A deterministic model in biomathematics. asymptotic behavior and threshold conditions

A DeterministicModel in Biomathematics. Asymptotic Behavior and Threshold Conditions GUNNAR ARONSSON* UppsaIa University, Department Thunbergtigen

of Mathematics,

3, S-752 38 Uppsala, Sweden

AND INGVAR MELLANDER Chalmers iJnioersi@ of Technoiogv and the Uniwrsi@ Fack, S-412 96 Giiteborg 5, Sweden Receiwd

of Giiteborg,

18 April 1979; revised 22 October 1979

ABSTRACT We study a determinis tic SIS model, which takes into account seasonal variations as well as heterogeneities in the population. The appropriate threshold condition is found (the condition for the disease to remain endemic).

1.

INTRODUCTION

In this paper, we study a deterministic multigroup model in theoretical epidemiology. The model is of SIS type, i.e., incubation and immunity are neglected. Since our model is a multigroup model, we can take into account heterogeneity in the population. The model is a refinement of a gonorrhea model studied by A. Lajmanovich and J. Yorke [lo], in that we take into account seasonal variations. We derive a threshold condition for the disease to remain endemic, expressed in terms of the Perron-Frobenius eigenvalue of a certain “monodromy” matrix of a linear periodic differential system. In the periodic endemic case, we find a nontrivial periodic solution, which is globally asymptotically stable. Clearly, this corresponds to a stable seasonal fluctuation of the disease. Asymptotic behavior is studied in various, also nonperiodic cases.

*Supported by the Swedish National Science Research Council (NFR), 2569407. MATHEMATICAL

BIOSCIENCES

49~207-222

(1980)

contract

F

207

@Elsevia North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017.

0025~5564/80/040207+

16$01.75

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We use spectral properties of positive matrices and holomorphic matrix functions and some classical Floquet theory. Further, the monotonicity property of the solution operator turns out to be very useful. It is usually believed that theoretical models can lead to a better understanding of the epidemiology of infectious diseases. The present work is intended to be a contribution in that direction. The model may be applied, with some judgment, to gonorrhea and other diseases of SIS type (see [8, p. 6081, and [9, p. 3361). 2.

BASIC

PROPERTIES

In most theoretical

epidemiological

models the population

is assumed to

be homogeneous. One usually also assumes homogeneous mixing, i.e., the probability of transferring the disease from one infected to one susceptible person in a given time interval is independent of the particular pair chosen. While these assumptions may be realistic in many cases, they are definitely unrealistic in others. Clearly, the mixing cannot be uniform if the population is spread over a large area,

for instance

containing

cities at great

distances. Also, inhomogeneity may be caused by social differences, age differences, etc. Thus it is often necessary to consider multigroup models taking such differences into account. Among such models we want to mention the work by Baroyan, Rvachev et. al. (1967). They considered a deterministic model for the spread of influenza between 128 of the largest cities in the Soviet Union. For references and details, see Bailey [2, pp. 18, 350-3571. There is also work on stochastic multigroup models; see references Bailey [2, pp. 75, 1271. See also Warren, Foster, and Bleistein [14].

in

One disease for which the mixing is strongly inhomogeneous is gonorrhea. A deterministic multigroup model for gonorrhea was constructed and studied by Lajmanovich and Yorke (L&Y) in Ref. [lo], published in 1976. They studied questions of asymptotic stability and showed that the asymptotic properties of the model were determined by the spectrum of a certain matrix A. We will consider here a nonautonomous version of the model of L&Y, and in particular we allow for seasonal uariutions. Also here, the spectrum of a (different) matrix is crucial for the threshold condition and we will see how the results of L&Y are included as special cases of our results. We will formulate below the nonautonomous model, and the model of L&Y simply corresponds to the case of constant coefficients. The one-group case of our problem was studied by Hethcote [8], where he allows for seasonal variations and some other complications as well. We divide the population into n groups, G,,Gz,..., G,, each of constant size. Each group is assumed to be homogeneous, but different groups may have different data. We will neglect incubation and immunity, i.e., we

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consider a so-called SIS model (as usual for gonorrhea; see L&Y [IO, pp. 221-2221). For a group Gi, let ci be its total size, xi the number of susceptibles, yi the number of infectives, and q(t) the recovery rate. Further, let &(t) be the “contact rate” of infectives in Gj with susceptibles in G,. This suggests the differential system * = - a.(t)y. + i p-( f)y.x. dt ’ ’ i_, ” J ”

i=l2 , )..., n.

Since xi +yi = ci, we find l;i = - ai(t)Yi

+ (ci -Yi)jgl

pidt)Yj9

i=l2 , )..., n.

(1)

The functions q(t) and hi(t) are nonnegative and defined for all t, or possibly for t 2 to. We assume them to be piecewise continuous, i.e. to have at most a finite number of discontinuities in every finite time interval, and to have one-sided limits at the discontinuities. We also assume that there are positive numbers CYand E, such that cui(t)> (r, i = 1,2,. . . , n, for all t, and for a pair (j,i), either j!,(t)=0 or Pi(t) >ei for all t. If &(t) >E,, we say that Gj infects Gi. Let S=(Q) be the n x n matrix defined by sii= 1 if Gj infects Gi, ~-0 otherwise. We make the natural assumption that Gj infects Gi if and only if G, infects Gj. We say that the set of groups {G,}; is connected if we cannot divide it into two nonempty subsets with no cross-infection. If { Gi); is not connected, it can be divided into maximal connected subsets. These subsets are completely independent, and we can study them separately. We assume that such a reduction has already been done, and hence we assume that { Gi}l is connected. This implies that the matrix S is irreducible; see [12, pp. 9-131. We are primarily only interested in solutions to (1) that belong to E={yEW”; O
I

Let A(t) be a piecewise continuous matrix function on [0, W) such that %(t) > 0 for i#j, t > 0. Let Q(t) be that fundamental matrix of the system li = A(t)x for which @(O)= I. Then @(t) z 0 for t > 0. Let z(t) be another matrix function of the above type satisfying A(t) > A(t) on [0, 00). Then m(t) > a(t) for t 2 0, where m(t) is the fundamental matrix corresponding to A< t).

Proof Q(t) > 0 for t > 0 follows trivially from the well-known fact that the first orthant is positively invariant for such systems. Let T((t) and x(t) be

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corresponding columns in 8(t) and Q(t), and let y(f) = f(t) - x(t). We want to show that y(t) z 0 for t > 0. But y(t) is that solution of the system 9 = A(t)y + [x(t) - A(t)]z(t) which satisfies y(0) = 0. Hence y(t) = J@(t)0-‘(~)[A<~)-A(~)]X(~)d~>O, since Q(t)@-‘(T)>0 for t>T by the first part of the lemma. From [ 1, p. 1151, we have the following LEMMA

2

Let A(t) and Q(t) be as in Lemma 1, and let A(tc) irreducible for some t,, > 0. Then 0(t) > 0 for t > t,. LEMMA

(right-hand limit) be

3

E is positively invariant for (1). Approximate (1) by a system i, = g(y, t,~), where E > 0 and -ai(t)yi+(ci-yi)[ZjPji(t)yj+e] for i=1,2,...,n. Clearly E is positively invariant for this system, and the lemma follows in an obvious manner by making E tend to zero. Proof.

g(Y,t7E)i=

LEMMA Let

4

y,(t)

and y*(t)

be solutions

yl(tO)
to (1) in E for t > t,,. Assume

Then yl(t)
says that the Proof. Fix t > to. The lemma yo+y(t, t,,y,), y. E E, is strongly isotone. For y. E E and r > to we write y(r) =y(r, to,yo). Then -%=-(ai(7)+

that

t >tW solution

operator

~~i(7)Yj~T))xi+[ci-Yi(7)]~P*i(T)x~ i

is the variational system along y(r). Obviously y,(r) to, 1 to and has nonnegative off-diagonal elements. By Lemma 2 the fundamental matrix (P(T), satisfying @(to) = I, is positive for r = t. Thus all the partial derivatives of the solution operator above are positive in E. By applying the mean-value theorem to each component of the solution map, we obtain the result. Remark. isotone.

It follows from [3, Theorem

10, p. 291 that the solution

map is

Epidemiologically the lemma means that a sudden increase of the disease level in some group immediately affects all other groups, and in the same direction. COROLLARY

Zfy,EE,y,#O,

theny(t)=y(t,t,,y,)EintEfor

t>t,.

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Proof: We already know that y(t)t,. follows by applying Lemma 4 toy(t) and the trivial solution.

The rest

This corollary is stronger than Lemma 3.2 in L&Y [ 10, p. 2261.

Remark.

Fory,zER”,y>O,

z>O, let .

Obviously @(z,y)=@(y,z) > 1, and f?(z,y)= 1 if and only ify=z. see that if y and z E E, then izk-Yki

<

ce-

It is easy to

bk-

(2)

LEMMA 5 Let y(t)

and z(t)

p(t) = e(z(t),y(t)).

and let 6(t) absolutely

be two positive

solutions

to (1) in E for t > to, and let

Let

be defined

continuous,

similarly.

Then p(t)

is nonincreasing

and local&

and

D ‘y(t)

< -

where D + standr for right-hand

[ /4t) - WWW)~+yW)~ derivative.

Proof. Being the maximum of 2n functions which are locally Lipschitzian, p(t) itself has that property. Thus p(t) is locally absolutely continuous, and

where Sl(t)={m; n

1
and&(t)=

{m; 1
andy,(t)/zAt) = At)]-

Let us fix t and omit it below. If m E SI [ = S,(t)], then p = z,/y,,, i.e. yi/ym >zj/z,,,, 1
and

>zi/yj,

212

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ARONSSON AND INGVAR MELLANDER

gives D +z, _ D +Y, <(YKZ,)X&& G Y, i

m

Hence D+~=~D+(log~)c(l-$)~p,z~~-(P-l)~z. m In the same follows.

way D +(y,/z,)

Q -(p-

m

i

l)$

if m E S,, and

the lemma

In the rest of this section and in the next, the functions ai and Pji(t) are assumed to be periodic with period T > 0. For tE R we introduce the period maps 5: E+ E by fi(ro)=u(i+

Y~EE.

T,~,Y,),

Clearly &+ r =h, and we write f, =f. Define Pk, k=0,1,2 ,..., by P,,=c=(c ,,..., c,)’ and Pk=f(Pk_I), k> 1. By the corollary to Lemma 4, P, E int E, and hence 0 P, > P, > . . . >O. Thus lim,,, Pk = Q exists and belongs to E, and Q is a fixed point off. There are two possibilities, namely Q =0 or Q #O. Which case occurs is in fact independent of the somewhat arbitrary choice i=O in our definition off, and it depends only on our differential system. Namely, assume that Q = 0. Theny(t)=y(t,O,c) tends to 0 as t-+oo. Lety(t) be an arbitrary solution to (1) in E for t > to, and let m be an integer such that me T z t,. Then, by Lemma 4, y(t)=y(t,mT,y(mT))mT. Hence y(t)+0 as t-co. This means that the disease dies out naturally for every initial state if Q = 0. hand, z(t) =y(t,O, Q) is a nontrivial positive

If Q # 0, on the other periodic solution in E. Hence,

if Q #O, the disease may be habitually

present

in all groups of the

population.

In the latter case we will show (Theorem 2) that z(t) is globally asymptotically stable in E \{O}. Using Lemma 4, we make the following OBSERVATION Ify

andzEintE,

DEFINITION

For O
then e(f(y),f(z))
unlessy=z.

PERIODIC SOLUTIONS IN AN EPIDEMIC MODEL

213

Each H, is a convex cone. LEMMA

6

There exists a positive nondecreasing function r on (0,~) y(t, to,yO) E Ft,- 10) for t > to

such that

if yore.

Proof. Since E is positively invariant, we construct r(7) for 0<7 < 1, and put r(r)=r(l) for T> 1. Now (O,l]= U~_,(2-V,2-u+‘]r IJ ,“-,ZO. If we ifyoEE and (t, - to)E canfindp,>O,v=1,2 ,..., such thaty(t,,t,,y,)EH, Z,, then the desired function will be r(T)=min{p,; 2-“+i >T}. Thus, we assume now that (tl - to) E Z,. An easy indirect argument, using the fact that the system is periodic, gives the result for Z,, if 11yolJ >8, for an arbitrary 8 > 0. We thus assume IIyoJI < 8, where 8 > 0 is at our disposal. The system is 9 = A(t)y + N(t,y) (see below); let O(t) be that fundamental matrix of j = A(t)y for which @(to) = I. Clearly (by Lemma 2 and an indirect argument) there is a 6 = 6(v) > 0 such that 6 < (Q(t,))iJ < 6 -I for 1 < ij < n, provided t, - toEI,. For any E >0, there is 8 >0 such that (Iy(t,, teyO)@(t,)yoll
THE THREE

MAIN CASES

Let us write the system (1) as 9 = A (t)y + N(GY), where

and

Let a’(t) be that fundamental matrix of the linearized which satisfies Q(O)= I. It follows from Lemma 2 that matrix C=@(T) is positive. Since the Perron-Frobenius important here, we cite it for the reader’s convenience. For

system j = A(t)y the monodromy theorem will be a proof, see [ 131.

THEOREM Let

B > 0

be a square irreducible matrix. Then

(1) B has a positive eigenvalue equal to its spectral radius, p(B). (2) To p(B) there corresponds a positive eigenvector, w.

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(3) p(B) increases when any entry of B increases. (4) p(B) is an algebraically simple eigenvalue of B. Applying this theorem to C and CT, we find positive eigenvectors w and o, respectively, both corresponding to the eigenvalue X = p( C) = p( C’). Further let w and w be chosen so that w rw = 1. As mentioned before, L&Y treated the case of constant cui(t) and &(t) (no seasonal variation in recovery rate and infection transmission). The equation (1) is then autonomous and assumes the simpler form j=Ay+N(y).

(3)

Let s(A) =maxhE,+)Reh be the “stability modulus” of A. L&Y found (not surprisingly) that this number was crucial for the asymptotic behavior of solutions to (3). We now cite their main theorem (see [lo, p. 2271). THEOREM For the Jystem (3) there are two possibilities. Either s(A) < 0 and then y = 0 is globally asymptotically stable in E, or s(A) > 0 and then there exists a constant solution k E E \ (0) such that k is globally asymptotically stable in E\(O). Let us treat (3) as a special case of (1) and choose any “period” T >O. Obviously Q(T) = C = e” ‘, and since the eigenvalues of eAT are exponentials of those of AT (see for example [5, Chapter V, $11) we have h- eT’N”). Note that X > 1 if and only if s(A) > 0. In Theorems l-3 we will generalize the above theorem.

3.1. THE CASE h< 1 THEOREM I Assume that h < 1. Then there is a constant K such that if y( t) E E satisfies (1) for t > t,, then 1)y(t)\/ < KX(‘-‘d/TII y(t,,)l\ for t > t,,. The theorem says that the disease prevalence zero if X< 1, regardless of initial conditions. Proof.

By Lemma

y(t) =@(t)w’(to)y(to)

decays exponentially

to

1, @(t)@-‘(s) > 0 for t >T, and hence +

If~(t)~-‘(T)N(T,y(T))dT < Q(t)@-‘(toMto). IO

Now @(kT)= Ck if k is an integer, and hence it is sufficient to prove that ]lCkll < K,Ak for some constant K,. But this follows for instance from Theorem 1.2, p. 7 in [12].

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THE CASE A> 1

3.2.

ForE>OweputE,={yEE; LEMMA Let

c~Ty>e}

(recallthat

C~O=AW>O).

7 the equation

(1) be such

that X > 1. Then f: E,+E,

if E is

sufficient&

smaN. Proof. We have oTf(y)=~~[Cy+o(y)]=A~~+o(y). Thus wTf(y)> wry if y E E\E, for some nonempty Ee, 0 > 0. Further, f(E,& c Ee, for some 8, >O. Choose E GO,, and the lemma follows. THEOREM Let

2

the equation (1) be such that A > 1. Then:

(i) there is a unique nontrivial periodic solution z(t) to (1) of period T; (ii) there is a positive constant a with the following property: for every compact F c E\(O) there is a constant Kr such that y, E F implies 11y(t, t,,yJ - z(t)ll < Kre -“(r-‘~ for t > tW Epidemiologically, this means that the disease prevalence will approach a periodic (seasonal) fluctuation, provided present initially.

in each group the disease is

Choose an E > 0 as in Lemma 7. Obviously, the functionf has at Proof. least one fixed point in E,c E\(O). The uniqueness follows from the observation after Lemma 5. This point corresponds to a nontrivial periodic solution, z(t). For x,, lint E, let GxO= {y E E; y >x,,}. We will first prove a weaker version of (ii) in which we assume that F is contained in some G,,. For y0 E F, to E W, put y(t) =y(t, tO,yO) and let p(t), t > to, be as in Lemma 5. Since {z(t); t E W} is a compact subset of int E, it is contained in some G,?, za lint E. Let M be the maximum of fI on Gx, X G,,. Then p(t,,)
IC;(fVAf). Hence -&(t)+10g[p(‘)-1]}=~~

p(t)-1

dt

< -#Jt)

a.e.,

and thus log[ p(t) - 11G p(t) + log[ p(t) - 11 < p(t,,) + log[ p(to) - l] $:,&(r)dr. NOW j;)&(7)d7=at+p(t), p(t) is periodic and bounded. Hence

where a=(l/T)J~&(7)dT>O p(t)- 1 t0,

and where

Ilr(O - 4011m G BY (9, M, = exp[M + log(M - 1) + 2 w,IAOIl. maxkck[ p(t)- 11, and our weaker version of (ii) is proved.

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Now the solution operator y(t,t,,ys) is jointly continuous in all three arguments (Theorem 2.1 in [7, p. 941). Thus, by the corollary of Lemma 4, the mapping (ta,y,,)+y(t,, + 1, t,,y,) from R x E into E maps [0, T] X F into some G;,. Let k be chosen according to the first part of the proof, referring to G;,. Then lly(t)-z(t)]/ to+ 1, and obviously IIy(t) - z(t)11 d llcll for rat,,. Let KF= e”max (g, IlcII), and the proof is complete. 3.3

THE CASE X= 1

THEOREM Let

3

the equation (1) be such that A= 1. Then:

(i) there is a constant K such that ify(t)

IlY(‘)IIG (ii) there is a positive i = A(t)z such that

where y(t)

is an arbitrary

K

l+(r-t,)

periodic

nontrivial

solution

E E satisfies for

+(t)

(1) for t > to, then

t > t,;

of the linearized

system

solution to (1) in E.

Thus, also in the limit case A- 1 the disease will die out naturally for every initial stage, and the rate of decay is like l/t. This is in full agreement with Hethcote’s results for the case n= 1 (i.e. a one-group model), in the papers [8, p. 6101, and [9, p. 3381. Proof.

We will first show that O< -lim t]]y(t)]] < lim tl]Y(‘)]] < coo. I--r00 ,*C0

(4)

ro. Then t(t) satisfies the linearized Let &t)=@(t)w, and let n(t)=[@-‘(t)] system i = A(+, and n(t) the adjoint system, i = -A r(t)z. Clearly, [ and q are periodic with period T. Since @(t) is nonsingular and nonnegative, 6 is positive. It follows from Lemma 1 that @-‘(t)=@(O)@-‘(t)> 0 for t (0. Hence, by its periodicity, q(t) is also positive, and each qi(t) is bounded away from zero and infinity. Thus it is enough to prove (4) with ]ly(t)ll replaced by u(t) = q r( t)y(t). We have Ii=

-q=A(t)y+q1TA(f)y+q7TN(t,Y)=q1TN(t,Y)

=-

7

Ilr(l)( T

&(r)r,(c)u,tr))

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Recalling our restrictions on the functions &(t) and Lemma 6, we see that K,~~yJ~Z<-li~Kz~~~~~2fort>t,+1andsuitableconstantsK,>0andK,> 0. Thus K3u2 < - ti Q K4u2 for t > to+ 1 and some positive K3 and K.+. Finally,

and (4) follows by integration. Proof of(i).

By Lemma 4 (we may assume 0 =Gt, < T), [I+ ct - to)Mt)

< (I+ tMt) = (I+ ‘)u(t, T,Y( T)) =G(I+ t)v(t, TJ)

But this is bounded

for

t > T.

by above, and (i) follows.

Proof of (ii). In order to prove (ii) we decompose y(t) into two components, one parallel to t(t) and one orthogonal to q(t). It will further be convenient to change variables in (1). Let z(t) = t*y(t). Then i=A(t)z+f[z+N(t,z)],

(5)

since N(t,y) is homogeneous in y of second degree. The matrix C = @(T) is real and nonsingular. Then there exists a real logarithm of C2. For this, we refer to Yakubovich and Starzhinskii [16, p. 561. Thus we find a real matrix B such that eB”== C2. We then define a matrix function P(t) by B(t)=

P(t)eB’,

(6)

and P(t) is then real, periodic with period 2 T, and nonsingular (we leave aside the question of finding a real logarithm for C). As in classical Floquet theory [6, p. 1191, we change variables: z(t)= P(t)x(t). The equation for .x(e) is then i=

Bx+;

+ f P-‘(t)N(t,P(t)x).

(7)

Now the eigenvalues of B.2T are logarithms of those of C2 [5, Chapter V, $!I]. Hence B has an eigenvalue k-vi/T where k is an integer. Since B is real, - k.ai/ T is also an eigenvalue. If k# 0 this means that the eigenvalue 1 of C2 is not simple, in contradiction to the Perron-Frobenius theorem. Hence B is a singular matrix, and the eigenvectors of B and BT with

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eigenvalue 0 are (multiples of) the Perron-Frobenius eigenvectors of C* and (C’)r, respectively. Thus Bw = 0 and B *a = 0. Put M = {x E I%“; o rx = O}. Then B maps Iw”into Decomposition of x(t). M. Let us write x[ = x(t)] = uw + m, where u = o rx and m E M (recall that wTw=l). If we write P-‘(t)N(t,P(t)x(t))=iT(t)w+Ci(t), firm, then from (7) dw+ti=Bm+f[u(t)w+m(t)+o(t)w+m(t)]

and hence d= f [u(t)+iqt)],

(8)

ti=Bm+f[m(t)cm(r)].

(9)

The function m(t). By (4), z(t) and x(t) are bounded. Hence m(t) = x(t) -a Tag and E(t) are bounded. Now the matrix C2 is strictly positive and hence primitive [13, pp. 35, 401. Hence the n - 1 eigenvalues of C2 different from one have modulus strictly less then one. Thus B IM has all its n - 1 eigenvalues in the left half plane, and since the inhomogeneous term in (9) is 0(1/t), t+co, it is easy to see that m(t)=

t-boo.

0(1/t),

The function u(t).

homogeneity

We know that v(t) is bounded. in y of N(t,y),

(10) Hence by (10) and the

5(t)=wTP-‘(t)~N(t,P(t)[v(t)w+m(t)]) = -‘P(r).[u(t)]2+

0(1/Q,

t-w,

where q(t)= -orP-‘(t)N(t,P(t)w) is periodic. Now oTB=O, Bw=O, and hence ~T.e-B’=~T, eB’.w=w. Thus act)= -wTe-E’P-‘(t)N(t,P(t)eB’W)= -q =(t)N(t,&t)) is positive. From (8) d=f[v-o%j(t)]+0(1/12). tu is confined between By (4), u=o=x=o=P-‘.Px=q=z= for large t. The same holds for q = 1/v, and hence

+_A=q2

1 _1 t( 4

$6(Q)

+ o(l/t2),

positive bounds

t+m.

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219

ThUS

-4=f[h4f)]+0(+) and But S$&T)~T = a,*t + O(l), periodic). Hence

where

tq+q=-$fq)=@)+O(f).

a, > 0 (recall

tq=ta,+O(logt)

that

$ is positive

and

and

ThUS

and

by (10). But t*y(t)=z(t)=P(t)x(t)

o( F),

= J&[(t)+ Thus (ii) holds with +(t) = (l/a,)&

4.

THE NONPERIODIC

t+co.

t). This completes

the proof.

CASE

In this section, we will derive some results for the general, nonperiodic case. We are not aiming at a complete discussion. We will use an intuitively obvious comparison method. The conditions on the coefficients q(t) and b,,(t) will be as before, except the condition of periodicity. Let ji = -

ai(

r)Yi +

(Ci-yi)

x &(

t)yj,

1
(A)

1
(B)

and )ii = - q( t)Yi + (ci -Vi) Z &( ‘)Yj, j

be two systems of the type (1) with the same n and the same numbers I
ci,

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DEFINITION

If @(t) &(t) (B) mujorizes (A). The epidemiological LEMMA

f or all i, j, and t, we say that the system

interpretation

is obvious.

8

Let the Jystem (B) mujorize the system (A). Let y(t) andy(t) be solutions to (A) and (B), respectively, in E for t > to, and let y( to) >y(tJ. Then y(t) >y(t)

for t > t,, and ifY(t,)

#y(t,J,

then Y(t) >y(t)

for t >t,,.

Proof. The first statement follows from an easy result in the theory of differential inequalities, presented as Theorem 10 in [3, Chapter 11. The second statement follows from the first and Lemma 4. COROLLARY Let the equation (B) mujorize (A), and let both be periodic with period T. Zf (A) has a nontrivial periodic solution z(t) in E (i.e. AA > l), then (B) also /UZY one, L(t), and z(t) >z(t) for all t, unless the equations are identical.

Proof. By Lemma 1 and part (3) in the Perron-Frobenius theorem we have hB > X,. Next F(t) >z(t) follows from the above lemma and Theorem 2. Further, either T(t)>z(t) for all t, or .F(t)=z(t), and the last statement follows by inspection of the systems. DEFINITION

By an infinitely old solution to our differential equation (1) in E, we mean a solution defined and belonging to E on an interval of the form (- cc,u). It follows from Theorem 2 that if the equation is periodic and X> 1, then the periodic z(t) is the only infinitely old solution in E which is bounded away from zero. If h< 1, on the other hand, then Theorems 1 and 3 show that y E 0 is the only infinitely old solution in E. For an equation of the type (1) (not necessarily periodic) and for s < 0, let y(s) =y(O,s,c). It follows from Lemma 4 that y(s) is a strictly increasing function. Hence R = lim,,_, y(s) exists and belongs to E. It is easy to see that the solution through (0, R) is an infinitely old solution in E. We see that y E 0 is the only infinitely old solution in E if and only if R = 0. Let the equation (B) majorize the equation (A). It follows from Lemma 8 that yB(s) > y”(s), and thus RB > RA. Hence (B) has a nontrivial infinitely old solution if (A) has. Thus, if (B) is periodic with &, Q 1, then y=O is the only infinitely old solution to (A) in E. Further, if (A) is periodic with AA> 1, then (B) has a nontrivial infinitely old solution z(t) E E, which is componentwise bounded away from zero. We now have the following

PERIODIC

SOLUTIONS

IN AN EPIDEMIC

221

MODEL

THEOREM4 Let the equation (1) majorize

(i) equation

a periodic

(1) has an infinitely

equation with A > 1. Then:

old solution

z(t)

in E, which is (even

componentwise) bounded away from zero; (ii) if y(t) is an infinite& old solution to (1) in E, such that lim,,_,ll >O, then y(t)-z(t); (iii) there is a positive compact F c E \{O}

constant

a with the following

there is a constant

Kr such that y,~

property

y(t)11

for every

F implies

Proof: Statement (i) is obvious from the preceding. We denote the majorized equation by (P), its period by T, and its periodic solution by x(t). Obviously z(t)>x(t) for all t. In (iii), we can put a=(l/T)$T$JT)dT and, for any compact F c E \{O}, choose Kr as in the proof of Theorem 2, applied to (P). In that proof we can replace (P) by (1); the fact that we are studying a majorizing equation offers no extra difficulties. The crucial estimates for the majorizing equation are simply better than those for (P). Further, by Lemma 8, the set G;, occurring in the proof also contains y(to+ 1) if y(t) is a solution to (1) and y(t,)E F. Now (ii) follows from (iii).

Remark.

Statement

(ii) is no longer true if the condition

lim,~-mllY(f)ll>O is dropped. We have thus the following nice triple of “endemic”

cases:

1. If (1) is autonomous and s(A) > 0, then there is a unique equilibrium in E. Further, k > 0 and y-k is globally asymptotically stable in

k#O

E\(O).

2. If (1) is periodic and A > 1, then there is a unique nontrivial periodic solution z(t) in E. Further, z(t) > 0 and z(t) is globally asymptotically stable in E\(O). 3. If (1) majorizes a periodic equation for which h > 1, then (1) has an infinitely old solution z(t) in E, which is componentwise bounded away from zero. It is the only infinitely old solution in E which does not tend to zero for t+ - 00. Further, z(t) is globally asymptotically stable in E \{O}. The reader will notice the far-reaching analogy between the three cases. One may consider the particular solution z(t) in case 3 as a generalization of the constant and periodic solutions in cases 1 and 2, respectively.

222

GUNNAR

ARONSSON AND INGVAR

MELLANDER

5. COMMENTS The multigroup model for gonorrhea, studied by A. Lajmanovich and J. Yorke, is autonomous and leads to a constant endemic equilibrium. Thus, seasonal variations are not taken into account. However, there are seasonal oscillations in the incidence of gonorrhea, at least according to U.S. statistics. The order of magnitude of these oscillations is 10-2096. For this, we refer to [4], [ 17, pp. 53-551, and [15]. Allowing the coefficients & and ai to have seasonal (periodic) variations should therefore add some further realism to the model, and that is what we have done here. The regular seasonal peak of gonorrhea incidence in the third quarter (U.S. statistics) is usually explained as due to seasonal variations of human behavior. Also other possible contributing factors are suggested in the literature, e.g. an observed seasonal variation of the virulence of the gonococci. These questions are discussed in some detail in [4], [15], and [ 171. Our analysis shows that the main features of the Yorke-Lajmanovich model carry over in a natural manner to the case of seasonal variations. REFERENCES 1 2

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci. 38: 113- 122 (1978). N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications 2nd ed., Griffin, London, 1975.

3

W. A. Coppel, Stability Boston, 1965.

Equations,

Heath,

4

C. E. Cornelius III, Seasonality of gonorrhea in the United States, HSMHA Rep. 86:157-160 (1971).

Heafth

5 6 7

F. R. Gantmacher, Matrizenrechnung, VEB Verlag, Berlin, 1958. J. Hale, Ordinary DifferentiaI Equations, Wiley, New York, 1969. P. Hartman, Ordinav Differential Equations, Wiley, New York, 1964.

8

H. W. Hethcote, Asymptotic behavior in a deterministic epidemic model, Bull. Math

and Asymptotic

Behavior

of Differential

Biology 35607-614 (1973). 9 H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci. 28:335-356 (1976). 10 A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28:221-236 (1976). 11 W. C. Rheinboldt, On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows, J. Math. Anal. Appl. 32:274-307 (1970). 12 E. Seneta, Nonnegatioe Matrices, Allen and Unwin, London, 1973. 13 R. S. Varga, Matrix Iteratiw Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. 14 P. Warren, J. Foster, and N. Bleistein, A stochastic model of a nonhomogeneous carrier-borne epidemic, SIAM J. Appl. Math. 31:569-578 (1976). I5 R. A. Wright and F. N. Judson, Relative and seasonal incidences of the sexually transmitted diseases, British J. Venereal Diseases 54:433-440 (1978). 16 V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Wiley, New York, 1975. 17 J. A. Yorke, H. W. Hethcote, and A. Nold, Dynamics and control of the transmission of gonorrhea, J. SexuatIy Transmitted Diseases 5: 5 l-56 (1978).