 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 ¼ c, then equation (3) becomes dM=dt ¼ 2M 2=K, which has one stable equilibrium
M ¼ 0.
. If b – c, then equation (3) can be written as dM/dt ¼ (b 2 c)M[1 2 (M/K(b 2 c))]. The
equation then has one stable equilibrium M ¼ 0 if b , c; and the equation has two
equilibria if b . c: M ¼ 0, which is unstable, and M ¼ 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 ¼ 6,(Ms, MI) ! (1, 21)
When n ¼ 4,(Ms, MI) ! (1, 21)
When n ¼ 7,(Ms, MI) ! (1, 21)
(Ms, MI) ! (10, 10)
K ¼ 40, a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1 M0s ¼ 200, M0
I ¼ 230.
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Using the equilibria M ¼ 0 and M ¼ 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 ¼ 0, MI . 0, so by equation (1), dMs/dt ¼ bMI . 0 and the
flow is into the first quadrant. On the positive Ms-axis, MI ¼ 0, Ms . 0, so by equations (1)
and (2), ðdMs=dtÞ ¼ ðb2 cÞMs 2 ðM2s=KÞ, and (dMI/dt) ¼ 0, so the flow stays on the Ms-axis
and 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) ¼ Ms þ MI $ 0 in the first quadrant. Note
that the derivative of V through equations (1) and (2) is
dV
dt¼ ðb2 cÞV 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 þ MI ¼ K(b 2 c) with 0 , Ms # K(b 2 c). By uniqueness of solutions, the limit set
must 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 þ MI ¼ K(b 2 c) in
equation (2), we have
dMI
dt¼ 2bMI þ aMsMI ¼ MIðaMs 2 bÞ:
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:
Jð0; 0Þ ¼b2 c b
0 2c
!; JðKðb2 cÞ; 0Þ ¼
2ðb2 cÞ c2 aKðb2 cÞ
0 2bþ aKðb2 cÞ
!;
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and
Jb
a;Kðb2 cÞ2
b
a
� �¼
2aKðb2 cÞ þ b 1 2 1aK
� �2 b
aK
aKðb2 cÞ þ baK
2 2bþ c 2b 1 2 1aK
� �þ c
0@
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 ¼ c, then (0, 0) is a degenerate node, and the global analysis above show that it is still
attracting 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 þ aK(b 2 c). If
K , b/(a(b 2 c)), then both eigenvalues are negative and (K(b 2 c), 0) is a stable node.
If K ¼ b/(a(b 2 c)), then (K(b 2 c), 0), is a degenerate node, and the global analysis shows
that 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) þ 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) þ 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. Density–time profile for the case K . Kc: K ¼ 40, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.
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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 ¼ (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* ¼ max{2 (b 2 c),
2b þ 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 ¼ 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 ¼ b/(a(b 2 c)), any mouse
population, 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 ¼ 40, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 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 rules of [7], which
are also formalized in [6]:
Definition 1 A finite difference scheme is said to be non-standard if one of the following
occurs:
(i) In the discretized expression of r(y) for the equation (dy/dt) ¼ r(y), all non-linear
terms are approximated in a non-local way by a suitable function of several mesh
points.
(ii) The usual denominator Dt ¼ h of the discrete derivative is replaced by a non-negative
function f(h), with f(h) ¼ h þ O(h 2) as h ! 0þ.
The results of implementing these rules are discrete models of equations (1) and (2), which
are stable in the following sense:
Definition 2 A finite difference scheme is said to be qualitatively stable with respect to a
property P of the exact solutions of the original differential equation(s), or P-stable, if the
discrete solutions replicate P for all step sizes.
3.1 NSFD scheme I
To construct the first scheme with the desired properties, we proceed as follows. We use a
forward-difference approximation for the derivatives. We rewrite the term bM 2 cMs in
equation (1) as (b 2 c)Ms þ bMI, and approximate the linear terms Ms, MI by Mns ;M
nI .
For the non-linear terms in equation (1), we approximate the terms MsM by Mnþ1s Mn
and MsMI by Mnþ1s Mn
I ; while for equation (2), we approximate the terms MIM by Mnþ1I Mn
and MsMI by MnsM
nþ1I . Note that the resulting scheme will be non-standard in the sense of
Definition 1 (i). Moreover, since the scheme is implicit, it has superior stability over any
explicit finite difference scheme.
The resulting discrete model for equations (1) and (2) is the following:
Mnþ1s 2Mn
s
h¼ ðb2 cÞMn
s þ bMnI 2
Mnþ1s Mn
K2 aMnþ1
s MnI ð4Þ
Mnþ1I 2Mn
I
h¼ 2cMn
I 2Mnþ1
I Mn
Kþ aMnþ1
s MnI ð5Þ
here and elsewhere Mns ¼ MsðtnÞ, M
nI ¼ MIðtnÞ, and h . 0 is the time step.
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It is not difficult to see that addition of equations (4) and (5) yields the following consistent
discretization of equation (3), with obvious notation:
Mnþ1 2Mn
h¼ ðb2 cÞMn 2
MnMnþ1
K; ð6Þ
which is known to possess the same stability as in equation (3) [9].
The implicit equations (4)–(6) can be solved to give the following explicit formulation of
the NSFD scheme:
Mnþ1s ¼
K½Mns þ ðb2 cÞhMn
s þ bhMnI �
K þ aKhMnI þ hðMn
s þMnI Þ
; ð7Þ
Mnþ1I ¼
K{ð1 2 chÞK þ ½ð1 2 chÞð1 þ aKÞ þ abhK�hðMns þMn
I Þ�}MnI
½K þ hðMns þMn
I Þ�½K þ aKhMnI þ hðMn
s þMnI Þ�
; ð8Þ
Mnþ1 ¼K½1 þ ðb2 cÞh�Mn
K þ hMn: ð9Þ
While it is clear that the right-hand sides of equations (7) and (9) do not admit negative
iterations for Mns and M n for b . c (the only biologically relevant case) and positive initial
conditions, this is not obvious of MnI for equation (8). However, we can see easily that MI will
be non-negative if we restrict the step size to h # (1/c). In fact, the scheme (7)–(9) displays
the correct dynamics for all h # (1/c). Observe that the condition h # (1/c) implies that the
lower the death rate, the larger the value of allowable step size for this scheme. Numerous
numerical simulations of equations (7)–(9) have shown that the scheme possesses the
required positivity and convergence properties even for very large step sizes (figures 5(b),
6(a),(b)).
Figure 3. Density–time profile for the case K ¼ Kc: K ¼ 20, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.
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3.2 NSFD scheme II
The positivity restriction h # (1/c) imposed by equation (8) may be eliminated by using
Definition 1 (ii) as follows. Since e2ch ¼ 1 2 chþ Oðh2Þ for 0 , h p 1, or
h ¼ ðð1 2 e2chÞ=cÞ þ Oðh2Þ, we see that the function
fðhÞ ¼1 2 e2ch
cð10Þ
clearly satisfies the condition in Definition 1 (ii). Replacing h by f(h) in equations (7)–(9),
we have the following NSFD scheme for (1)–(3):
Mnþ1s ¼
K½Mns þ ðb2 cÞfðhÞMn
s þ bfðhÞMnI �
K þ aKfðhÞMnI þ fðhÞðMn
s þMnI Þ
; ð11Þ
Mnþ1I ¼
K{ð1 2 cfðhÞÞK þ ½ð1 2 cfðhÞÞð1 þ aKÞ þ abfðhÞK�fðhÞðMns þMn
I Þ�}MnI
½K þ fðhÞðMns þMn
I Þ�½K þ aKfðhÞMnI þ fðhÞðMn
s þMnI Þ�
; ð12Þ
Mnþ1 ¼K½1 þ ðb2 cÞfðhÞ�Mn
K þ fðhÞMn: ð13Þ
Note that the scheme (11)–(13) is non-standard in terms of both (i) and (ii) of Definition 1.
Moreover, since 1 2 cf(h) ¼ e2ch . 0 for all h . 0, we see that the choice of f(h) given
by equation (10) in equations (11)–(13) yields a scheme which is stable with respect to the
properties (P1)–(P3).
4. Numerical experiments
In this section, we describe the results of numerical experiments conducted to assess the
competitiveness of the schemes presented. We used the same parameter values as in [1]:
a ¼ 0.1, b ¼ 1, c ¼ 0.5. The threshold capacity is Kc ¼ b/(a(b 2 c)) ¼ 20.
All schemes produce nearly identical profiles and phase portraits for small step size h and
initial data near the global attractor.
For the case K . Kc, figures 1 and 2 give the typical density–time profile and phase
portrait.
For the case K ¼ Kc, figures 3 and 4 give the typical density–time profile and phase
portrait.
For the case K , Kc, figures 5 and 6 give the typical density–time profile and phase
portrait.
From figures 1–6, it seems that the standard methods work well for small step sizes.
However, we find that they do not produce true approximations for even small step sizes and
some relatively large initial data. For example, let K ¼ 40 (K . Kc) and h ¼ 0.1, for the
initial data M0s ¼ 200;M0
I ¼ 230, it takes only a few iterations (n) for the Euler and RK
methods to diverge, while the NSFD scheme produces data approaching the correct
equilibrium (10, 10) (table 1).
Further, while for small step sizes, the Euler and RK methods can approximate the
dynamics of equations (1) and (2) for initial data near the attractor, the simulation yields false
dynamics if the step size is increased (figures 7, 8). Figure 7 shows that for the same initial
data M0s ¼ 15 and M0
s ¼ 7:5, RK-4 method with h ¼ 0.1 gives correct time–density profile
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Figure 5. Density–time profile for the case K , Kc: K ¼ 10, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.
Figure 4. Phase portrait for the case K ¼ Kc: K ¼ 20, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.
Figure 6. Phase portrait for the case K , Kc: K ¼ 10, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1. Phase portrait.
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that shows the convergence to attractor (10, 10), but RK-4 method with h ¼ 3.5
produces a false asymptotically 2-periodic solution. The RK-2 method displays the similar
phenomena.
Figure 8 shows that the Euler method yields irregular dynamics with step size h ¼ 1.
As a comparison, with the same step sizes h ¼ 1 and h ¼ 3.5, the NSFDI still provides
truthful approximation (figures 9 and 10).
It is interesting that, even if we let h ¼ 10, h ¼ 1000, the NSFD-I still gives the similar
global attractor, but the eigenvectors slightly changed (figures 11 and 12).
Notice that the NSFD-II scheme is obtained by replacing h in NSFD-I by
f(h) ¼ (1 2 e2ch)/c, and 0 , f(h) , (1/c) for all h . 0, thus the NSFD-II scheme
corresponds to a NSFD-I with the effective step-size less than (1/c) and therefore preserves
positivity and give the correct dynamics for all h . 0. To illustrate this, we include a time–
density profile and a phase portrait using NSFD-II with K ¼ 40 and step size h ¼ 1000
(figures 13 and 14).
Figure 7. Density–time plot for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5, initial data M0s ¼ 15 and M0
s ¼ 7:5,using RK4 method. The plot shows correct density–time profile with h ¼ 0.1 but displays the appearance of a falseperiod-2 solution with h ¼ 3.5.
Figure 8. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 1 using Euler methods.
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Figure 9. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step size h ¼ 1.
Figure 10. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 0.1, c ¼ 0.5 using NSFD-I scheme and the step sizeh ¼ 3.5.
Figure 11. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step size h ¼ 10.
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Figure 12. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step sizeh ¼ 1000.
Figure 13. Density–time profile for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-II scheme and the step sizeh ¼ 1.
Figure 14. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-II scheme and the step sizeh ¼ 1000.
M. Chen and D. P. Clemence898
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5. Conclusion
The stability of a hantavirus epidemics mouse population model has been studied using
various mathematical techniques and different ways of forming discretization of the
differential equations. The biological significance of the various mathematical results was
discussed, particularly with respect to the basic reproduction number and the extinction or
persistence of epidemics, Ross threshold Theorem. In view of the difficulties that are often
encountered in employing standard finite difference methods forcalculating numerical
solutions to differential equations, we constructed and analyzed two NSFD schemes to
overcome the issues raised by these difficulties. Both schemes are O(h) convergent and were
found to be P1-stable and P2-stable as determined by the mathematical properties of these
discrete schemes and verified by numerous simulations. It should be noted that while NSFD-I
appears to have a natural ‘time scale ’and this leads to a restriction on the step-size, it, along
with NSFD-II, both possess the required stability with respect to P1 and P2. Our general
conclusion is that, with regard to the mouse hantavirus epidemic model, NSFD schemes
provide superior numerical integration methods as compared to standard procedures such as
the Euler and Runge–Kutta schemes.
Acknowledgements
We thank Professor Gregory Gibson for his help with the figures; we are also grateful to
Professors Abba Gumel and Ronald Mickens for useful discussions on the topics
investigated. We also wish to thank Professor for his advice and analysis of his NSFD
methods.
The work of D. P. Clemence was partially supported by NASA grant NAG 9-1402; he is
also grateful for the hospitality of the Africa Institute for Mathematical Sciences where some
of the work was completed under a Victor Rothschild Fellowship.
References
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[2] Castillo-Chavez, C., Feng, Z. and Huang, W., 2002, In: C. Castillo-Chavez, P. Van den Driessche, D. Kirschnerand A.-A. Yakubu (Eds.) On the Computation R0 and its Role on Global Stability, Mathematical Approachesfor Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125 (Springer-Verlag),pp. 229–250.
[3] Chen, B. and Solis, F., 1998, Discretization of nonlinear differential equations using explicit finite ordermethods. Journal of Computation and Applied Mathematics, 90, 171–178.
[4] Diekmann, O., Hesterbeek, J.A.P. and Mets, A.J., 1990, On the definition of basic reproduction ration R0 inmodels for infections disease in heterogeneous populations, Journal of Mathematical Biology, 28, 365–382.
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