An epidemic equation with immigration

An Epidemic Equation with Immigration KENNETH L. COOKE Departmentof Mathematics,PomonaCollege, Claremont,CalijI91711 Communicated by S. Karlin

ABSTRACT We formulate a model of single-species population growth in which there is imrnigration into the population at any prescribed rate and with any prescribed age distribution. Births are assumed to be density-dependent but not dependent on the age distribution in the population. Deaths may be according to any age distribution. The model results in a non-linear, non-homogeneous integral equation with delay. This equation is also shown to be a model for growth of capital and for certain epidemics. Behavior of solutions as t tends to infinity is investigated, with the aid of theorems of Levin, Shea, Londen, and Karlin, when the forcing term tends to a constant limit.

1.

INTRODUCTION The author

and J. A. Yorke

x(t)=c+ where

c

and

L.

f

I t-L

the scalar integral equation

[l] have studied

t >

P(t-s)g(x(s))h

are real constants

t,

(1.1)

and L is a positive constant,

the equation as a model for certain processes. The principal mathematical

interpreting growth and epidemic result obtained was this: deterministic

THEOREM A Assume that g(x) is a continuously differentiable function and that P(t) is

continuously

differentiable,

non-increasing.

and non-negatiue

on the interval

0~ t < L. Let x(t) be any solution of Eq. (l.l), and let [to-L,o] be its maximal interval of existence, where to
(9 (ii) (iii)

x(t)FmJ x (t)--tconstant x (t)+ - 00

as t-o as t-+o as t-m

MATHEMATICAL BIOSCIENCES 29, 135-158 (1976) 0 American Elsevier Publishing Company, Inc., 1976

135

136

KENNETH

L. COOKE

Zn particular, any bounded solution on t, < t < 00 approaches a limit as t+ co. In this paper, we obtain extensions equations such as x(t)=/’

of these results to non-homogeneous

P(t-s)g(x(s))ds+f(t),

(1.2)

-4O=~’f’(t-s)g[x(s)+f(s)lds,

(1.3)

I-L

I-L

where the limit

is assumed to exist. Our objective is twofold. First, in Sec. 2, we show that these equations arise as models of population growth or the spread of infection under conditions similar to those in [l], but taking account of immigration into the population. Next we derive results on the asymptotic behavior of solutions. These are deduced from theorems due to Levin and Shea [2], Londen [3], and Karlin [4]. 2.

GROWTH POPULATION

AND EPIDEMIC

MODELS

WITH

IMMIGRATION

MODEL

Consider a population (of animals, cells, etc.) in which all individuals are of the same species or class. Assume that there is immigration into the population at the rate of F(t) individuals per unit time at time t, and let f(t) = number

of immigrants

alive at time t,

y(t) = number

of native-born

individuals

x(t) = total number

of individuals

alive at time t,

alive at time t.

We make the assumption that the number of births per unit time at time t is a function of the total population at t, g(x(t)). That is, the births are density-dependent but not dependent on the distribution of ages within the population. One of two assumptions is made concerning deaths (removals): Case 1. Assume that every individual exactly, where L is a positive constant. native-born individuals alive at t is just the interval [t-L, t]. We assume that we are

dies or is removed at age L, In this case, the number of sum of all births over the time examining a long-continuing

137

AN EPIDEMIC EQUATION

process, so that any “initial

population”

rw=J’I-L x(r)=

This last equation

has disappeared.

Thus we have

g(x(J>)b

(2-l)

1I-Lf dx(s>)ds-+-f(t).

(2.2)

is a special case of Eq. (1.2). We also have

uw=J’ f-L

d.Y(s)+f(s)l&

(2.3)

which is a special case of Eq. (1.3). In this model, we have made no assumption about the distribution of ages within the immigrant group, but in any event we can assert that

o< f(t)<

I1-Lf F(s)&

(2.4)

since the number of immigrants alive at time t cannot exceed the total number who have arrived over [t - L, t]. In many cases, it is a good approximation to assume that the immigration rate F(t) tends to a constant as t+cc and also that the age distribution within the immigrants tends to a stationary distribution. Suppose we let F (t, u) du = rate of entry at time t of immigrants

of ages u to u + du

Then

F(r)=~LF(t,~)d~.

(2.5)

0

Moreover,

since the inner integral represents the number of immigrants arriving at time t--s who are still alive at time t. If we assume that there is an

KENNETH L. COOKE

138 integrable

function

h(u) such that lim F(t,u)=h(u) ,+CC

uniformly

(2.7)

on 0 < u < L, then

lim F(t)= 1G+oO

J’A(u)du.

(2.8)

0

Equations (2.6) and (2.7) imply thatf(r) tends to a constant as f--zoo. Thus, when the assumption (2.7) is made, the forcing function f in Eqs. (2.2) and (2.3) satisfies f(t)+constant and the theorems below can be applied. Case 2. Now we assume that there is a distribution spans) or removal. Let P(s) = proportion

surviving

(or probability

Assume that P is non-negative, this case, we have

non-increasing,

of ages at death (life

of survival)

to age at least S.

and P(s) =0 for s > L. In

(2.9)

since the integrand P (t - s) g( x (s)) ds represents the number of native-born at times s to s + ds who are still alive at time t. The total population is

(’

x(t)=

P(r-s)g(x(s))ds+f(t),

Jt-L

and also

u(t)=~’ P(t-s)g(y(s)+f(s))ds. I-L

(2.11)

These equations are of the form in Eqs. (1.2) and (1.3), respectively. case, if we define F (t, u) as above, we have

ss L

j(t)=

L-s

ds

0

0

F(t-s,u)

P(u+s) P(u)

duT

In this

(2.12)

AN EPIDEMIC

EQUATION

139

since the inner integral represents the number of immigrants arriving at t - s who are stiil alive at time t. [P (U + s)/ P (w) is the probability of survival to age u + s, given that age u was attained; for u= L, this is interpreted as zero.] If we again assume that F satisfies Eq. (2.7), we see that f(t) tends to a constant as t+cc and the theorems below are applicable. ECONOMICMODEL

The equations of this section have an economic interpretation in terms of the growth of a capital stock when there is importation of capital. Let x(t) denote the value of capital at time t. Assume that the production of new capital within the economy depends only on x(t), and denote this rate of production by g(x( t)). Also let F(t)

= rate of importation

of new capital at time t,

f(t) = value of imported

capital at time t,

y(t) = value of internally

produced

capital at time t.

Assume that equipment depreciates over a time L (the lifetime of the equipment) to value 0. Assume depreciation to be independent of the type of equipment, or that the model refers to only one type of equipment. Let P(s) denote the value at age s of a unit of equipment, that is, of equipment which had value one at age zero.’ Assume that there is a fixed value C of non-depreciating assets. Then u(f)=/*

P(t-s)g(x(s))d, f-L

since P(t - s)g(x(s))d.v represents produced at time s. Therefore

x(r)=lt

the value

at time

t of new capital

P(t-s)g(x(s))dr+C+f(r),

(2.13)

P(t-s)g[y(s)+C+f(s)lcis.

(2.14)

f-L

uw=jt I-L

If we let F(t, u)du denote the number

of units of capital equipment

of ages

‘Or, assume that equipment retains its full value until breakdown, and P(s) is the probability of survival to age at least S.

KENNETH L. COOKE

140

u to u+ du entering at time t, an equation like Eq. (2.12) is again valid. Also,f(t)+constant as t-+co if F satisfies Eq. (2.7). In the special case in which all imported equipment (immigrants) is of zero age, Eq. (2.12) should be replaced by

f(t)=joLF(t-s)P(s)ds=Ij~LF(s)P(t-s)L,

(2.15)

where F(t) is the rate of entry of equipment (immigrants). If F(t) tends to a constant as t+m, thenf(t) tends to a constant. If F(t)+O, thenf(t)+O. EPIDEMIC

MODEL

If immigration is allowed in the model of spread of infection of Cooke and Yorke [ 11, equations of the form in (1.2) and (1.3) arise. Consider an infection for which (a) having the infection instills negligible immunity and (b) the incubation period (time between exposure and becoming infectious) is negligibly short. For example, gonorrhea approximately fits these assumptions, as explained in [l]. Furthermore, immigrants and travelers are apparently an important factor in the spread of sexually transmitted diseases; see Van Parijs [7]. Let C be a fixed constant and let N (t) = Cn (t) = total population, x(t) = number

of infectious

individuals

per C individuals,

y(t)

number of infectious individuals = contracted the disease by contacts not infectious immigrants),

(per C individuals) within the population

f(t)

number of individuals (per C individuals) who immigrated with = the infection and still have the disease at time t (infectious immigrants).

Assume that the number of new cases per unit time population is a function only of the relative density the form g(x(t)). (Under very crowded conditions, need modification.) Note that x(t)=y(t)+f(t), and total number of “native-born” infectives andf(t)n(t) infectives who immigrated with the disease and still t.

who (are

per C individuals in the of infectives, that is, of this assumption may also that y(t)n(t) is the is the total number of have the disease at time

Case 1. Assume that every individual is cured after exactly time L and returns at once to the susceptible population. Then for the number of

141

AN EPIDEMIC EQUATION

“native-born”

infectives

we have

Y(t)n(o= y

f-L

hence x(t)=

t s

g(x(s)>n(s)

h+f(t)

t s

(2.16)

9

n(t)

I-L

u(t)=

g(x(s)Ms)dJs;

dY(s)+f(sMs)

ds

(2.17)

n(t)

I-L

We may allow the possibility that the infectious immigrants have had the disease for different lengths of time. Let F(t,u)du denote the rate of entry at time t of infectious individuals who have had the disease for a time u to u+du when they enter. Then

I

L-s

F(t-s,u)du

0

represents the number of infectives still infectious at time t. Therefore j(t)=

who immigrated

at time t-s

--$~Lds(L-sF(t-s,u)du.

and are

(2.18)

0

Of particular increasing,

interest is the subcase in which the population n(t)=noea’

In this case, the equations

(ix >O).

is exponentially (2.19)

take the form (2.20)

y(t)=

/’

g[y(s)+f(s)]e-““-“)ds,

(2.21)

1-L

f(t)=e-“’

/0

Lds

J0

L-sF(t-s,u)du.

(2.22)

142

KENNETH

L. COOKE

Equations (2.20) and (2.21) are of the form of Eqs. (1.2) and (1.3). Moreover, if we assume that F satisfies Eq. (2.7) then f(r) tends to zero as t+co. If we assume (2.23)

lim F(t,u)e-“‘=h(u), r-+m

that is, the number of infectious immigrants is increasing at the same rate as the total population and is approaching a stable age distribution, then f(t) tends to a constant as t-+co. Case 2. Assume that there is a distribution of times until cure, rather than a fixed cure time. Let P(s) denote the probability of having the infection for a time at least s after infection, and take P(s)=0 for s > L. Then the number of “native-born” infectives is

y(t)n(t)=J’

g(x(s))n(s)P(t-s)ds. t-L

Therefore x(t)=

f s

n(s) dx(s))P (t - 3)n(r) ds+f(t>,

I-L I

v(t)=

4s) dY(S> +f(s))P (t - 3)q ds.

(2.25)

/ t-L If n(t) has the form in Eq. (2.19), these equations are of the form2 in Eqs. (1.2) and (1.3). If F(t,u) is defined as before, then L-S F(t-s,u)

P(u+s) p(u)

du.

(2.26)

These considerations indicate that various problems in the growth of populations and the spread of infection lead to equations of the form in (1.2) and (1.3) where f satisfies (1.4). The rest of the paper presents some mathematical deductions concerning the asymptotic behavior of solutions of these equations and certain more general equations. In Sec. 3 there is a brief epidemiological interpretation of some of the mathematical results. 21n general, the substitution ;(t)=x(r)n(t), i(r)=u(t)n(r), similar equations in which g(x) must be replaced by g(x,s).

j(r)=f(r)n(f)

results in

143

AN EPIDEMIC EQUATION

3.

ASYMPTOTIC

BEHAVIOR

In this section, we shall derive several theorems on the asymptotic behavior of solutions of Eqs. (1.2) and (1.3), and certain generalizations of these. We are particularly interested in conditions under which lim,,,x(r) exists. Our theorems are simple consequences of difficult theorems on integral equations due to various authors. First, we shall apply a general theorem published by Levin and Shea [2]. The Levin-Shea theorems are stated for integral equations of the form

Z(t)+

I-mmg(Z(t-s))dA(s)=J(t)

(--w
(3.1)

and so we first show that our Eq. (1.2) can be put in this form. As in [2], we use the notation 1sof bounded

NBV(Z)={A(t)JA(t)

on Z = [t,, tz],

variation

(- cc < I, < t, f co), left-continuous

L m (I ) = { f(t)] f (t) is Lebesgue measurable

on

(t,, tzl,A (t,) =0}

and essentially

bounded

on Z }

B(I) = {x(t)]x(t) is Bore1 measurable on I}

The following lemma is analogous

to Lemmas 2.2-2.6 in [2].

LEMMA Let g be continuous, let A be in NBV[O, co), A(0) =O, f~ L”(0, w), and assume that f(m) exists. Let x(t) be a function in B (0, co)nL-(0, 00) which satisfies3

x(t)+

1Ls(X(t-w-w=Sw

(OG t
0

x(t)=&)(t)

‘The integral is to be interpreted as the integral over [0, L].

(-L
(3.2)

144

KENNETH

where o(t) is continuous.

Then

Define k,x_% by

f(t)

(t 2 L),

f(r)+g(x(o))[d(L)-A(tSLg(w(t-s))dA(s) I+ x(O)+A(~)dx(O))

(0<1
1

fO)=

L. COOKE

(t
a(t)ENBV(-co,co),

co,w)nLm(-m,co), and?

~(t)~L~(-co,c~), f(w)=f(co), satisfies Eq. (3.1) for tE(-co,co).

I(t)EB(-

ProoJ: The idea is to extend A and x in a simple way and then choose f so that Eq. (3.1) is satisfied. Let k and 2 be defined as stated in the lemma. (In general f does not equal w on [ - L,O), but this will not affect our subsequent conclusions.) Then for t B L we have

JLg(X(w~(4-f(o=w)+~Lg(S(1-wm-f(f)

0=x(t)+

0

0

s

m g(Z(t-s))dA”(s)-f(t). -cc

=2(t)+

Hence f satisfies Eq. (3.1) on t > L if we define O
f”(t)=f (t) for t > L. For

0=x(t)+J’+g(x(t-s))dA(s)+I^Lg(x(r-s))d4(s)-f(l) 1+

0

=2(t)+

j’+d~(~-~))~(4+~Ld4W~NM4 0

=2(t)+ +

Irn g(i.(t-s))d&s)+(P(m))dA(s) -m ,+

Ld4+WA(+f(4 / I+

=2(t)+ +

*+

/

mg(a(r-s>>~(s)-g(x(O))[A(L)-A( --m

Lg(4~-w+)-f(~).

/ I+

i

145

AN EPIDEMIC EQUATION

Hence ? satisfies Eq. (3.1) on O< t < L if we definef(t)

as in Eq. (3.3). For

t G 0, we have

Z(t)+

lrn

g(i(t-s))~~(s)-f(t)=x(O)+ILg(~(t-s))dA(S)-~(I) 0

-03

=x(O)+g(x(O))[A(L)-A(O)l-J(t) and so we take f”(t) = x(0) + g( x (0))A (L). Clearly k,?, P, have the indicated properties. n We are interested

dA(s)=

in the special case in which

A(t)=

- P(s)ds,

-j”‘P(s)ds

(O
(3.4)

0

Then Eq. (3.2) has the form in Eq. (1.2),

x(t)=jLg(x(t-s))P(s)dSff(t)=I’

g(x(s))P(t-s)ds+f(t) I-L

0

(0 < t < 00)

(3.5)

By the lemma, we may extend x,f, P to .Z,f, 2 so that Eq. (3.1) is satisfied. The Levin-Shea theorems relate bounded solutions of Eq. (3.1) to bounded solutions of the “limit equation” u(t)+S”

--m

g(y(t-s))d(s)=f(c=o)

For our case, the limit equation

(-w
(3.6)

is simply

r(t)=f(w)+ILg(y(t-s))P(s)ds 0

g(y(s))f’(t-s)ds

(-w
(3.7)

By combining the information on solutions of Eq. (3.7) given by Theorem A with theorems of Levin and Shea, we can cow obtain results for Eq. (3.5). We first recall some of the notation and concepts from [2]. If {t,} is an increasing sequence of real numbers which satisfy lim,,,(t,,, - t,,,_ ,) = co,

KENNETH

146

then a “$-sequence” associated with {t,,,} is a sequence functions {&(t)}, #,,,(f)~ Cm(- cc, co), such that

L. COOKE

of real-valued

(-m
41(t)= 1 (I

Q t,>,

L(t)=0

(t< tm-,

l&(t)>0

(t,-,sttrJ,

G,(t)=0 and

(h

Q

t),

t> r,,,,,),

J/{(t) s 0 (t, s t Q tJ, An,,(L) = 13

s 0 (t, s t s t,,,).

q;(f)

Theorem 1 concerns a class af functional differential equations more general than (1.2) or (1.3). Corollary 1 deals with Eq. (1.2) itself. We let C [ - L. 0] denote the Banach space of continuous real-valued functions on [ - L,O] with the supremum norm. For each continuous function x(s) on s E[ - L, co) we define elements x, in C[ - L,O] for each t > 0 by x,(O)=x(t+O)

(@Et-L,Ol).

That is, x, is a translation of the “segment” of x(s) over the interval [t - L, t]. Finally, we let G be a map of [0, co) x C[ - L,O] into the real numbers, and we consider the equation

x(t) = G (t>x,> THEOREM

(t > 0).

(3.8)

I

Assume that G is continuous and that there is a constant c (independent of $) such that

lim G(t,(P)=c+ t-+m

I0

for all + in C [ - L, 01, with uniform C[ - L,O]. Assume that g and P satisfy

‘P(s)g(+(-s))ds convergence

(a) g(x) is continuous/y differentiable, (b) P(t) is non-increasing and non-negative onO
(3.9) on bounded subsets of

and P’(t) is continuous

AN EPIDEMIC

147

EQUATION

where 72-L=

sLP(s)ds. 0

Then :

(i)

ZfS

is empty, Eq. (3.8) has no continuous solution which is bounded as

t+c0.

(ii) Zf S is non-empty, then for each bounded continuous solution x of Eq. (3.8) there exist sequences {t,,,}, {E,}, {&,,}, and {y,} such that t,- t,_, +w, E,,, decreases to zero, {&,,} is a q-sequence in the sense of Leuin and Shea, and { y,} is a sequence of functions chosen from the set T = { y(t) :y is uniformb continuous on ( - 00, co), llyllm < /Jx(lW, andy

y(t)=c+

satisfies (3.10)},

I Lws(Ytt-s))~ (-w
(3.10)

0

Moreover,

sup Ix(O-ym(t)l G-z< r< t,+,

< &??I

(m=3,4,...),

lim _rl(t)=O. t-m

(3.11)

(3.13)

Each function y in T has a limit k in S as t+ co. Proof. Let x be a bounded

continuous

solution

of Eq. (3.8). Define

(t > 0).

f(t)=G(t,x,)+(s)g(x(t-s))ds 0

Clearly f is continuous. Also, since x, lies in a bounded subset of C [ - L, 01, it follows from the uniform convergence in (3.9) that f is bounded and that f(t)+c as t+w. Furthermore, x satisfies

xtt)=ftt)+~=Pts)~txtt-s))ds 0 which is Eq. (3.5). Thus, assumption that f(t)+c.

it suffices

to consider

Eq. (3.5), under

the

148

KENNETH

L. COOKE

To complete the proof, we now write Eq. (3.5) in the form in Eq. (3.2). Then by the Lemma, we extend x,A, f to Z,a, f to get Eq. (3.1). We can then apply Theorem lb in [2] to Eq. (3.1); it is only necessary to verify the one additional hypothesis that the Tauberian condition lim (.?(t+r)-z?(t)]=0 t-+cc

(T)

T-+0

is fulfilled. However, in our case, A(s) is absolutely continuous and then, as pointed out in [2], (T) follows from the other hypotheses of Theorem lb. This is easily verified in our case, since

G If(t+7)-f(t)l+

i::I, g(x(s))P(t+vs)ds-J’ g(x(s))P(t-s)ds I-L

and since g(x(s)) is bounded when x(s) is bounded. Theorem lb gives Eqs. (3.11), (3.12), (3.13) immediately. By Theorem A, each functiony in r has a limit, which clearly must be in S. H COROLLARY

Assume that: (a) g(x) is continuously differentiable. (b) P(t) is non-increasing and non-negative, and P’(t) is continuous on O
~Lp(4~(9(-~))ds. 0

Then Eq. (3.8) reduces to Eq. (3.5). Moreover, G is continuous (3.9) holds. Therefore, Theorem 1 applies. n Example .

in (t,+), and

Consider the equation

x(t)=c+

I LP(s)h(t-s,x(t-s))dr, 0

(3.14)

149

AN EPIDEMIC EQUATION

and assume that lim h(t,r)=g(r) 1-_)00 uniformly

on bounded

(3.15)

sets in R. Define

I LP(s)h(l-s,+(-s))ds.

G(t,+)=c+

0

Then Eq. (3.9) is satisfied, with uniform convergence on bounded subsets of C [ - L, 01. Consequently, Theorem 1 may be applied. As a special case, we may consider Eq. (1.3) where it is assumed that f(t)+c, as t-co. Let h(s,r)=g(r+f(.r)). Then lim h(t,r)=g(r+c,)zf *-Pm

g(r)

uniformly on bounded sets in R, and Eq. (1.3) has the form of Eq. (3.14) with c = 0. In this case, the limit equation has the form

Y(r)=jLP(S)g(Y(f-s)+C1)dr,

(3.16)

0

and the set S is

Even though each y,,,(t) approaches a solution k in the set S, Eqs. (3.11) and (3.12) do not permit us to make a very explicit statement about the asymptotic behavior of x(t)’ as t-+co. Even in the case in which S consists of isolated points, we cannot assert that lim,,,x(t) exists, because every translate of a solution ym is another solution. The following theorem of Londen [3] will be used to obtain addditional information. REMARK

THEOREM

2. (LONDEN)

Assume that a(t) is non-negative and non-increasing on [0, co), a(O) is finite, and a(t) is in L’[O,co). Let h(x) be continuous on --oo
J0

‘a(t-s)h(x(s))ds=f(t),

O
(3.17)

150

KENNETH

L. COOKE

such that x(t) is in Lm(O, cc). Then x(t) is slowly varying in the sense that for any positive constant T one has lim f-cc

sup

x(s)-

inf

x(s)

f-T
I-T
1

=o

(3.18)

A Iso lim dist(x(t), f_OO

L) = 0,

(3.19)

where L is defined by

j0

ma(s)ds=f(rm),

limx(t)

< x < lim x(t) I--to3 -

,-+*

.

(3.20)

1

In order to apply Londen’s theorem, Eq. (3.5) must be put in the form of Eq. (3.17). Since it has been assumed that P(s)=0 for s > L, Eq. (3.5) may be written as x(t)=

jfg(x(s))P(t-s)ds+J(t)

(OG t
(3.21)

0

where g(x(s))P(t-s)ds

(0~ t
(3.22)

(Ldf
E L”(0, oo), then f(t) E Lm(O, cc)), provided x(t) is bounded on the interval [ - L,O] and P(t) is bounded. Also, f(t) and j(r) have the limit as t-+w. Identifying - g with h, we conclude that Theorem 2 for Eq. (3.5). Thus we have

THEOREM

3

Assume that P(t)>O,

P (t) is non-increasing, for

P(t)=0

P(t)EL’[o, g(x) is continuous f(t)

E L”(0, co),

and

P (0) is finite,

t > L,

11,

on - cc
= c exists,

AN EPIDEMIC

151

EQUATION

Let x(t) be a solution of Eq. (3.5) with x(t) in L”(O,oo). varying in the sense of Eq. (3.18) and satisfies

Then x(t) is sfowly

(3.23)

!izdist(x(t),S,)=O, where s, =

k:k=c+rLg(k),

nL=

LP(s)ds.

k< lim x(t) , I ,+UZ

limx(t)< 1-m

(3.24)

It is possible to generalize Theorem 3 to equations of the form (3.8) when (3.9) holds, by the same device as in Theorem 1. We omit the statement of the result.

REMARK

CONDITIONSFOREXISTENCEOFLIMIT ANDBOUNDEDNESSOFSOLUTIONS

If the set S consists of isolated points, and if x(t) is a bounded solution of (3.5), then the set S, contains a finite number of points. From (3.23) it is then easy to see that lim x(t)= ,-bCO

k,

where k is one of the points in S,. Since Theorems 1 and 3 deal with bounded solutions, it is of interest to give conditions under which all non-negative solutions are bounded. The following theorem is a slight extension of a result of this type given by Brauer [6]. THEOREM4

Consider the equation x(t)=

Assume that Ik(s,t)l

’ g(x(s))P(t-s)k(s,t)ds+f(t). / I-L < c, for t- Lgs<

g(x)

tin, lim sup x-m

X

<1,

t,O< t, and

7TL=

s0

LP(s)dF.

Assume also that H,: f(t) is continuous and If(t)1 Q M, HP : P (t) is non-negative, P(t) G 0 for t > L, Ha: g(x) is continuous on (- 00, oo), g(x) > 0 on x > 0.

(3.25)

152

KENNETH L. COOKE

Then euely non-negative

solution is bounded on 0 < t < 00.

Prook Choose p < 1 and K so that c,?r,g(x) < px for x > K. Define

Z,={ulx(~)>K}, We have f(t) < M, and Hence for any t > 0,

xw=m+/

Z2={uIx(u)<

since g is continuous,

K}.

g(x) < K2 for 0 < x < K.

g(x(s))P(t-s)k(s,t)ds

1,

t-L
/ 12 r-L
s(x(s))P(t-s)k(s,t)ds

GM+

P(t-s)ds

~x(S)P(‘-S)lk(S,t)ld~+K2c,S

s 11

12 I-L
f-LCSCI


sup

P(t-s)ds+

x(s)

K2c,l’

rLl-L
< M+p

Therefore,

P(t-s)ds I-L

sup x(s)+ OCS
K2c,rL.

for any T > 0, sup

x(t)<

O
Since this bound

is independent

M+ K,c,r= l-p .

of T, the theorem is proved.

n

This result can be applied to Eq. (2.24) even when n(t) is not of the form noear. We only need assume that In(s)/n(t)l < c, for t-L < s < t, which is reasonable in the applications. REMARK

INTERPRETATION

FOR EPIDEMIC

MODEL

We shall give an interpretation of Theorem 3 for the epidemic model of Eq. (2.24), where we assume that n(t) = noear, (Y> 0. Then Eq. (2.24) is of the form in Theorem 3 with P (t) replaced by P (t)e-=‘. If F (t, u) < h(u), where F(t, u) is the rate of entry of infectious individuals who have had the disease for a time U, then from Eq. (2.26) we see that f(t) tends exponentially to zero. Then in Theorem 3 we have c=O. In practice, we may assume that

153

AN EPIDEMIC EQUATION

g(x) is bounded as x-+co, and then certainly every non-negative solution is bounded, and

(3.25) holds. Consequently

where now L

7TL,=

1P

(t)eC”‘dt.

0

For many functions g(x), S, contains finitely many numbers, and from Theorem 3 we conclude that all solutions tend to one of these values. Thus, the possible asymptotic (endemic) levels for x(t), which is the rate of prevalence of infection, are given by the intersection points of the curves y = g(k) and y = k/vrL (see Fig. 1). Ordinarily, g(0) = 0 and therefore one of these intersections is at k =O. Brauer, for a slightly different equation (of Volterra type), found a condition such that no solution which is not identically zero for large t can tend to zero as t+co. A similar linearization for our equation could no doubt be used to show that a point k in S, is locally unstable if g’(k) > l/rL and stable if g’(k) < 1/rL. For example, in Fig. 1, points A and C are stable but B is unstable.

FIG.1 On the other hand, if F(t, u) is growing with t at the same rate as n(t), say F(t, u) = eYi(u), then f(t) tends to a positive constant c. The points in S, are the intersection points of the curves y = k/rL and y = g(k) + c/vrL. The latter is a translation of y = g(k) by c/rrL. If g(k) has the shape in Fig. 1, we note that as c increases, the stable level at k =0 shifts to positive k,

KENNETH L. COOKE

154

and if C is large enough, the intersectoon points A and B disappear entirely, leaving only the point C (shifted to the right). Thus, in such a case, a small immigration of infectives might create a large jump (a bifurcation) to a high endemic level of disease. However, for various other shapes of g(k), no such large qualitative change can occur. It therefore appears to be of practical importance to determine the shape of the graph of g(k) for any disease for which this model may be applicable. 4.

THE LINEAR

EQUATION

In growth or epidemic processes, it is unlikely that the function g(x) will be linear. Nevertheless, it may be of interest to indicate some of the additional information which is available in the linear case, since this is useful in local-stability analysis as in the preceding paragraph. We take g(x)= ax in Eq. (1.2). Consider first the case in which a~~= 1 andf(t) tends to zero as t-+ co. Theorem 3 is of no use, since the set S, is an interval. THEOREM

5

Consider the equation x(t)=f(t)+njgLx(t-s)P(s)ds=f(t)+aJ’

x(s)P(t-s)dr I-L

(O
(4.1)

with initial condition x(t) = w(t) for - L < t Q 0. Assume that

P(t) B 0, def

a7rL= a

L

/0

P(s)ds=

a

I

(hence

1

a > 0),

L

tP(t)dt=m#O,

0

lim f(t)=O. 1-r* Then ifx(t)

is a bounded Borel-measurable

solution of Eq. (4.1),

lim x(t)= ,-tm

0

AJm_f(t)dt+ 0

$lLdvlou(T)P(s)d7. --s

(4.2)

155

AN EPIDEMIC EQUATION

Proof. We shall deduce this from Karlin’s form of the renewal theorem [4], quoted as Theorem 6b by Levin and Shea [2]. Let x(t) be a bounded solution of Eq. (4.1). Define A(t) for 0 =Gt G L by Eq. (3.4), so that x(t) satisfies

x(t)+a Extending

I0

LX(t-s)u!A(s)=f(t)

(0 < t < Cc).

A, f, and x as in Eq. (3.3) we deduce from the Lemma that

Jmqt-s)dA(s)=f(t)

Z(t)+a

(-w
-m

We have =-adA(t)=uP(t)dt>O

(O
=o

(t
&u&t)]

Oz d[-d(t)1 I -co

t>Jc)

=u/LP(t)dt=u?i,=l. 0

Since J(t)=x(O)-mLx(0)=O nally, f(t) tends to 0 as t-co,

for t
we have

JT,\_f(t)ldt
Fi-

Jrn td[-u~(t)l=a~~tP(t)dt=miO. -cc 0

Since all hypotheses

of Karlin’s theorem are satisfied, it follows that

Thus lim

,f(t)=x(O)+

-!Jmf(t)dt

1-00

0

=x(O,+JJo =x(O)+

f(t)&

0

gyt)dt

This gives the conclusion

f

- ~j%.J~“[X(o)-W(t-s)lP(s)dt

0

0

1 m =- m o f(t)&+ s

9x(O)-w(t-s)lP(s)ds

- ;I‘?q

0

~~~dqsw(t-s,P(s,dt 0

of Theorem

0

5.

n

156

KENNETH

L. COOKE

If the assumption limf(t) =0 is dropped, one cannot in general assert that limx(t) exists, but it is possible to prove that the convolution of x with any integrable function has a limit. See [4,5]. We now turn to the case in which arrL# 1. The following is an immediate corollary of Theorem 3 and of (3.25). COROLLARY

2

Assume that P and f satisfy the hypotheses of Theorem 3 and that g(x) = ax and an, # 1. Zj x(t) is a solution of Eq. (4.1) with x(t) in L-(0, w), then lim x(t)=(l-asp,)-’ ,-+CC

lim f(t). ,402

In particular, if arL < 1, then every non-negative satisfies (4.3). Proof The equation

defining

(4.3)

solution is bounded and

S is k = c + arLk, and there is a unique root

k=(l

-asrL)-‘c.

By Theorem 3, any solution x(t) in Lm(O, cc) approaches this unique limit. Moreover, if an, < 1, then condition (3.25) is satisfied and every nonnegative solution is bounded. n If ar= > 1, Corollary 2 is not helpful, because solutions general, bounded. The following result is more useful. COROLLARY Assume

are not, in

3

that x(t) is a non-negative

solution of Eq. (4.1) and

P(t) > 0,

Let a be the unique real root of the equation L

a

e-“‘P(t)dt=

/0

1.

(4.4)

Assume that

I

D3

e-O’lf(t)ldt
(4.5)

0 L

a

te-“‘P(t)dt+O,

(4.6)

lim e-“ff(t)=O, ,+CC

(4.7)

/0

157

AN EPIDEMIC EQUATION and that x(t)e-“’

is bounded as t+oo.

Then

lim ~(t)e-~‘= *+OO

(4.8)

constant.

Proof. Let x,(t)=e-“‘x(t),

P,(t)=e-“‘P(t).

fi(t) = e-“Y(t),

Then from Eq. (4.1) follows xl(t)=j,(t)+aJLx,(t-s)P,(s)ds. 0

For this equation, all hypotheses a constant as t-+ 00.

of Theorem

5 are fulfilled, so x,(t) tends to n

REMARK Corollary 3 shows that solutions which are O(eO’) are in fact asymptotic to a constant times eb’, provided f satisfies the stated assumptions on growth. If ar= > 1, then CJis positive, and the result shows that a large class of solutions are exponentially growing. If arL < 1, u is negative, and the result applies only when x(t) andf(t) are exponentially decaying to zero. For arL < 1, Corollary 2 is generally more useful.

5.

A NUMERICAL

EXPERIMENT

Several numerical experiments were performed to test the validity of the results presented above. An interesting and somewhat ambiguous case discovered is as follows. Let L= 1 and 0 2x

g(x) = I f(t)

2(l+e-l-e-“)

(x GO), (0 < x < l),

(5.1)

(1 f x),

= e-‘/3,

(5.2)

(5.3)

In this case, f( cc) = 0 and

Since

the

condition

(3.25) is satisfied,

all non-negative

solutions

are

158

KENNETH

bounded. Moreover, the set S is

Theorem

3 is valid. However, the equation

L. COOKE

determining

2k = g(k), and we see that S= {k : 0 < k < l}. According to Theorem 3, every nonnegative solution is slowly varying and approaches the set S, but it is not clear whether limx(t) will exist. The results of Brauer do not immediately apply in this case, since g”(x)=0 for x near 0. In the numerical experiment, we chose x(t) initally equal to t + 1 on - 1 < t < 0. The solution x(t) is then continuous, and it was found that x(t) increased to a maximum of approximately 1.206 at t = 0.6875 and then decreased monotonically to one. Much of this work was done while the author was visiting the Istituto Matematico deN’Universitb di Firenze with the support of a Fulbright-Hays research grant. The author wishes to express his gratitude for this opportunity. The author wishes to thank James Yorke, S.-O. Londen, and J. J. Levin for helpful suggestions. Also, he wishes to thank S. E. List and J. Kinrich for performing the numerical experiments. REFERENCES 1

K. L. Cooke

2

epidemics, Math. Biosci. 16, 75-101 (1973). J. J. Levin and D. E. Shea, On the asymptotic some integral

3

(1972). S.-O. Londen,

and

J. A. Yorke,

equations,

Some

equations

modelling

behavior

growth

of the bounded

processes solutions

I, II, III, J. Math. Anal. Appl. 37, 42-82, 288-326,

On the asymptotic

behavior

of the bounded

solutions

and of

537-575

of a nonlinear

6

Volterra equation, SIAM J. Math. Anal. 5, 849-875 (1974). S. Karlin, On the renewal equation, Pac. J. Math. 5, 229-257 (1955). W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York, 1966. F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J.

7

M&h. Anal. 6, 312-317 (1975). L. G. Van Patijs, Nothing to hide, World Health, May

4 5

1975, pp. 2&25.