BULLETIN OF MATHEMATICAL BIOLOGY VOLUME 35, 1973

A S Y M P T O T I C B E H A V I O R I N A D E T E R M I N I S T I C E P I D E M I C M O D E L

�9 HERBERT W. HETHCOTE Department of Mathematics, University of Iowa, Iowa City, Iowa 52240

The effects of a periodic contact rate and of carriers are considered for a generalization of Bailey's simple epidemic model. In this model it is assumed that individuals become sus- ceptible again as soon as they recover from the infection so that a fixed population can be divided into a class of infectives and a class of susceptibles which vary with time. I f the contact rate is periodic, then the number of infectives as time approaches infinity either tends to zero or is asymptotically periodic depending on whether the total population size is less than or greater than a threshold value. The behavior for large time of the number of infectives is determined for three modifications of the model which involve carriers.

1. Introduction. Determinis t ic models have a long h i s to ry of use in the des- cr ipt ion of the spread of an infection. A basic reference is the m o n o g r a p h of

Bai ley (1957) which contains a descr ipt ion of bo th s tochast ic and determinis t ic epidemic models. A more recent rev iew of m a t h e m a t i c a l cont r ibut ions to the

descr ipt ion of the spread of epidemics is b y Dietz (1967). A widely used model

assumes division of a fixed popula t ion into three disjoint classes: susceptibles, infect ives and individuals who are r e m o v e d f rom the suscept ible infect ive

in te rac t ion b y isolation, dea th or p e r m a n e n t i m m u n i t y due to previous infec- tion. This mode l (hereafter referred to as the S I R model) is not app rop r i a t e for diseases where infect ive individuals become susceptible again as soon as t h e y

recover . Consequent ly , we consider the following model which is a special case of a mode l s tudied b y K e r m a c k and McKendr ick (1932) and generalizes the

s imple epidemic model p roposed b y Bai ley (1950). 607

608 H. W. H E T H C O T E

Let a fixed population N be divided into a class of infeetives I(t) and a class of susceptibles S(t) which vary with time t such that S(t) + I(t) = N and

I ( o ) = I o # o. (1)

Let the rate of increase of infectives due to new infections be/3I(t)S(t) where/3 is a constant called the infection or contact rate. Thus the rate of new infec- tions is proportional to the number of infectives and the number of susceptibles. Assume that infection does not give immunity and let the rate at which infec- tives recover and become susceptible again be ~I(t) where the recovery rate is a constant. Thus the differential equation for 1(0 is

I '( t) =/3I( t )S( t ) - yI(t) . (2)

The number of susceptibles can always be found from I(t) by using S(t) = N - I(t). The solution and asymptotic behavior of this model (which we call the SIS model) and of the corresponding stochastic model were determined by Weiss and Dishon (1971).

The first modification of the SIS model considered is a periodic contact rate /3(0. I f the population size N is greater than the average relative recovery rate t~ (defined below), then I(t) is asymptotic as t --> co to a periodic function. I f N _ t~, then I(t) tends to zero in an oscillatory manner. The other modifica- tions of the SIS model involve dissemination of the infection by carriers. I f the infection is spread either by both infectives and a constant number o f carriers or by carriers only, then I(t) is asymptotic as t --> co to a positive constant. I f the number of carriers decreases as an exponential function of time, then I( t) tends to zero.

These modified models of SIS type might be appropriate for respiratory diseases when infection gives only type specific immunity or when immunity is rapidly lost. Periodic variation in the number of infectives has been observed in illnesses transmitted via the respiratory tract such as common cold, influenza, pneumonia, streptococcal sore throat and meningitis (Rogers, 1963). Carriers are a mode of transmission in meningitis and streptococcal sore throat (Benen- son, 1970).

2. Periodic Contact Rate. Let the contact or infection rate in the differential equation (2) be a positive continuous periodic function fl(t) with period p and let the recovery rate ~ be constant. For seasonal variation, the period p would be 1 year. Using S(t) = N - I(t) in (2), the differential equation for I(t) is

I ' ( t ) = [ 3 ( t ) N - r ] I ( t ) - 3(t)F(t). (3)

For linear differential equations with periodic coefficients, the solutions have

ASYMPTOTIC BEHAVIOR IN A DETERMINISTIC EPIDEMIC MODEL 609

periodic factors, usually in a product with an exponential. Since the coefficients

in the nonlinear differential equation (3) are periodic, one would expect periodic

factors in the solution. The periodic and exponential components of l(t) are identified below and then the asymptotic behavior as t --> co of I(t) is given. The results presented in this Section and in Section 3 are proved in Section 4.

Le t k = 1V~ - y where f~ = l ip f~ fl(u) du. The funct ion

= - P] du

is easily shown to be periodic with period p. The unique solution for t > 0 of (1) and (3) is

exp [N ft o fl(u) du - yt] (4)

I(t) = ft ~ fl(v) exp [N fo fl(u) du - 9ev] dv + 1/I o

exp [a(t)] = O(t) + e x p ( - k t ) / I o' (5)

where c t

O(t) = Jo exp [ - k(t - v)]fi(v) exp [a(v)] dv. ( 6 )

The behavior of l( t) is clarified by considering its asymptot ic behavior. For t > _ p ,

~ (p(t) - D e x p ( - k t ) / ( e x p (kp) - 1) k # 0 (7) o ( t ) =

[.x(t) + [t/p]D k = 0 (8)

where

and

(p(t) = D exp { - k ( t - [t/pip - p)} + tg(t - [tip]p) (9) exp (kp) - 1

x(t) = O ( t - [t/pip). (10)

The symbol It/p] is used to denote the greatest integer t h a t does not exceed tip. The constant D is defined in Section 4. I t can be verified directly t ha t ~0(t) and x(t) are periodic functions with period/9.

The behavior of I(t) for large t wi th k < 0, k = 0 and k > 0 is obtained by using (5)-(10). Complete asymptot ic expansions (Erd~lyi, 1956) have been obtained, bu t are not included here. I f k = hr/~ - Y < 0, the infective class I(t) --> 0 in an oscillatory manner as t -+ ~ . For k = N~ - y > 0, the asymp- tot ic representation for I(t) is exp [a(t)]/rp(t) which is a periodic function with

610 H. W. H E T H C O T E

period p. I f we call ~ = ~/]~ the average relative recovery rate, then a princi- pal result of this paper can be s ta ted as follows: I ( t ) is asymptot ical ly periodic i f h r > ~5 and I ( t ) tends to zero if N _< ~. The average relative recovery rate fi could be called a threshold value since the asymptot ic behavior of the epidemic depends on whether the tota l populat ion N is less t han or greater t h a n ft. This result is similar to the Kermaek and McKendriek (1927, 1932) Threshold Theorems. Graphs which il lustrate the asymptot ic behavior of the epidemic are given in Figures 1 and 2.

1.0| ] #=2-1B COS 5t

0.9t N =1 u 0.8 K=I

0.7

H 0,6 u~

__0.5 U

0.4 : 7

0.3

0.2

0.'1

0.( TIME

10

Figure 1. Solution curves of (3) are shown with k = 1 > 0 and Io = 0,2, 0.4, 0.6 and 0.8. The curves are asymptotically periodic

with period 27r/5

I f fl is a constant , then the above results simplify considerably. The unique solution of ,(1) and (2) is

[ exp = l|fl(ekt 1 - 1)/k 4- 1/ I o ]c # 0 (11)

1(0

The behavior of I ( t ) for large t wi th k < 0, k = 0 and k > 0 is obtained directly from (11). Since k = Nfi - ~, i f w e let p = ~/fl, t h e n I ( t ) - - - > N - p as t-->oo

1.0

0.9.

0.8 �84

0.7

~0.6

>--0.5

L) Ld ~-0.4 Z

O.3

0.2

0.1

0.0 0

A

2 3 4 5 TIME

A S Y M P T O T I C B E H A V I O R I N A D E T E R M I N I S T I C E P I D E M I C M O D E L

B=2-1.8 C05 5 t N=I

=2.5 K=-0.5

F i g u r e 2. S o l u t i o n c u r v e s o f (3) w i t h k -- - 0.5 < 0 a n d Io = 0.2, 0.4, 0.6 a n d 0.8 a r e s h o w n . T h e c u r v e s a p p r o a c h z e r o i n a n osc i l la -

t o r y m a n n e r

611

for N > p and I( t ) - -> 0 as t--> 0o for N _< p. These results were obtained earlier by Weiss and Dishon (1971). Asymptotic expansions of I(t) have also been obtained, but are not presented here.

If the contact rate fl is constant or periodic in the SIR model, I(t) - ~ 0 as t --> 0o for all values of N and p, and S(t) approaches a positive constant. Re- current behavior in an SIR model modified to allow a constant growth rate for the susceptible class and an equal death rate in the removed class has been studied by using perturbations near the equilibrium solution. These and other results on recurrent epidemics are described in Bailey (1957, Chap. 8). The birth of new susceptibles is essential in obtaining a nonzero equilibrium point for the differential equations and periodic behavior in the SIR model. Birth and death within the fixed total population would not affect the results in the SIS model of this section unless a significant portion of deaths were due to the in- fectious disease. I f one assumes that the total population grows exponentially in the SIS model so that S(t) + I(t) = N exp (at) and that all individuals con- tributing to the growth are susceptible, then I ( t ) - -> ~ as t--> ~ . The same conclusion holds for linear growth of the total population.

3. Carriers. Assume that the infection is spread both by infeetives and by a

612 H . W . HETI-ICOTE

constant number of carriers C. Then the differential equation given by (2) is changed to

1' = / 3 ( 1 + C)S - y1

= tiCN + ti(N - C - p)I - tiI~ (12)

where p = y/ti . As before, the suseeptibles can be found by using S( t ) =

N - l ( t ) . We remark that the rate of new infections t iCS(t) in (12) could be due to an inanimate carrier such as a polluted water supply which infects at a rate proportional to the number of suseeptibles.

The unique solution for t > 0 of (1) and (12) is

1 a l ( t i I o + a2)eal t + a2(til o -- a l ) e - % t (13) I ( t ) = 73 ( t i I o + a2)e:, ~ - ( t i I o - a O e - % t

where R

al. 2 = ~{[(N - C - p)2 + 4CN]112 • ( N - C - p)}

with a 1 and 32 corresponding to the plus and minus signs, respectively. From (13) we see tha t the asymptotic representation of I ( t ) is al / t i so tha t I ( t ) is asymptotic to a positive constant for all values of N and p. This is reasonable since one would expect the carriers to keep the number of infectives from tend- ing to zero. An asymptotic expansion of I ( t ) has also been obtained.

I f the infection in the SIS model is spread only by a constant number of carriers C, then the differential equation (2) becomes I ' ( t ) = t i C ( N - I ) - ~ I which is a first order linear differential equation. The solution is easily found to be

( + tieN I ( t ) = I o t iC + y ] t iC -~ y

so that I ( t ) is asymptotic to the positive constant t iCN/ ( t iC + y) . In the SIR model, when the infection is spread either by both infectives and a constant number of carriers or by carriers only, both S( t ) and I ( t ) approach zero as t approaches infinity; i.e. the total population becomes removed since the carriers eventually infect everyone.

Weiss (1965) considers the deterministic and stochastic behavior of a model (of SIR type or of SIS type with y = 0) in which only carriers spread the disease and the number of carriers decreases exponentially with time as the carriers are identified and eliminated. Using these assumptions, the differential equation (2) becomes

r ( t ) = t i ( c e - a ~ ) S - ~X

= t i O N e - a t - ( t i c e - a t + r ) I .

ASYMPTOTIC BEI tAVIOR IN A DETERMINISTIC EPIDEMIC MODEL 613

The solution of this linear first order equation and (1) is easily found to be

f iCN f~ exp ( - a v - t i c e-aria + ~,v) dv + I o e -~c/a I ( t ) =

exp ( - [3C e - at/a + ~t)

I t can then be shown that I( t) ~ 0 as t --> oo. We would expect the number of infectives to tend to zero as the carriers are eliminated since only carriers spread the infection.

4. Proofs of the Results in Sections 2 and 3. I t can be verified directly that (4) and (13) are global solutions of (1) and (3) and (1) and (12). Since the right- hand sides of the differential equations (3) and (12) are locally Lipschitzian, solutions are locally unique by the Picard theorem (Birkhoff and Rota, 1962). Hence (4) and (13) are the unique global solutions of (1) and (3) and (1) and (12). The solutions (4) and (13) were found by using I = w'/f lw in the Riccatti differential equations (3) and (12) and then solving the resulting second order linear differential equations for w.

The equation (5) for I( t) is obtained from (4) by using 0(t) as defined by (6) and

f~ fi(u) du - ~t = N ~ t + a(t) - ~t = N let -t- (z(t).

By dividing the integral O(t) into the sum of an integral from 0 to p and an integral from p to t, and then changing variables in the latter integral, one ob- tains the recursion relation

for t > p where 0(t) = D e x p ( - k t ) + O ( t - p )

PP D = Jo ekVfi(v) e~(V) dv.

For k r 0, we use the recursion relation to find

0(t) = D e-kt(1 + e kp + . . . + e k~u/pl] + O(t - [tip]p)

= D e -kt exp {kp([t/p] + 1)} - 1 e ~ - 1 + O(t - [ t / p ] p ) .

This yields (7) with r defined by (9). For/c = 0, a similar method yields (8) with x(t) defined by (10). I f f l is constant, then (11) follows from (4).

Bailey, N. T. J. 1957.

LITERATURE

1950. "A Simple Stochastic Epidemic." Biometrica, 37, 193-202. The Mathematical Theory of Epidemics. London: Charles Griffin.

614 H. W. HETHCOTE

Benenson, A. S. 1970. Control of Communicable Diseases in Man, l l t h Edn. New York: American Public Heal th Association.

Birkhoff, G. and Rota, G.-C. 1962. Ordinary Differential Equations. Boston: Ginn & Co.

Dietz, K. 1967. "Epidemics and Rumours: A Survey." J. Roy. Statist. Soc. Ser. A, 130, 505-528.

Erd61yi, A. 1956. Asymptotic Expansions. New York: Dover Publications. Kermack, W. O. and McKendriek, A. G. 1927. "Contributions to the Mathematical

Theory of Epidemics, Part I . " Proc. Roy. Soc. Ser. A, 115, 700-721. - - and - - 1932. "Contributions to the Mathematical Theory of Epidemics,

Par t I I . " Ibid., 138, 55-83. Rogers, F. B. 1963. Epidemiology and Communicable Disease Control. New York:

Grune &Stra t ten . Weiss, G . H . 1965. "On the Spread of Epidemics by Carriers." Biometrics, 21, 481-

490. Weiss, G. H. and Dishon, M. 1971. "On the Asymptotic Behavior of the Stochastic and

Deterministic Models of an Epidemic." Math. Biosci., 11, 261-265.

RECEIVED 6-5-72