A Positivity-preserving Mickens-typeDiscretization of an Epidemic Model

S.M. MOGHADASa,*, M.E. ALEXANDERb, B.D. CORBETTa and A.B. GUMELa

aDepartment of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2;bInstitute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba,Canada, R3B 1Y6

(Received 4 February 2003; In final form 30 April 2003)

Dedicated to Professor Ronald E. Mickens on the occasion of his 60th Birthday

A deterministic model for the transmission dynamics of two strains of an epidemic in the presence of a preventivevaccine is considered. Theoretical results on the existence and stability of the associated equilibria of the model aregiven. A robust, positivity-preserving, non-standard finite-difference scheme, having the same qualitative features asthe continuous model, is constructed. The theoretical and numerical analyses of the model enable the determinationof a threshold level of vaccination coverage needed for community-wide eradication of the epidemic.

Keywords: Basic reproductive number; Mickens-type discretization; Epidemic model; Preventive vaccine

INTRODUCTION

Over the last few decades, scientists and public health officials have often resorted to the use

of mathematical models to gain deeper insights into the mechanisms of disease transmission

and to evaluate control strategies (see Refs. [1,7] and the references therein). The resulting

compartmental models, mostly deterministic and non-linear in nature, are generally solved

(simulated) using standard numerical integrators (such as explicit Runge-Kutta methods

[4,6,8]). These methods, however, are known to exhibit scheme-dependent instabilities

and/or converge to spurious solutions for certain values of the discretization and model

parameters [4,6,8].

In this paper, a robust Mickens-type finite-difference method will be constructed and used

to solve a deterministic model for the transmission dynamics of two strains of an epidemic in

the presence of a preventive vaccine. The model, which is a very slight modification of the

model in Ref. [7], consists of the following equations:

dS

dt¼ ð1 2 pÞP2 bwIwS 2 brIrS 2 mS; ð1Þ

dV

dt¼ pP2 bwð1 2 fwÞIwV 2 brð1 2 frÞIrV 2 mV; ð2Þ

ISSN 1023-6198 print/ISSN 1563-5120 online q 2003 Taylor & Francis Ltd

DOI: 10.1080/1023619031000146913

*Corresponding author. E-mail: moghadas@cc.umanitoba.ca

Journal of Difference Equations and Applications,

Vol. 9, No. 11, November 2003, pp. 1037–1051

dIw

dt¼ ð1 2 qÞbwIwS þ bwð1 2 fwÞIwV 2 ðmþ gÞIw; ð3Þ

dIr

dt¼ qbwIwS þ brIrS þ brð1 2 frÞIrV 2 ðmþ gÞIr; ð4Þ

dR

dt¼ gIw þ gIr 2 mR; ð5Þ

where S, V, Iw, Ir and R represent the populations of susceptible, vaccinated, wild-type strain,

vaccine-resistant strain and recovered individuals, respectively. The model presented in

Ref. [7] monitors a constant total population size N ¼ S þ V þ Iw þ Ir þ R by assuming

equal net birth and death rates (mN). In this paper, the model Eqs. (1)–(5) considers a varying

total population containing a susceptible class generated via recruitment (either by birth or

immigration). Furthermore, unlike in Ref. [7] (where a proportion of susceptible individuals

were vaccinated), vaccination is administered to a fraction of recruited individuals (see Ref.

[3] and the references therein). The quantity P represents the recruitment of individuals

(considered susceptible) into the population; p is the coverage level of preventive vaccine;

bw and br are the transmission probabilities of infection from wild-type and vaccine-resistant

strains, respectively; fw and fr are the fractions of vaccinated individuals in whom vaccine

induces protection against wild-type and vaccine-resistant strains, respectively; m is the

natural death rate; and g is the recovery rate of infected individuals. The quantity q represents

the fraction of individuals infected by the wild-type strain who develop vaccine resistance

(mutated wild-type infections). Further details on the description of the model and its

associated parameters are given in Ref. [7]. Since the model monitors human populations, it is

assumed that all the state variables and parameters of the model are non-negative. Therefore,

we consider the model in the following space of parameter values and state variables:

D :S $ 0; V $ 0; Iw $ 0; Ir $ 0; R $ 0;

P [ Zþ; 0 # p;bw;br;m;fw;fr; q; g # 1:

(

Since the aim of this study is to construct a non-standard finite-difference scheme which

preserves the qualitative features of the model (e.g. stability, positivity, etc.), we shall start by,

first of all, investigating the existence and stability of the associated equilibria of the

continuous model. This is done in the second, third and fourth sections. A positivity-

preserving scheme is then constructed, analyzed and simulated in the fifth, sixth and seventh

sections. It should be emphasized that no theoretical analysis or numerical method for solving

Eqs. (1)–(5) is given in Ref. [7].

DISEASE-FREE EQUILIBRIUM (DFE)

In the absence of infection (where Iw ¼ Ir ¼ R ¼ 0Þ; the model has a unique disease-free

equilibrium (DFE) given by

DFE ¼ð1 2 pÞP

m;pP

m; 0; 0; 0

� �:

S.M. MOGHADAS et al.1038

The local stability of the DFE is established by evaluating the Jacobian of the model at the

DFE. This leads to the following eigenvalues l* ¼ 2m , 0 (of multiplicity 3) and

l1 ¼bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P

m2 ðmþ gÞ; l2 ¼

brð1 2 pfrÞP

m2 ðmþ gÞ:

Thus, the DFE is locally asymptotically stable if l1 , 0 and l2 , 0: Letting,

Rw ¼bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P

mðmþ gÞ; Rr ¼

brð1 2 pfrÞP

mðmþ gÞ;

and defining R0 ¼ max{Rw;Rr}; it follows that l1 and l2 are both negative if and only if

R0 , 1: This result is summarized below.

Lemma 2.1 The DFE is locally asymptotically stable if R0 , 1 and unstable if R0 . 1:

The threshold quantity R0 is defined as the basic reproductive number of infection [1].

Lemma 2.1 determines the long-term asymptotic behavior of the solutions of the model that

initiate in the basin of attraction of the DFE when R0 , 1: Consequently, it is instructive to

investigate the long-term behavior of solutions that initiate outside the basin of attraction of

the DFE when R0 , 1 (as follows):

Theorem 2.1 If R0 , 1; then the DFE of Eqs. (1)–(5) is globally asymptotically stable

on D.

Proof Adding Eqs. (1)–(5) gives ðdN=dtÞ ¼ P2 mN: Thus, N ! ðP=mÞ as t !1: Here,

V ¼ ðS;V ; Iw; Ir;RÞ : S þ V þ Iw þ Ir þ R #P

m

� �;

is a positively invariant region for the model (1)–(5). Consequently, every solution of the

model that initiates in the positive region {ðS;V ; Iw; Ir;RÞ : S;V ; Iw; Ir;R $ 0} remains

there and eventually enters V.

It follows from Eq. (1) that dS=dt # ð1 2 pÞP2 mS ( for S; Iw; Ir $ 0). Defining

S1 ¼t!1lim

u$tsup SðuÞ;

it is easy to see, using the Comparison Theorem (see Ref. [9], that a solution of the equation

ðds=dtÞ ¼ ð1 2 pÞP2 ms; with sð0Þ ¼ Sð0Þ $ 0 is an upper solution of S(t) (that is, sðtÞ $

SðtÞ for all t $ 0). Since sðtÞ! ð1 2 pÞP=m as t !1; it follows that for a given e . 0; there

exists a t0 . 0 such that SðtÞ # sðtÞ # ðð1 2 pÞPÞ=mþ e for t $ t0: Thus, S1 #

ðð1 2 pÞPÞ=mþ e : Letting e ! 0, we have

S1 #ð1 2 pÞP

m: ð6Þ

Similarly from Eq. (2), it can be seen that V 1 # ð pPÞ=m; where V 1 ¼ limt!1 supu$tVðuÞ:

Using the upper bounds of S(t) and V(t) for t $ t0 in Eq. (3) gives

dIw

dt#

bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P

m2 ðmþ gÞ þ bw½ð1 2 qÞ þ ð1 2 fwÞ�e

� �Iw

¼ ðmþ gÞ ðRw 2 1Þ þbw½ð1 2 qÞ þ ð1 2 fwÞe�

ðmþ gÞ

� �Iw: ð7Þ

MICKENS-TYPE DISCRETIZATION 1039

The assumption R0 , 1 implies that Rw , 1 and Rr , 1: Thus, for sufficiently small e,

it follows that the coefficient Iw in inequality (7) is negative (since Rw , 1). Consequently,

Iw ! 0 as t !1:

Similarly, using the upper bounds of S(t), V(t) and Iw(t) ( for t $ t0 sufficiently large) in

Eq. (4) gives

dIr

dt#

bwqð1 2 pÞPe

mþ

brð1 2 pfrÞP

m2 ðmþ gÞ

� �Ir

¼bwqð1 2 pÞPe

mþ ðmþ gÞðRr 2 1ÞIr: ð8Þ

Since Rr , 1; it follows that for sufficiently small e, the coefficient of Ir in Eq. (8) is negative.

Hence, Ir ! 0 as t !1: Using Iw, Ir ! 0 in Eq. (5) implies that R ! 0 as t !1:

Furthermore, it follows from Eq. (1) that ( for t $ t0)

dS

dt$ ð1 2 pÞP2 ebwS 2 ebrS 2 mS:

Thus, using the Comparison Theorem, it can be shown that

S1 $ð1 2 pÞP

m; ð9Þ

where S1 ¼ limt!1 infu$t SðuÞ: Consequently, from Eqs. (6) and (9), it follows that

SðtÞ! ðð1 2 pÞPÞ=m as t !1: By using Iw, Ir ! 0 in Eq. (2), we have V1 $ ð pPÞ=m;

where V1 ¼ limt!1 infu$tVðuÞ: Hence VðtÞ! ð pPÞ=m; as t !1: In summary, we have

shown that if R0 , 1; then (S, V, Iw, Ir, R) ! DFE as t !1: Thus, the DFE is globally

asymptotically stable on D. A

Remark It should be noted, from the expression for Rw and Rr (written as functions of p)

that

R0

wð pÞ ¼bwðq 2 fwÞP

mðmþ gÞ; R

0

rð pÞ ¼2brfrP

mðmþ gÞ:

It follows that if q 2 fw , 0; then Rw is a decreasing function of p. Furthermore, it is clear

that Rr is always a decreasing function of p. It is easy to see that there are critical vaccination

thresholds

pw ¼bwð1 2 qÞP2 mðmþ gÞ

bwðfw 2 qÞP; pr ¼

brP2 mðmþ gÞ

brfrP;

for which Rwð pwÞ ¼ 1 and Rrð prÞ ¼ 1: Let pc ¼ max{pw; pr}: Therefore, for the case

q 2 fw , 0; it can be seen that if p . pc; then R0 , 1: Thus, the disease will be eradicated

from the community whenever the vaccination coverage level exceeds this threshold (that is,

p . pc).

BOUNDARY EQUILIBRIUM (BE)

In order to find the condition(s) for the existence of possible boundary equilibria (BE) of the

model Eqs. (1)–(5) where either Iw ¼ 0 or Ir ¼ 0; we consider G ¼ bwIw and H ¼ brIr as

S.M. MOGHADAS et al.1040

the expressions for the force of infection of the model (see Ref. [1]). Thus, at equilibrium,

Eqs. (1)–(5) become

S* ¼ð1 2 pÞP

mþ G* þ H*; ð10Þ

V* ¼pP

ð1 2 fwÞG* þ ð1 2 frÞH* þ m; ð11Þ

I*w ¼

ð1 2 qÞð1 2 pÞPG*

ðmþ gÞðmþ G* þ H* Þþ

ð1 2 fwÞpPG*

ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð12Þ

I*r ¼

qð1 2 pÞG* þ ð1 2 pÞH*�

P

ðmþ gÞðmþ G* þ H* Þþ

ð1 2 frÞpPH*

ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð13Þ

R* ¼1

m

g

bw

G* þg

br

H*

� �: ð14Þ

Substituting Eqs. (12) and (13) into the expressions for G and H, respectively, gives

G* ¼bwð1 2 qÞð1 2 pÞPG*

ðmþ gÞðmþ G* þ H* Þþ

bwð1 2 fwÞpPG*

ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð15Þ

H* ¼br qð12 pÞG* þ ð12 pÞH*�

P

ðmþ gÞðmþG* þH*Þþ

brð12frÞpPH*

ðmþ gÞ ð12fwÞG* þ ð12frÞH* þm� : ð16Þ

The fixed points of Eqs. (15) and (16) give the equilibria of the model. Clearly,

ðG* ;H* Þ ¼ ð0; 0Þ is a fixed point of Eqs. (15) and (16) which corresponds to the DFE.

Substituting G* ¼ 0 (as a fixed point of Eq. (15)) into Eq. (16) gives

H* ¼brð1 2 pÞPH*

ðmþ gÞðmþ H* Þþ

brð1 2 frÞpPH*

ðmþ gÞ ð1 2 frÞH* þ m� : ð17Þ

When H* – 0; a positive fixed point of Eq. (17) is a positive solution of the equation

FðHÞ2 1 ¼ 0 where

FðHÞ ¼brð1 2 pÞP

ðmþ gÞðmþ HÞþ

brð1 2 frÞpP

ðmþ gÞ ð1 2 frÞH þ m� :

Let FðHÞ ¼ FðHÞ2 1: It is easy to see that Fð0Þ ¼ Rr 2 1: Thus, if Rr . 1 (so that

R0 . 1 and, consequently, DFE is unstable), then Fð0Þ . 0: Differentiating F(H) gives

F0ðHÞ ¼ 2brð1 2 pÞP

ðmþ gÞðmþ HÞ22

brð1 2 frÞ2pP

ðmþ gÞ ð1 2 frÞH þ m� 2

, 0:

This implies that F(H) is a decreasing function on (0,1). It can also be

seen that limH!1 FðHÞ ¼ 21: Therefore, F(H) has a unique positive root, namely H0.

MICKENS-TYPE DISCRETIZATION 1041

Thus, (0, H0) is a positive fixed point of Eq. (17). Substituting this fixed point into Eqs. (10)–

(14) gives a unique BE point of the model (in which Iw ¼ 0 but Ir – 0). This result is

summarized below.

Theorem 3.1 The model has a unique boundary equilibrium if Rr . 1 ð p , prÞ and no

boundary equilibrium if Rr # 1 ð p $ prÞ:

Remark It should be mentioned that since the BE cannot be expressed in closed form, its

stability using traditional methods (such as considering the characteristic equation of the

corresponding Jacobian) is generally difficult to establish.

ENDEMIC EQUILIBRIUM (EE)

Since, like the BE, the endemic equilibrium (EE) of the model (where all the state variables

of the model are positive) cannot be expressed in closed form, the determination of

conditions for its existence is generally very tedious. However, in this section, conditions for

the existence of a unique endemic equilibrium for a special case where bwð1 2 fwÞ ¼

brð1 2 frÞ are derived.

Multiplying Eq. (15) by H* and Eq. (16) by G* (and equating the respective right hand

sides) gives (for G* – 0 and H* – 0Þ :

G* ¼½bwð1 2 qÞ2 br�H*

qbr

ð18Þ

Thus, if bwð1 2 qÞ2 br . 0; then substituting Eq. (18) into Eq. (16) gives the following

quadratic equation:

ð1 2 qÞð1 2 fwÞðmþ gÞðbw 2 brÞ2

ðqbrÞ2

ðH* Þ2

2bw 2 br

qbr

nð1 2 qÞð1 2 fwÞbwP2 mðmþ gÞ ð1 2 fwÞ þ ð1 2 qÞ

� oH*

2 m2ðmþ gÞðRw 2 1Þ ¼ 0: ð19Þ

This equation has a unique positive solution (for H*) if Rw . 1 and no positive solution if

Rw # 1: Substituting the positive solution of Eq. (19) into Eq. (18) gives a unique fixed point

of Eqs. (15) and (16). It can also be shown that if bwð1 2 qÞ2 br , 0; then no positive fixed

point of Eqs. (15) and (16) exists. Therefore, we have the following theorem.

Theorem 4.1 Suppose bwð1 2 fwÞ ¼ brð1 2 frÞ and bwð1 2 qÞ2 br . 0: Then the

model has a unique EE if Rw . 1:

Remark It should be noted that if bwð1 2 fwÞ ¼ brð1 2 frÞ, then Rw . Rr: Since in this

case, the basic reproductive number of infection of the wild-type strain (Rw) exceeds that of

the vaccine resistant strain (Rr), the wild-type strain will always dominate the vaccine-

resistant strain. Furthermore, if Rr . 1; then it follows from Theorems 3.1 and 4.1 that the

model has a unique boundary equilibrium and a unique endemic equilibrium. Biologically-

speaking, this means (at the very least) that an epidemic of vaccine-resistant strain emerges.

S.M. MOGHADAS et al.1042

It is important to note that if Rw . 1 . Rr; then the model has no boundary equilibrium and

an epidemic of both strains occurs (see Theorem 4.1). Numerical simulations in the Section

“Simulation”, together with some of the theoretical results of this paper, allow us to propose

some conjectures on the existence and stability of the boundary and endemic equilibria of the

model (see Section “Conclusion”).

Remark Note that in the case R0 ¼ 1 ðRw ¼ Rr ¼ 1Þ; the DFE is the only equilibrium of the

model. Since V is a positively invariant region for the model, the global stability of the DFE

can be established by proving the nonexistence of certain types of solutions such as periodic

orbits, homoclinic orbits or polygons. The non-existence of such solutions may not be easy to

show, especially in high dimensional models such as Eqs. (1)–(5). For models of dimension

less than three, the Dulac criterion or Lemma 3.1 in Ref. [2] may be applied to prove the

nonexistence of these types of solutions.

MICKENS-TYPE NUMERICAL METHOD (NSFD)

To construct a Mickens-type numerical scheme that preserves the positivity property of the

model (see Refs. [6,8]), the derivatives in Eqs. (1)–(5) are approximated by their first-order

forward-difference approximations and the right-hand side functions are approximated in

ways which lead to the following implicit methods:

Snþ1 2 Sn

‘¼ ð1 2 pÞP2 bwIn

wSnþ1 2 brInr Snþ1 2 mSnþ1; ð20Þ

V nþ1 2 V n

‘¼ pP2 bwð1 2 fwÞI

nwV nþ1 2 brð1 2 frÞI

nr V nþ1 2 mV nþ1; ð21Þ

Inþ1w 2 In

w

‘¼ ð1 2 qÞbw 2In

w 2 Inþ1w

� Snþ1 þ bwð1 2 fwÞ 2In

w 2 Inþ1w

� V nþ1

2 ðmþ gÞInþ1w ; ð22Þ

Inþ1r 2 In

r

‘¼ qbwInþ1

w þ brð2Inr 2 Inþ1

r Þ�

Snþ1 þ brð1 2 frÞð2Inr 2 Inþ1

r ÞV nþ1

2 ðmþ gÞInþ1r ; ð23Þ

Rnþ1 2 Rn

‘¼ g Inþ1

w þ Inþ1r

� 2 mRnþ1; ð24Þ

where ‘ . 0 is the time-step. Note that, to ensure positivity, the Mickens non-local

approximations Inw ! 2In

w 2 Inþ1w and In

r ! 2Inr 2 Inþ1

r have been used for Inw and In

r in

Eqs. (22) and (23), respectively. Rearranging the implicit formulations in Eqs. (20)–(24)

gives, respectively,

Snþ1 ¼‘ð1 2 pÞPþ Sn

1 þ ‘ bwInw þ brI

nr þ m

� ; ð25Þ

V nþ1 ¼‘pPþ V n

1 þ ‘ bwð1 2 fwÞInw þ brð1 2 frÞI

nr þ m

� ; ð26Þ

MICKENS-TYPE DISCRETIZATION 1043

Inþ1w ¼

2‘bwð1 2 qÞSnþ1 þ 2‘bwð1 2 fwÞVnþ1 þ 1

� In

w

1 þ ‘ bwð1 2 qÞSnþ1 þ bwð1 2 fwÞV nþ1 þ mþ g� ; ð27Þ

Inþ1r ¼

Inr þ ‘bwqSnþ1Inþ1

w þ 2‘brSnþ1In

r þ 2‘brð1 2 frÞVnþ1In

r

1 þ ‘½brSnþ1 þ brð1 2 frÞV nþ1 þ mþ g�; ð28Þ

Rnþ1 ¼‘g Inþ1

w þ Inþ1r

� þ Rn

1 þ ‘m: ð29Þ

Since the right hand sides of Eqs. (25)–(29) are positive (for 0 # p; q, fw, fr # 1), it is clear

that for any positive initial data, the above non-standard scheme Eqs. (25)–(29) will give positive

solutions which are located in the feasible region. In other words, the scheme Eqs. (25)–(29)

satisfies the positivity property of Eqs. (1)–(5). Furthermore, it is worth mentioning

that although this numerical scheme is implicit by construction, Eqs. (25)–(29) enable the

solution of Eqs. (1)–(5) to be computed explicitly via a Gauss–Seidel-type sequential process

(by substituting Eqs. (25) and (26) into Eq. (27) to compute Inþ1w ; followed by the computation of

Inþ1r using Eqs. (25)–(27) and finally substituting Eqs. (27) and (28) into Eq. (29) to determine

R nþ1). It should be noted that integrating Eqs. (1)–(5) with explicit Runge-Kutta methods will

lead to discrete models that admit negative terms which may, consequently, give solution profiles

that exhibit scheme-dependent numerical instabilities (such asconvergence to spurious solutions)

for large values of step-size. However, these instabilities may generally be removed by using

sufficiently small step-sizes [3,4,6].

FIXED-POINT ANALYSIS OF THE NUMERICAL METHOD

The aim here is to check whether the numerical scheme Eqs. (25)–(29) has the same stability

property as the original model Eqs. (1)–(5). Since we have only the DFE in closed form, we

shall restrict the fixed point analysis to the DFE accordingly.

It is easy to see that the

DFE ¼ð1 2 pÞP

m;pP

m; 0; 0; 0

� �is a fixed point of the numerical scheme Eqs. (25)–(29). Furthermore, it can be seen, after

some tedious manipulations, that the Jacobian of Eqs. (25)–(29) at the DFE has eigenvalues

h1 ¼ h2 ¼ h3 ¼ ð1=ð1 þ ‘mÞÞ and

h 4 ¼mþ 2‘brð1 2 pfrÞP

mþ ‘ brð1 2 pfrÞPþ mðmþ gÞ� ;

h 5 ¼mþ 2‘bw ð1 2 pfwÞ2 qð1 2 pÞ

� P

mþ ‘bw ð1 2 pfwÞ2 qð1 2 pÞ�

Pþ ‘mðmþ gÞ:

Clearly, for ‘ . 0; jhij , 1 for i ¼ 1; 2; 3: Here, we shall show that if R0 , 1; then

jh4j , 1 and jh5j , 1: Consider h4 ¼ h4ð‘Þ as a function of ‘ for ‘ [ ð0;1Þ: It is easy to

see that lim‘!0h4ð‘Þ ¼ 1: Differentiating h4(‘) gives

dh4ð‘Þ

d‘¼

m brð1 2 pfrÞP2 mðmþ gÞ�

mþ ‘ brð1 2 pfrÞPþ mðmþ gÞ� � 2

:

S.M. MOGHADAS et al.1044

Since R0 , 1 (and, consequently, Rr , 1), it follows that dh4ð‘Þ=d‘Þ , 0 for all ‘ . 0:

Thus, h4(‘) is a decreasing function on (0,1). Noting that h4ð‘Þ . 0 for all ‘ . 0; it follows

then that h4ð‘Þ [ ð0; 1Þ for ‘ . 0: Similarly, it can be shown that since R0 , 1 ðRw , 1Þ;

jh5ð‘Þj , 1 for all ‘ . 0: Therefore, we have established the following theorem.

Theorem 6.1 If R0 , 1; then the numerical scheme (25)–(29) will converge to the DFE

for every ‘ . 0:

This theorem is of public health significance since epidemiological models, such as

Eqs. (1)–(5), are generally monitored over a long-term period. Therefore, it is desirable to

construct numerical methods, such as Eqs. (25)–(29), that preserve the physical properties of

the model for large step-sizes. Theorem 6.1 shows that the numerical method Eqs. (25)–(29)

will always converge to the correct steady-state solution (DFE), irrespective of the step-size

used in the simulations, whenever R0 , 1:

It is worth mentioning that in order to remove the scheme-dependent instabilities that often

arise when standard numerical integrators, such as the RK methods (whether adaptive or

otherwise), are used to solve non-linear initial-value problems (such as Eqs. (1)–(5)), these

schemes must be simulated with sufficiently small step-sizes [3,4,6,8]. Consequently, the

extra computing cost associated with examining the long-term behaviour of the dynamical

system (model) being investigated may be substantial.

SIMULATIONS

Experiment 1. In order to illustrate the theoretical results of the paper, the model Eqs. (1)–(5)

was simulated using the numerical scheme Eqs. (25)–(29) constructed in Section “Mickens-

type numerical method”. In the first part of the simulations, the following parameter values

(estimated in Ref. [7]) were used: P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009; m ¼ 0:02; g ¼ 26;

fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001: With these parameter values, the critical vaccination rates

are pw ¼ 0:863 and pr ¼ 0:843 (note that pw . pr). In this case, the critical vaccination rate

needed for community-wide eradication of the disease is pc ¼ max{pw; pr} ¼ 0:863:

The numerical method was then simulated using an arbitrarily-chosen initial condition

X0 ¼ ð1000; 500; 5; 5; 10Þ [ D with ‘ ¼ 0:1 and values of vaccination coverage level ( p)

chosen for three scenarios namely: p , pr; p . pw and pr , p , pw: The simulation results

obtained are tabulated in Table I. This table shows which equilibrium point is reached

for different values of p. Figures 1 and 2 depict the profiles of the model variables for

p ¼ 0:8 , pc and p ¼ 0:87 . pc; respectively. In the case p ¼ 0:8 , pr; the unique endemic

TABLE I Numerical results generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009;m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001; X0 ¼ ð1000; 500; 5; 5; 10Þ and various values of p

P Rw Rr Equilibrium solutions Comments

0.8 1.34 1.04 DFE ¼ ð10000; 40000; 0; 0; 0Þ EE is stableBE ¼ ð9463; 38896; 0; 1; 1639ÞEE ¼ ð7015; 39164; 3; 0:003; 3818Þ

0.844 1.10 0.99 DFE ¼ ð7800; 42200; 0; 0; 0Þ EE is stableEE ¼ ð6788; 41917; 1; 0:002; 1205Þ

0.87 0.97 0.98 DFE ¼ ð6500; 43500; 0; 0; 0Þ DFE is stable

MICKENS-TYPE DISCRETIZATION 1045

FIG

UR

E1

Pro

file

sof

the

model

var

iable

sg

ener

ated

usi

ng

NS

FD

wit

hl¼

0:1;P

¼1

00

0;b

w¼

0:0

02

9;b

r¼

0:0

00

9;m¼

0:0

2;g¼

26;f

w¼

0:9

5;f

r¼

0:5;

q¼

0:0

00

1;

p¼

0:8

ðp,

prÞ

and

X0¼

ð10

00;5

00;5;5;1

0Þ.

S.M. MOGHADAS et al.1046

equilibrium is stable (boundary equilibrium exists but is unstable). Further simulations for

the case pr , p ¼ 0:844 , pw show similar profiles to those obtained in Fig. 1. In this case

ð p ¼ 0:844Þ; the model has no boundary equilibrium. However, when p was increased to

p ¼ 0:87 . pc; the model has only the DFE which is stable (see Fig. 2). Thus, with this

choice of parameter values, community-wide eradication of the epidemic is feasible if at least

87% of the recruited individuals are vaccinated. This is in line with Theorem 2.1.

Experiment 2. Here, the parameter values in Experiment 1 were used with exception of fr

which is now decreased to fr ¼ 0:44: In this case, pw ¼ 0:863 , pr ¼ 0:958 and the critical

FIGURE 2 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001; p ¼ 0:87 ð pw , pÞ and X0 ¼ð1000; 500; 5; 5; 10Þ:

MICKENS-TYPE DISCRETIZATION 1047

vaccination rate for disease eradication is pc ¼ 0:958: Numerical results for various values of p

are tabulated in Table II and Figs. 3 and 4 (for p ¼ 0:8 and p ¼ 0:9; respectively). Figures 3 and 4

show that if p , pw ð pw , p , prÞ; the unique EE (the unique BE) is stable. However, by

increasing p to p ¼ 0:96 . pc; the DFE becomes stable (see also Fig. 2). Note that, in this case,

the DFE is the only existing equilibrium of the model. Thus, here, the vaccination coverage level

must be at least 96% to ensure community-wide eradication. In other words, the reduced value of

fr (vaccine protection) has, expectedly, led to an increase in the threshold vaccination coverage

level needed for eradication (from 87% whenfr ¼ 0:5 to 96% whenfr ¼ 0:44). Overall, these

simulations are consistent with the theoretical results in Theorems 2.1 and 3.1.

The effect of step-size (‘) on the convergence of the Mickens-type numerical method

was monitored by simulating the method with large step-sizes (such as ‘ ¼ 10; 103, 106)

for the case R0 , 1 ð p . pcÞ: It was observed that, in this case, the method always

converges to the DFE (consistent with Theorem 6.1). It should be mentioned that although

the use of large step-size in numerical computations is desirable (for monitoring long-term

dynamics), the transient behaviour of the dynamical systems may not always be captured

for large ‘.

For comparison purposes, the model was simulated using RK45 method (the Dormand-

Prince pair formula, relative error tolerance 1023 and absolute error tolerance 1026 [5])

with the parameter values used in Experiments 1 and 2 and various values of the

vaccination coverage ( p). With the parameter values used in Experiment 1, the RK45

method gave profiles that converge to the correct steady-state solution for p ¼ 0:8; 0.844,

0.87. Thus, for this choice of parameter values, the RK45 method gave results that are

consistent with those obtained using the Mickens-type numerical method (tabulated in

Table I). However, the RK45 method fails to converge to the correct stable equilibrium

(BE) when the parameter values in Experiment 2 were used. The simulation results show

that the RK45, unlike the Mickens-type numerical method, does not converge to the BE

when 0:863 ¼ pw , p , pr ¼ 0:958: For example, the profiles of Iw, Ir and R for p ¼ 0:9;

depicted in Fig. 5, reveal that the RK45 fails not only to converge to the BE, but also to

preserve the positivity property of the model. Thus, for this choice of parameter values, the

RK45, unlike the Mickens-type method (see Table II), gave profiles that exhibit scheme-

dependent instabilities involving negative values (see Fig. 5). For population models, such

as Eqs. (1)–(5), these negative profiles are, of course, biologically unrealistic. It should be

noted that the theoretical results in section “Disease-free Equilibrium” show that

TABLE II Numerical results generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009;m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; X0 ¼ ð1000; 500; 5; 5; 10Þ and various values of p

P Rw Rr Equilibrium solutions Comments

0.8 1.33 1.12 DFE ¼ ð10000; 40000; 0; 0; 0Þ EE is stableBE ¼ ð8511; 36430; 0; 4; 5056ÞEE ¼ ð7022; 39020; 3; 0:17; 3955Þ

0.85 1.07 1.08 DFE ¼ ð7500; 42500; 0; 0; 0Þ BE is stableBE ¼ ð6667; 39721; 0; 1; 3609Þ

0.9 0.8 1.04 DFE ¼ ð5000; 45000; 0; 0; 0Þ BE is stableBE ¼ ð4670; 43288; 0; 2; 2041Þ

0.96 0.49 0.99 DFE ¼ ð2000; 48000; 0; 0; 0Þ DFE is stable

S.M. MOGHADAS et al.1048

the feasible region V is positively invariant. Thus, any solution with initial condition in V

(such as X0) must remain there for all t . 0:

CONCLUSION

A Mickens-type finite-difference method was constructed and used to solve an epidemic

model for the transmission dynamics of two strains of a childhood disease. Critical

FIGURE 3 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:8 ðp , pwÞ andX0 ¼ ð1000; 500; 5; 5; 10Þ:

MICKENS-TYPE DISCRETIZATION 1049

vaccination rates needed for community-wide eradication of the disease were obtained

for different scenarios. Theoretical results, together with numerical simulations of

the Mickens-type numerical method, confirm that:

(1) this method always converges to the correct steady-state solution;

(2) this method always preserves the positivity property of the model.

FIGURE 4 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:9 ð pw , p , prÞ andX0 ¼ ð1000; 500; 5; 5; 10Þ:

S.M. MOGHADAS et al.1050

Simulation results using the RK45 method reveal that:

(1) the RK45 method does not always converge to the correct steady-state solution.

(2) the RK45 method does not always preserve the positivity property of the model.

Based on the theoretical and numerical results of this paper, the following conjectures are

proposed:

(i) The model has an unique EE if Rw . Rr and Rw . 1; and no endemic equilibrium if

Rw # Rr: If such an equilibrium exists, then it is globally asymptotically stable.

(ii) The unique boundary equilibrium is globally asymptotically stable if Rw # Rr and

Rr . 1:

Acknowledgements

This work was supported in part by the Natural Sciences and Engineering Research Council

of Canada (NSERC). The authors are grateful to the referees for their comments which have

improved the paper.

References

[1] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, London/New York,1991.

[2] S. Busenburg and P. Van den Driessche, Analysis of a disease transmission model in a population with varyingsize, J. Math. Biol., 28 (1990), 257–270.

[3] A.B. Gumel, R.E. Mickens, B.D. Corbett, A non-standard finite-difference scheme for a model of HIVtransmission and control, J. Comput. Methods Sci. Eng., To appear.

[4] J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, Wiley,Chichester, UK, 1991.

[5] Matlab, The language of Technical Computing, by The MathWorks, Inc., 1997.[6] R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comp.

Appl. Math., 110 (1999), 181–185.[7] A. Scherer and A. R. McLean, Mathematical models of vaccination, Brit. Med. Bull., 62 (2002), 187–199.[8] A. Serfaty de Markus and R. E. Mickens, Suppression of numerically induced chaos with nonstandard finite

difference schemes, J. Comp. Appl. Math., 106 (1999), 317–324.[9] G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed., McGraw-Hill 1991.

FIGURE 5 Profiles of Ir and R generated by RK45 with P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009; m ¼ 0:02; g ¼ 26;fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:9 ð pw , p , prÞ and X0 ¼ ð1000; 500; 5; 5; 10Þ:

MICKENS-TYPE DISCRETIZATION 1051