Journal of
J. Math. Biology 9, 37--47 (1980)
Mathematical
Biology
9 by Springer-Verlag1980
Integral Equation Models for Endemic Infectious Diseases*
Herbert W. Hethcote 1 and David W. Tudor 2
1 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
2 Department of Mathematics, The College of Charleston, Charleston, South Carolina 29401,
USA
Summary. Endemic infectious diseases for which infection confers permanent
immunity are described by a system of nonlinear Volterra integral equations of
convolution type. These constant-parameter models include vital dynamics
(births and deaths), immunization and distributed infectious period. The models
are shown to be well posed, the threshold criteria are determined and the
asymptotic behavior is analysed. It is concluded that distributed delays do not
change the thresholds and the asymptotic behaviors of the models.
Key words: Epidemiology - - Endemic infectious diseases - - Deterministic
models - - Thresholds - - Distributed delays - - Stability.
1. Introduction
Many infectious diseases are endemic in a population, i.e., present for several years.
I f there is no inflow of new susceptibles into a population and if the infection
confers permanent immunity, then the infectious disease will always die out. The
source of new susceptibles which allows many diseases to remain endemic is the
birth of new susceptibles. Consequently, in order for a model to describe an
endemic infectious disease over a time period longer than a year or two, it should
include vital dynamics (births and deaths). A model without Vital dynamics is only
suitable for describing an epidemic, i.e., an outbreak of an infectious disease such
that the incidence both increases rapidly and then decreases within a short time
period (such as one year).
The models which we derive and analyse here have four distinguishing features.
First, they include vital dynamics. Second, they assume that infection confers
permanent immunity. Third, they include immunization of newborns and also of
susceptibles of all ages. The fourth and most significant feature of the models is
that they allow any biologically reasonable probability distributions for the exposed
period and the infectious period.
Some features of our models have been included in previous models. The
*
This work was partially supported by NIH Grant AI 13233.
0303-6812/80/0009/0037l$02.20
38
H.W. Hethcote and D. W. Tudor
asymptotic behavior for infectious disease models without vital dynamics is significantly different since the disease always dies out and the final susceptible population is always positive [7, 13, 19]. Ordinary differential equations, functional
differential equations and integral equations have been used for models without vital
dynamics [13, 1, 10, 19, 11, 20]. Although some models consider the age structure of the population [11, 18], the models here do not.
Models for which the infection confers no immunity have been studied [3, 6, 7, 14].
Some models have assumed that the immunity is only temporary [12, 7] and
conditions under which models of this type can have periodic solutions have been
determined [9]. Models with immunization have been analysed [10, 8, 18]. Models
which have an exponentially distributed infectious period reduce to ordinary
differential equation models [13, 1, 7]. Models with a constant infectious period
have been considered [3, 11, 19, 20].
The population being considered is divided into disjoint classes. The susceptible
class S contains those who can become infected, the exposed class E contains those
who are exposed but not yet infectious, the infective class I contains those who are
infectious, and the removed class R contains those who have permanent immunity
either from immunization or previous infection. The class E is not included in some
models whenthe period of exposure is short or can be ignored. The flow between
the classes is used in naming the models; here we consider SIR and SEIR models.
The SIR model is derived in Section 2. It is a system of nonlinear Volterra integral
equations of convolution type. This model is shown to be well-posed and special
cases of the model are identified. In Section 3 the threshold criterion is identified
for the SIR model, the equilibrium points are determined and their stability is
analysed. It is proved that the disease always dies out if the threshold is not exceeded
and that the endemic equilibrium point is locally asymptotically stable if the
threshold is exceeded.
The SEIR model is derived in Section 4 and its asymptotic behavior is analysed in
Section 5. Conclusions are given in Section 6. The principal conclusion for these
models with permanent immunity is that the threshold quantity, the equilibrium
points and their stability depend only on average values and do not depend on the
distributions of the exposed and infectious periods. This implies that the simpler
models are sufficiently general for use in studying specific endemic diseases with
permanent immunity. Some of the methods of proof used here are also used in
[9].
2. The SIR Model
Assume that the population size is constant and that the population is uniform and
homogeneously mixing. Divide the population into disjoint classes which change
with time t and let S(t), I(t) and R(t) be the fractions of the population that are
susceptible, infectious and removed, respectively. The constant contact rate/3 is the
average number of contacts (sufficient for transmission) of an infective per unit time.
Thus the susceptibles are transferred at a rate equal to/3S times the number of
Integral Equation Models for Endemic Infectious Diseases
39
infectives. Let P ( t ) be the probability of remaining infectious t units after becoming
infectious. Assume that P ( t ) is a noninereasing function with P ( 0 ) = 1 and
P(ov) = 0 and that P ( t ) is dominated by a decaying exponential. These conditions
allow many different P ( t ) such as those corresponding to a constant infectious
period, an exponentially distributed infectious period and a gamma distributed
infectious period.
Let Q ( t ) be the probability of being alive at time to + t given that an individual is
alive at time to. We use Q ( t ) = e -st since this Q ( t ) is the only probability which
is independent of the age of the individual. The function e -st is also the only Q ( t )
for which the model is translation invariant, i.e., a semiflow. We assume that the
average lifetime 1//~ is finite since otherwise there is no vital dynamics and the
disease always dies out. Since the population size is constant, the birth rate must
be equal to the death rate/z. The death rate is the same for susceptibles, infectives
and removed individuals. The assumptions that the lifetimes are exponentially
distributed and that the birth and death rates are equal constants (independent of
the disease state) are only approximations to reality. However, they are the simplest
way to introduce vital dynamics; other more realistic assumptions lead to more
complicated models [11, 18].
Active immunization by means of vaccine or toxoid is incorporated into the model.
The fraction ~o of newborns are immunized so that the flow rate of immunized
newborns into the removed class is ~otz. This newborn immunization is meant to
include the usual immunization against diphtheria, whooping cough, tetanus,
measles, rubella, mumps, and poliomyelitis that is given to children before the age
of 18 months. Children have some maternally transferred passive immunity for 6
to 12 months. Immunization of susceptibles at a rate OS(t) is also assumed. This
immunization corresponds to a general immunization of susceptibles of all ages.
Let the initial susceptible and removed fractions be So > 0 and Ro >/ 0 and let
Io(t) e -st be the fraction of the population that was initially infectious and is still
alive and infectious at time t. The function Io(t) is a nonincreasing function,
Io(0) > 0 and Io(t) <~ Io(O)P(t) so that Io(t) approaches 0 as t approaches or.
Because of deaths the average effective infectious period f o P ( t ) e - ~ dt is slightly
less than the average infectious period z = f o P ( t ) dr.
The integral equation for I ( t ) is
I ( t ) = Io(t) e -ut +
flS(x)I(x)P(t
- x ) e -"<t-x> d x
(2.1)
where the second term is the sum of those who become infectious in the time
interval [0, t] and are still alive and infectious at time t. The removed fraction satisfies
R ( t ) = R o e -ut + [Io(0) - Io(t)] e -us
+
+
t~S(x)I(x)[1 - P ( t - x)] e -"<t-~) d x
~o(1 - -
e -~t) +
OS(x) e -~r
dx.
(2.2)
40
H . W . Hethcote and D. W. Tudor
The second term represents those initial infectives who have recovered and are still
alive at time t, and the third term represents those who became infectious in the
interval [0, t] and are still alive at time t, but are no longer infectious. The fourth
term represents the newborns in [0, t] who were immunized and are still alive at
time t, and the fifth term represents susceptibles immunized in [0, t] who are still
alive at time t. The assumption that the population is constant leads to
S ( t ) + I ( t ) + R ( t ) = I.
(2.3)
The equations (2.0, (2.2), (2.3) can be combined to give the following differential
equation for S ( t ) .
S'(t) = -~SI
+ / z ( 1 - 9) - I~S -
OS.
(2.4)
When the waiting time in the infective class is exponentially distributed, then
P ( t ) = e -t/" and I t ( t ) = It(O) e -t/'. The system (2.1), (2.2), (2.3) reduces to the
system of ordinary differential equations:
S ' ( t ) = - f l S 1 + (1 - 9)/z - OS - I~S
I'(t) = [3SIR'(t) =I]r
I/r - Id
(2.5)
+ 9tz + OS - tzR
S+I+R=I.
Global stability results for this model have been proved [8].
When the infectious period is constant, then P ( t ) is 1 on [0, r] and is 0 otherwise.
Then the system (2.1), (2.2), (2.3) reduces to a system of ordinary differential
equations on [0, r], and on [z, m), it reduces to the system of delay-differential
equations:
S ' ( t ) = f l S ( t ) I ( t ) + (1 - 9)/~ - OS(t) - i z S ( t )
I'(t) = i~S(t)I(t) - ~S(t - r)I(t - r)e -~ - td(t)
(2.6)
R ' ( t ) = f l S ( t - r ) l ( t - r ) e - ~ + 91~ + OS(t) - t~R(t)
S ( t ) + I ( t ) + R ( t ) = 1.
Local stability results for this model without immunization have been obtained [5].
By standard theorems [17] there is a unique solution of (2.1), (2.2), (2.3) and initial
conditions which exists on a maximal interval and depends continuously on the
parameters and initial data. The theorem below shows that the solutions remain
bounded between 0 and 1 so that by a standard theorem [17], the maximal interval
is [0, m). The theorem below also shows that the model is epidemiologicaUy
reasonable so that the model is both mathematically and epidemiologicaUy well
posed. The proof is omitted since it is similar to (and easier than) the proof of
Theorem 4.1.
Theorem 2.L The triangular subset B o f the plane S + I + R = 1 with S, I and R
nonnegative is positively invariant with respect to the s y s t e m (2.1), (2.2), (2.3).
Integral Equation Models for Endemic Infectious Diseases
41
3. Stability Analysis of the SIR Model
The contact number o = [3fo P(t) e -"t dt is the average number of contacts of an
infective during the infectious period. The (S,/, R) coordinates of the equilibrium
points of system (2.1), (2.2), (2.3) are (S*, 0, 1 - S*) and
(Se, le, Re) = (1 , ( a S * -
1)(/~ +
/3
0), 1 - s o - i , )
(3.1)
where S * = ( 1 - ~0)/z/(tz + 0). The threshold quantity aS* which determines
whether the disease dies out (aS* <~ 1) or remains endemic (aS* > 1) is the average
number of susceptibles infected by an infective during the infectious period when
the susceptible fraction is S*. If aS* > 1, then near the equilibrium point where
I = 0, each infective infects more than one susceptible during the infectious period
so that the disease does not die out.
For all values of aS*, the side of the triangle B where I = 0 is part of the stable
manifold for the equilibrium point (S*, 0, 1 - S*). If aS* > 1, then (S*, 0, 1 - S*)
is a saddle point and there is a line in B which is part of the unstable manifold for
the linearization of (2.1), (2.2), (2.3). For aS* ~< 1, the only equilibrium point in B
is (S*, 0, 1 - S*). As shown in the following theorem, B is an asymptotic stability
region for this equilibrium point if ~S* < 1 (we conjecture that this is also true for
aS* = 1). The proof is omitted since it is similar to (and easier than) the proof of
Theorem 5.1.
Theorem 3.1. I f aS* < 1, then all solutions of(2.1), (2.2), (2.3) approach ( S *, O, 1 - S *)
as t approaches ~ .
To analyse the local stability of the equilibrium point (3.1) when orS* > 1, we first
translate this equilibrium point to the origin by letting I = Ie + Vand R = Re + W.
Since le and Re satisfy
P 0
Ie
|
flS, I e P ( - y ) e "~ dy
d -oo
Re =
f2
/3sje[1 - P(-y)]
e"~
ay + ~ + 0/~,
the system (2.1), (2.2), (2.3) becomes
[f2(t)J
t [/3P(t - x) e -"(t-x)
+
f
~0 [/311 - P ( t -
x)] e -"r
0
e-"(t-x)
]
-0
[ s e v - IXV + w) - v(v + w)] dx
(3.2)
V+W
x
where
[I~
e -ut - f - t
f l S f l e P ( - y ) e "~ dy
r f,( = |Ro e-"' + [Io(0)- Io(')] e-Ut
YI/j~,t./
/A(t)]
nL
- f -'~ /3Sele[1 -- e ( - - y ) ]
e"~ d y -
0 e-"t/~
- ~ e -"~
42
H . W . Hetheote and D. W. Tudor
The nonlinear Volterra integral equation system (3.2) can be written in matrix form
as
x(t) = F(t) + f A(t
-
y)G(X(y))dy.
(3.3)
The characteristic equation of the linearization of (3.3) is
det(Identity-f/e-~tA(t)Jdt)
=0
(3.4)
where J is the Jacobian of G evaluated at the origin. The following lemma follows
from results of Miller [16].
Lemma 3.2. I f solutions of (3.3) exist on [0, oo) and are bounded, F(t) e C[0, oo),
F(t) -~ 0 as t --~ ~ , A(t) eLl[0, oo), G(X) e C1(R2), G(O) = O, J is nonsingular and
the characteristic roots of (3;4) have negative real parts, then the origin is locally
asymptotically stable for (3.3).
Most of the conditions in Lemma 3.2 are easily verified for (3.2). We now analyse
the characteristic roots. The characteristic equation of the linearization of (3.2) is
(A + ~ + 0)
f/
(1 - e-at)P(t)e -"t dt + eIe = 0.
(3.5)
It is impossible for a characteristic root to be real and nonnegative since then the
first term in (3.5) would be nonnegative and the second term would be positive.
Suppose ~ = x + iy is a root of (3.5) with x / > 0. Since roots occur in complex
conjugate pairs, we can assume that y is positive. Then the imaginary part of (3,5)
is
y
fo
(1 - e - ~ t c o s y t ) P ( t ) e - u t d t + (x + t~ + 0)
fo
s i n y t P ( t ) e - U t d t = O.
The first term above is positive and the second term is positive since the integral is
positive over the intervals [2br/y, 2(k + 1)~'/y] for k = 0, 1 , . . . . This contradiction
implies that all roots of (3.5) have negative real parts so that we have proved the
following theorem.
Theorem 3.3. I f aS* > 1, then the equilibrium point (3.1) is locally asymptotically
stable for the system (2.1), (2.2), (2.3).
Since (3.1) is the only equilibrium point in B except for the saddle point (S*, 0,1 - S*),
we conjecture that B minus the boundary where I = 0 is an asymptotic stability
region for the endemic equilibrium point (3.1) when aS* > 1.
4.
The SEIR Model
This model is similar to the SIR model except that we now have a class E of exposed
individuals. Let PE(t) be the probability of remaining exposed t units after becoming
exposed and P1(t) be the probability of remaining infectious t units after becoming
Integral Equation Models for Endemic Infectious Diseases
43
infectious. Let o) > 0 be the average period of exposure and let Eo(t) e - ~ be the
fraction of the population that was initially exposed and is still alive and exposed
at time t. The assumptions about [3, P~, Pz, Q, I~, ~, 9, o, Eo(t), Io(t), So and Ro are
the same as in Section 2 or analogous.
The integral equation for E ( t ) is
E ( t ) = Eo(t) e -"t +
f i S ( x ) I ( x ) P s ( t - x) e -"(t-x) dx
(4.1)
so that
fo'
[3S(x)I(x) e - " ( t - ~ dx[PE(t -- x)] --/zE,
E ' ( t ) = [3SI + E ; ( t ) e -"t -
where the integral is a Stieltjes integral with respect to PE(t - x) considered as a
function of x. Using the negatives of the second and third terms above as the inflow
to the I class, we find that the integral equation for l ( t ) is
I(0
-- I o ( t ) e -"~ + e -"~
f'
[-E;(x)lP~(t
-
x)dx
~'0
+
B S ( y ) I ( y ) e-"~t-~d~[PE(x - y)]Pz(t - x) dx.
(4.2)
A similar analysis leads to the integral equation for R(t):
R ( t ) = R o e -"t + [Io(0) - Io(t)] e -"t + e -"t
[ - E ; ( x ) ] [ 1 - Pl(t - x)] dx
,J0
flSIe-U(t-u)du[Ps(x - y)][1 - Pz(t - x)] dx + 9(1 - e -"t)
+
0
+
fo'
OS e -"(t-=) dx.
(4.3)
The constant population assumption leads to
S ( t ) + e ( t ) + I ( t ) + R(t) = 1.
(4.4)
The equations (4.1)-(4.4) can be combined to give the differential equation (2.4) for
s(t).
By standard theorems [17] there is a unique solution of (4.1)-(4.4) with initial data
which exists on a maximal interval and depends continuously on the parameters and
initial data. The theorem below shows that by a standard theorem [17], the maximal
interval is [0, ~ ) .
Theorem 4.1. The subset B o f the hyperplane S + E + I + R = 1 with S, E, I and
R nonnegative is positively invariant with respect to the system (4.1)-(4.4).
Proof. First we show that if PE(t) is a decreasing function, then the solution does
not hit the boundary S = 0, E = 0 or I = 0 in a finite time. Suppose T is the least
44
H.W. Hethcote and D. W. Tudor
positive t such that S ( T ) = O, E ( T ) --- 0 or I(T) = 0. Then R(t) >1 0 on [0, T] by
(4.3) so that S + E + I ~< 1 on [0, T]. Since S'(t) >>. -(13 + 0 + t~)S on [0, T] by
(2.4), then S(T) >1 So e -<B+~
> 0 so that S ( T ) cannot be zero. I f E(T) = O,
then
0 = E ( T ) = E o ( T ) e -"T +
ilS(x)I(x)P~(r- x)e -"(~-x~ dx.
Both terms on the right above are nonnegative so that S(x)I(x)PE(t - x) = 13 on
[0, T]. Since P E ( T - x) cannot be zero on all o f [0, T], S(x)I(x) = 0 for some
x < T; which is a contradiction. Suppose I ( T ) = 0, then
0 = I(r)
[-E;(x)lP~(r- x)ax
= l o ( r ) e -.~" + e - " ~
~'0
+
;of:
ilS(y)I(y) e-U~r-~)du[P~(x - y)]P~(T - x) dx.
Each term on the right above is nonnegative andPE is decreasing so that S ( y ) I ( y ) = 0
for some y < T; which is a contradiction. Since S(t) >i O, E(t) >1 0 and I(t) >>.0
for all t >t 0, it follows f r o m (4.3) that R(t) >>.0 for all t >/ 0. Since every nonincreasing PE(t) is the limit of a sequence o f decreasing PE(t) and solutions depend
continuously on the parameters, the t h e o r e m is also true for a nonincreasing PE(t).
5. Stability Analysis of the SEIR Model
The contact n u m b e r is ~ =/3(1 - ~ ) So Pz(t) e -"t dt where ~ = So PE(t) e -ut dt.
The ( S , E , I , R ) coordinates of the equilibrium points of (4.1)-(4.4) are
(S*, 0, 0, 1 - S*) and
(s~,
E~, I~, Re)
ille,a
-- ill)(
+ 0), 1 -- Se --
--
(5.1)
where S* = (1 - cp)/~/(/z + 0) as before. M a n y o f the results for this model
including the threshold quantity aS* are the same as for the SIR model. The
theorem below shows that B is an asymptotic stability region for (S*, 0, 0, 1 - S*)
when aS* < 1 (we conjecture that this is also true for a S * = 1).
Theorem 5.1. I f aS* < 1, then all solutions of(4.1)-(4.4) approach (S*, O, O, 1 - S*)
as t approaches ~ .
Proof. Since S'(t) = - i l l s + (l~ + 0)(S* - S), solutions starting in the subset o f
B where S > S* either move into the subset of B where S ~ S* or a p p r o a c h the
equilibrium point. We now show that solutions starting in the subset of B where
S ~< S* a p p r o a c h the equilibrium point.
Let J -- lim sup I(t) as t --~ oo and assume J > 0. T a k e e small enough so that
48 + aS*(J + e) < J. T a k e to > 0 so large that Io(to) e -~o < 8, E0(0) e -uto < 8,
fl~-e-Uto < 8 and fl(1 - / z ~ ) f -~~ P I ( - Y ) e U ~ dy < 8. T a k e tl > to so large that
Integral Equation Models for Endemic Infectious Diseases
I(t) < J + e f o r t
45
> tl. F o r t > to + tl,
f'
x ( t ) - - I 0 ( 0 e -"~ + e - " t
[-E;(x)lP,(t
-
x)dx
~0
+
~ s ( t + z)I(t + z ) e " % [ P . ( u - z ) l e , ( - u ) a .
t
t
< I0(t0)e -"~~ + e -"t~
+
i~
E~
t0
+
9
I2
[ - E ; ( x ) ] dx
[3S*(J + ,) e"~d,[PE(u -- z ) ] P , ( - u ) du
t0
fl e"~d,[PE(u -- z ) l P z ( - u) du
oo
+_
f:o
f-2~
z)]Pl(-u)du
< 2~ + s * ( J + 8)
fl e~%[PE(u - z ) l p , ( - u ) a u
oo
+
I_?
oo
fl e~(1 - tza)Pi(-u) du +
f;
fl e-~toPz(-u) du
to
< 48 + crS*(J + 8) < Jr,
but this contradicts the definition o f J so that J = 0 and l(t) --->0 as t --> oo.
Let 8 > 0 be arbitrary and take to > 0 so large that Eo(to)e -uto < 8/3 and
f l f - ~ e E ( - u ) e ~ du < 8/3. Take tx > to so that I ( 0 < 8/3/3~ for t > tl. F o r
t > to + tl,
E(t) = Eo(t) e -"t + fO flS(t + u)I(t + u ) P E ( - u ) e~" du
J.
< ~/3 +
fo
t
f l ( ~ / 3 ~ ) P R - . ) e "~ du + fl
f?
P . ( - u) e "~ ,tu < 8.
Thus E(t) --~ 0 as t --~ m.
I n the subset o f B where I = 0 and E = 0, (2.4) implies S---~ S* as t ~ oo so that
R ~ 1 - S*. Since solutions o f (4.1)-(4.4) are unique and c a n n o t intersect, the
p r o o f is complete.
The local stability analysis for the equilibrium point (5.1) whe~ eS* > I is similar
to that for the SIR model. We omit tt'~e detain leading to the characteristic equation
below for the linearization a r o u n d the equilibrium point (5.1).
(;~ + / z + O)
fff_"
oO
gO
(1 - e~Z)eU~d~[P~(u - z ) ] P , ( - u ) d u + ale = 0.
(5.2)
46
H. W. Hethcoteand D. W. Tudor
The analysis of this characteristic equation is similar to that for (3.5) so that all
roots of (5.2) have negative real parts. The equations (4.1)-(4.4) cannot be put in
the form (3.3) because of the double integrals so that Lemma 3.2 cannot be used;
however, the characteristic root analysis above seems to imply as before that the
equilibrium point (5.1) is locally asymptotically stable for the system (4.1)-(4.4).
This local stability result can be proved rigorously for special forms of PE(t) and
Pl(t). As before, we conjecture that B minus the boundary where E = 0 and I = 0
is an asymptotic stability region for the endemic equilibrium point (5.1) when
~S* > 1.
6. Conclusions
An infectious disease model is cyclic if there is some feedback from another class
into the susceptible class due to temporary immunity or temporary infectivity. In
Hethcote, Stech and van den Driessche [9], it is shown how some cyclic models
with distributed delays can have periodic solutions for some parameter values while
the corresponding models without delays do not have periodic solutions.
Here we have shown for noncyclic SIR and SEIR models with vital dynamics and
immunization that the asymptotic behavior of the models with distributed delays
is the same as that of the models without delays (the ordinary differential equation
models). That is, the threshold quantity oS*, the equilibrium points and their
stability depend on/3, ~r,/z, % 0 and a but do not depend on the form of the delay
probabilities PE(t) and P1(t). Consequently, if one is interested in the behavior near
the equilibrium points, or in the immunization necessary to cause the disease to die
out, or in how the equilibrium points change as the parameter values and immunization rates change, then the ordinary differential equation models are sufficient.
For noncyclic epidemic models without vital dynamics, other authors have shown
that the asymptotic behavior also depends only on average values and is not changed
by distributing the infectious period. For stochastic epidemic models, Ludwig [15]
showed that the distribution of final sizes of an epidemic depends on the total
infectiousness of an individual, but does not depend on the latent period or the
infectiousness of the individual during the temporal states of the infectious period.
For an age dependent SEIR model in which the infectiousness and removal rate
depend on the time since infectiousness began, Hoppensteadt [11, pp. 55-60]
showed that the final size depends on the threshold (the number of initial susceptibles expected to be exposed to each infective), but does not depend on the form
of the infectiousness and removal rate functions.
We conjecture that the introduction of distributed delays for other noncyclic
infectious disease models, with or without vital dynamics, would not change the
thresholds or asymptotic behavior of the models; that is, distributed delays would
not lead to periodic solutions. Consequently, we believe that the ordinary differential equation SIR models for a heterogeneous population [8] and for an age
structured population [18] include all of the essential epidemiological factors and
are sufficiently general to study the sensitivity and long range control of endemic
infections diseases with permanent immunity.
Integral Equation Models for Endemic Infectious Diseases
47
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Received April 24, 1979/Revised July 9, 1979