Asymptotic Theory Of An Infectious Disease Model

J. Math. Biol. (2006) 53:287–304
DOI 10.1007/s00285-006-0009-y Mathematical Biology
Asymptotic theory of an infectious disease model
Alan M. Whitman · Hashem Ashrafiuon
Received: 17 April 2004 / Published online: 30 May 2006
© Springer-Verlag 2006
Abstract In this paper, we present asymptotic theory as a viable alternative
solution method for infectious disease models. We consider a particular model
of a pathogen attacking a host whose immune system responds defensively,
that has been studied previously [Mohtashemi and Levins in J. Math. Biol. 43:
446–470 (2001)]. On rendering this model dimensionless, we can reduce the
number of parameters to two and note that one of them has a large value that
suggests an asymptotic analysis. On doing this analysis, we obtain a satisfying
qualitative description of the dynamic evolution of each population, together
with simple analytic expressions for their main features, from which we can
compute accurate quantitative values.
Keywords Asymptotic analysis · Infectious diseases · Pathogenic load
1 Introduction
Despite the development of antibiotics and vaccines, infectious diseases are still
one of the leading causes of human mortality, particularly in developing coun-
tries. The death toll in Africa from Malaria is still very significant and AIDS has
reached pandemic proportions. Other infectious diseases also persist around
the world such as TB, HIV, measles, lyme disease, different forms of hepatitis,
hantavirus pulmonary syndrome, legionnaire’s disease, etc. The short infection
period, relative to the life span of the host, generally causes acute pathology
in the hosts. When the pathogen establishes a site of infection, the immune
A. M. Whitman (B) · H. Ashrafiuon
Department of Mechanical Engineering,
Villanova University, 800 W. Lancaster Ave.,
Villanova, PA 19085, USA
e-mail: alan.whitman@villanova.edu

288 A. M. Whitman, H. Ashrafiuon
response may be inactive for some initial period. Meanwhile, the pathogen
starts multiplying, leading to an extensive increase in its antigenic mass. The
initial delay period of the stimulation of the immune response can be critical
in terms of host mortality. Thus for the more acute infections, estimation of the
time and extent of the critical levels of the pathogenic load may be of great
importance. For example, small levels of sporozoites can lead to clinically sig-
nificant malaria parasitization while large doses can cause death in a matter of
days [8]. In the case of HIV, studies with animals intravenously inoculated with
low level, less stabilized doses revealed a production of neutralizing antibod-
ies and high survival rates, while those inoculated with high level doses rapidly
developed clinical disease [4]. Similarly for measles, a large inoculum which may
occasionally result from airborne transmission has been suggested to increase
vaccine failure risk [11].
Mathematical models can be important tools in not only analyzing the spread
of infectious diseases in a population of individuals [5,12], but also in predict-
ing the timing and extent of infection and possible reinfection processes in
an individual [9]. While the former is generally used for planning, preven-
tion and control strategies, the latter can be effective in therapy/intervention
programs for treating the individuals who have been exposed to the partic-
ular pathogen. Such models can play an important role for treating acute
infections. Understanding the early dynamics of acute infections and antic-
ipating the time of occurrence and magnitude of the maximum pathogenic
load and the immune response can be critical in choosing effective intervention
schemes.
Deterministic mathematical models of infectious diseases are developed as
sets of coupled, first-order ordinary differential equations with given initial-
values [1,5,9,12]. These equations are nonlinear so they are usually solved
numerically; however, analytic approximations written in terms of the model
parameters are also useful due to the insights they provide. Thus, Mohtashemi
and Levins [9] used an averaging technique to derive analytical expressions
for the maximum pathogenic load and its time to peak, in terms of the ini-
tial inoculum, the initial period of immune activity, and other of their model
parameters. Analytical results can be useful even when an exact solution ex-
ists; an example is Kermack and McKendrick’s [7] approximate solution to
the classical epidemic model, which has an exact, though not explicit, solu-
tion [5]. Kermack and McKendrick’s work [7] produced one of the earliest
approximate solutions for a mathematical model of epidemics, and identified
the epidemic threshold, that minimum critical value of the density of sus-
ceptibles, for which an epidemic outbreak will occur. These approximations,
[7,9], depend on mathematical insights that are model specific. By contrast the
approximation we develop in the sequel is an instance of a general method
that has been applied to a host of problems in a wide variety of physical con-
texts.
Linearization of the equations is a first step toward obtaining approxima-
tions, but this method is severely restricted because it eliminates all interesting
nonlinear effects. It has been used to study repeated outbreaks of epidemics [3],

Asymptotic theory of an infectious disease model 289
but the results apply only to small fluctuations about equilibrium and so have
limited practical utility. Parameter perturbation is an alternative method for
generating approximate solutions to nonlinear differential equations and has
been widely used for that purpose in engineering [6,10,13]. The key to doing
this type of analysis is to identify a parameter for which, in some limit, the
model has a simple solution. If the solution makes sense, is useful, and can be
improved by recursion we say it is regular; otherwise it is singular. Two basic
singular perturbation methods have been developed, the method of multiple
scales, also known as the method of averaging, and the method of matched
asymptotic expansions. The former has been applied mainly to problems for
which regular perturbation schemes fail over long times (many cycles) due to
an accumulation of small errors incurred in each cycle of motion. The latter is
useful for problems in which each cycle of the motion has slow and fast parts in
which different physical mechanisms are important.
We suggest that singular perturbation theory provides an accurate solution
along with valuable qualitative insights for the transient pathogen-immune
response dynamics in infectious diseases. Here we present such an approximate
asymptotic solution, based on the method of matched asymptotic expansions
[6,10,13], to a model that was developed to describe the transient dynamics of
the interaction between an invading pathogen and its host’s immune system, [9].
In this previous work, the authors presented a model of pathogen and immune
system evolution that has six parameters; the rate of induction of the immune
system, the rate of decay of the immune system, the rate of removal of the path-
ogen by the immune system, the innate immunity, the reproductive rate of the
pathogen, and the initial pathogenic inoculum. They then tried to approximate
the first maximum of the pathogenic load, and the time it occurred, by means
of an averaging process using a hybrid quadratic, and an exponential model, of
the pathogen and immune response interaction.
In this paper, we first make the model equations, [9], dimensionless and find
that the resulting equations depend on three dimensionless parameters. We
then do an asymptotic analysis that is based on the smallness of two of these
parameters. As a result we obtain a solution that depends on only two parame-
ters. This solution yields very accurate values for the first peak of the pathogen
load and the time it occurs. It also gives the immune system peak value and its
time of occurrence, although not as accurately as that for the pathogen. How-
ever, these results are still more accurate than those obtained by the averaging
process in [9].
An interesting feature of this model, like many of this type, is that it predicts
multiple remission and reinfection periods. The asymptotic analysis presented
here is carried out as far as the second pathogen peak, for which we obtain an
estimate. It could be easily extended to include subsequent peaks as well. We
also indicate how the late time dynamics can be determined from the linearized
system. Finally, we present a uniformly valid solution that predicts the time
evolution of the pathogenic load and immune system response from the initial
infection through to the first reinfection time.

2 Model equations
We analyze the following dynamic model of an infectious disease attacking an
immune system [9];
dĪ
dt̄
= a0 − µĪ + k0H(t̄ − θ)P̄,
dP̄
dt̄
= (r − mĪ)P̄.
Here Ī is the immune system level, P̄ the pathogenic load, and t̄ the time. The
parameters a0, µ, k0, r and m are (positive) characteristic rates, while H denotes
the Heaviside function, which activates the immune system response only after
a delay, θ, from the initial infection time. The initial conditions are taken as
Ī(0) =
a0
µ
P̄(0) = p̄0
in which p̄0 represents the initial pathogenic load while the immune system is
initially at its uninfected level.
Now for times t̄ ≤ θ, the model equations are uncoupled; the solution of the
first is the constant Ī = Ī(0), and therefore the solution of the second is the
exponential P̄ = p̄0 exp{(r − mĪ(0))t̄}. Since the model is only interesting when
this initial pathogen behavior is growth, not decay, we have a condition
r −
ma0
µ
= µ 0 (1)
that applies to our subsequent discussion. With µ 0, this condition means
that 0.
2.1 Dimensionless representation
There are ten quantities in this model; they are the six parameters mentioned
previously (five rates and the delay time), plus p̄0, the two dependent variables,
and the independent variable. There are two dimensional quantities (number
and time) so eight dimensionless ratios can be formed [2]. These are, using 1/µ
for time and p̄0 for number
a0
µp̄0
k0
µ
r
µ
mp̄0
µ
µθ
Ī0
p̄0
P̄0
p̄0
µt̄. (2)
The last three of these are dimensionless variables and the other five are dimen-
sionless parameters (notice that according to the original definitions the dimen-
sions of a0 and m are directly and inversely proportional to number as well as

inversely proportional to time). Since any reduction in the number of param-
eters appearing in a model is a simplification, it is warranted in all but the
simplest case; in particular, it is useful here in reducing the number of indepen-
dent parameters from seven to five.
Defining dimensionless dependent and independent variables by
I =
1

mp̄0
µ

Ī
p̄0
−
a0
µp̄0

P =
1
2
k0
µ
mp̄0
µ
P̄
p̄0
t = µ(t̄ − θ) (3)
the model equations, for t ≥ 0, can be written as
dI
dt
= −I + P, (4)
dP
dt
= (1 − I)P. (5)
The initial conditions, applied at t = 0 (corresponding to t̄ = θ) where the
immune system is activated, are
I(0) = 0 P(0) =
p0α
2
= p. (6)
The dimensionless parameters, p0, and α, are defined by
p0 =
k0
µ
mp̄0
µ
α = eµθ
(7)
while is given by Eq. (1).
An unanticipated benefit of rendering the problem dimensionless in this way
is that of the five dimensionless parameters in the model, only the three, from
Eqs. (1) and (7), appear explicitly in the differential equations and initial condi-
tions; the other two occur only in the definition of the dimensionless variables,
Eq. (3).
2.2 Equilibria
The set of Eqs. (4) and (5) has two equilibrium points. They are the uninfected
equilibrium at I = 0 and P = 0, and an infected equilibrium at I = 1 and
P = −1. Straightforward linearization [3,9], shows that for 0 the unin-
fected equilibrium is stable and the infected equilibrium is unstable, while for
0 the former is unstable and the latter stable. At = 0 there is a bifurcation
analogous to that of the classic endemic SIR model of infectious disease [5]. As
mentioned before, this model is interesting only when the infected equilibrium
is stable, 0.

3 Large asymptotics
We have already noted that the case 0 corresponds to an initial growth
of the invading pathogen. Moreover, as becomes larger the growth becomes
more rapid. Thus there is a short time scale that characterizes this behavior in an
initial layer, while there is a longer time scale that characterizes the immune sys-
tem behavior after it has reacted to the invasion (the remission region in Fig. 1).
We will see further on that subsequently this process repeats with alternate peri-
ods of short time pathogen growth and long time immune system dominance.
Using singular perturbation theory, we can derive approximate solutions in each
of these regions, starting from the initial layer and moving forward by matching.
We consider the limit ≡ ǫ−1 → ∞, in an attempt to simplify the nonlinear
Eqs. (4), (5) and (6) sufficiently, so that we can obtain an approximation to the
solution that will be valid in this limit. Moreover, investigating this possibility
makes good physical sense given that the parameter value estimates proposed
in [9] produce = 6.99.
3.1 Initial layer (immune system response region)
We first note from Eq. (5), that I = O(1) in order that both terms on its right
hand side are the same order of magnitude. Then t = O(−1) in order that the
derivative term on the left is also of this order of magnitude. Thus the charac-
teristic time in this region is small compared with the characteristic relaxation
time for the immune system, µ−1. Continuing, we make P = O(1) so that the
derivative term in Eq. (4) is the same order of magnitude as the last term on
its right side, and is dominant. These arguments indicate an initial dynamics
2 4 6 8 10 12 14
0
0.5
1
1.5
2
2.5
x 10
4
time (days)
pathogenic
load
initial
layer
remission region
reinfection
layer
Fig. 1 The main regions of pathogenic load response

in which the populations P and I are finite, but vary over a short initial time
interval, t 1; they are consistent with the discussion at the beginning of
Sect. 3. The scaling to a new time variable that is O(1) in this region is then
T =
t
ǫ
. (8)
With this transformation Eqs. (4) and (5) become, with prime denoting differ-
entiation with respect to T
I′
= P − ǫI, (9)
P′
= (1 − I)P. (10)
The initial conditions of Eq. (6) are unchanged
I(0) = 0 P(0) = p
but now we require lim p0 → 0, in addition to lim ǫ → 0, in order to maintain
the initial value, p = O(1). This means that formally we are doing a double
limiting process
lim
ǫ→0
lim
p0→0
p0ǫ2
α1/ǫ
= O(1).
In this event, the problem only depends explicitly on two parameters, p, which
is finite, and ǫ which is small ( big). It does depend on both p0 and α, but only
in the combination indicated by Eq. (6). Recall that p0 and α are defined in Eq.
(7), with ǫ ≡ −1 defined in Eq. (1). Physically by means of this transforma-
tion, we are focusing our attention at the outset on an initial layer of times, near
t = 0, in which the pathogenic load rapidly increases to a maximum and then
decreases as a result of an equally rapid immune system response.
3.1.1 Asymptotic solution
We now expand the dependent variables in powers of ǫ
P = P0 + ǫP1 + O(ǫ2
) I = I0 + ǫI1 + O(ǫ2
) (11)
substitute into Eqs. (9), (10) and (6), and set to zero like powers of ǫ; then to
dominant order
I′
0 = P0 P′
0 = (1 − I0)P0, (12)
I0(0) = 0 P0(0) = p (13)

while to first order
I′
1 = P1 − I0 P′
1 = (1 − I0)P1 − P0I1, (14)
I1(0) = 0 P1(0) = 0. (15)
Dominant order term: We can obtain a first integral of the system given by
Eqs. (12) by substituting the first into the second, P′
0 = (1 − I0)I′
0, integrating,
and using the initial conditions, Eq. (13). We get
P0 =
s2 − (I0 − 1)2
2
, (16)
where we have written
s2
= 1 + 2p. (17)
Equation (16) indicates that this part of the phase plane trajectory, to dominant
order, is a parabola.
Now, on substituting Eq. (16) into the first of Eqs. (12), integrating and
solving for I0, we get
I0 = 1 + s tanh

s(T − T∗)
2

, (18)
where
T∗
=
2
s
tanh−1

1
s

=
1
s
ln

s + 1
s − 1

. (19)
Substituting Eq. (20) into Eq. (12) and differentiating then gives
P0 =
s2
2
sech2

s(T − T∗)
2

. (20)
Equations (18) and (20) are the dominant order solutions for the model. Clearly,
from Eq. (20), the scaled pathogen level starting from p reaches a maximum of
s2/2 = p + 1/2 at T = T∗ and thereafter subsides. The scaled immune level,
from Eq. (18), increases monotonically from 0 to 1 + s. It passes through 1 at
T∗ causing the pathogen level to decrease beyond that time, according to the
demands of the model [see either Eq. (5) or (10)].
First order term: We can derive a second order equation for I1 from Eqs.
(12) and (14) which, when integrated once, is
I′
1 + (I0 − 1)I1 = f0 ≡ I0 − s2
T + ln P0 − ln p. (21)

An integrating factor of Eq. (21) is P−1
0 = exp{

(I0 − 1) dv} so, using the initial
condition given in Eq. (15), its solution can be written as
I1(T) = P0(T)
T

0
f0(v)
P0(v)
dv. (22)
We can get P1 either from the second of Eqs. (14), noting that it has the same
integrating factor as Eq. (21), or from the first of Eqs. (14) and using Eq. (21)
for I′
1. These results are
P1 = −P0(T)
T

0
I1(v) dv = 2I0 − s2
T + ln

P0
p

− (I0 − 1)I1. (23)
Equations (22) and (23) are the first order corrections for the model.
3.1.2 Large T behavior
In order to continue the solution beyond this initial layer, we need to determine
its behavior at large T. We do this by substituting Eq. (8), T = t/ǫ and doing
the limit ǫ → 0 with t fixed. In this way we obtain from Eqs. (18) and (20)
I0 ∼ 1 + s P0 ∼ 2s2
esT∗
e−st/ǫ
, (24)
where the terms not written are exponentially small. Using these in Eq. (21)
gives the leading two terms as
f0 ∼ −s(1 + s)
t
ǫ
+ C, (25)
where C = 1+s+sT∗ +ln(2s2/p) and again the terms not written are exponen-
tially small. Noting that the value of the integral in Eq. (22) comes principally
from the vicinity of its upper limit, we use the second of Eqs. (24) and (25) in
it, make the substitution x = t/ǫ − v, and evaluate the integral to get
I1

t
ǫ

∼ −(1 + s)
t
ǫ
+
1 + s + C
s
. (26)
Finally, on reexpressing Eq. (26) as a function of T and using it together with
the first of Eq. (24) in Eq. (11), we get the two term representation of the two
term expansion as
Icom ∼ 1 + s + ǫ

1 + s + C
s
− (1 + s)T

. (27)

Here we have denoted its value in this limit, T → ∞, by the subscript com. We
obtain the expression for P1 in this limit by using Eq. (26) in the integral of Eq.
(23) and integrating. Combining the result with the second of Eqs. (24) in Eq.
(11) then gives
Pcom ∼ 2s2
esT∗
e−sT
1 − ǫ

1 + s + C
s
T −
1 + s
2
T2

. (28)
Equations (27) and (28) give expressions for P and I as they leave the ini-
tial layer in which the pathogen level peaks and then, due to the response of
the immune system, continually subsides. The immune system itself peaks after
the pathogen does and thereafter subsides as well. Clearly in this limit, P is
exponentially small while I = O(1).
A plot of Eqs. (11) using (18) and (22) for I and (20) and (23) for P is shown
in Fig. 2 where they are compared with those of a numerical integration of Eqs.
(6), (9), and (10), for ǫ = 0.1 and p = 1.5. The time scale used in the figure
is t rather than the initial layer time scale T. Clearly the approximate solution
leads to quantitatively accurate values for the populations throughout the initial
layer.
3.2 Main region (immune system relaxation region)
In this region, the time scale is t and P is exponentially small. Therefore, we
return to Eqs. (4) and (5), but make the substitution P = exp{−φ/ǫ}. Denoting
I = ı(t) and P = π(t) to distinguish these quantities here from their expressions
0 0.5 1 1.5 2
10
–1
10
0
10
1
Immune
level
Time
0 0.5 1 1.5 2
10
0
Pathogen
load
Time
numerical
asymptotic
numerical
asymptotic
Fig. 2 Comparison of dimensionless immune and pathogen levels from numerical integration with
the initial layer approximation for ǫ = 0.1( = 10) and p = 1.5

in the initial layer, the equations are
dı
dt
=
1
ǫ
e−φ/ǫ
− ı
dφ
dt
= ı − 1. (29)
The first term on the right in the first of Eqs. (29) is exponentially small as long
as φ 0 and O(1). In that event they are decoupled to all powers in ǫ, and can
therefore be integrated sequentially. Doing this produces
ı = A(ǫ)e−t
+
H(t − tı )
ǫ
t

tı
e−(t−v)−φ(v)/ǫ
dv, (30)
where
φ = B(ǫ) − A(ǫ)e−t
− t. (31)
The behavior indicated by Eqs. (30) and (31) means that initially the immune
system cannot see the pathogen and so it continually relaxes. However, when
its level falls below ı = 1, the pathogen again begins to grow exponentially
thereby producing a reinfection before the immune system can respond.
The constant tı is somewhat arbitrary. It can be any time when π is exponen-
tially small. We have taken it here as the time when ı = 1 so that from either
Eq. (30) or (31), tı = ln(A). The constants of integration A and B are power
series in ǫ that need to be determined. We do this by matching the expressions
we obtain from Eqs. (30) and (31) as they approach the initial layer, lim t → 0,
with those we obtained from the initial layer as they approach this main region,
Eqs. (27) and (28).
3.2.1 Small t behavior
We obtain the small t behavior by substituting Eq. (8), t = ǫT into a two term
expansion of Eq. (30), doing the limit ǫ → 0 with T fixed, and retaining two
terms. In this way we obtain
ı ∼ A0 + ǫ(A1 − A0T) (32)
and
φ ∼ (B0 − A0) + ǫ[(B1 − A1) − (1 − A0)T] + ǫ2

A1T −
A0
2
T2

. (33)
Now Eqs. (27) and (32) are just different representations of the same function in
the transition interval between the two regions, and so must match functionally

[6,13]. We see that this is indeed so for the following values of the constants
A0 = 1 + s A1 =
(1 + s + C)
s
. (34)
If we write the expression for P in this region, making use of Eq. (33), and
expanding for small ǫ, we obtain
π ∼ e−(B0−A0)/ǫ
e−(B1−A1)+(1−A0)T
1 − ǫ

A1T −
A0
2
T2

. (35)
With the following values of the constants Bk
B0 = A0 B1 = A1 − sT∗
− ln(2s2
). (36)
Equations (35) and (28) are identical.
3.3 Uniformly valid solution
Uniformly valid solutions for I and P can be obtained by adding the solutions
found in each of the two regions and subtracting their common part, [6,13].
Thus, for example,
Iu ∼ I(T) + ı(t) − Icom, (37)
where I(T) is given by Eq. (11) using Eqs. (18) and (22), ı(t) is given by Eqs.
(30) and (31) with Ak and Bk given by Eqs. (34) and (36), and Icom is given by
Eq. (27). The uniformly valid pathogen level is given in similar fashion by
Pu ∼ P(T) + π(t) − Pcom. (38)
These expressions are valid through order ǫ in the initial layer and main region,
but as t increases they ultimately become non-uniform due to the approach of
φ to 0, which results from the continual increase of the last term in Eq. (31).
3.4 Reinfection layer (immune system response region)
It is convenient to measure the recurrence of pathogen growth and the onset of
a second region of immune system response from the time that φ = O(ǫ). Thus,
we write
t = θr + ǫTr, (39)
where tr = ǫTr is the scaled time in this new layer with its origin at θr. Substi-
tuting Eq. (39) into Eq. (31), expanding e−ǫTr and retaining two terms, we find

for π
π = e−(B1−A1 exp{−θr})
esrTr , (40)
where we determine θr and sr by
θr − B0 + A0e−θr = 0 sr = 1 − A0e−θr . (41)
The first of Eqs. (41) is the requirement that φ = O(ǫ), while the second is
a definition of sr. From Eq. (40) it is clear that for sr 0 we again see an
exponential growth of the pathogen.
This layer is similar to the original one, the difference being that here there
are no initial conditions. Instead, we require that the layer solution, when pro-
jected backward in time, matches with the solution coming out of the main
region, an expression that we have just found in Eq. (40). In this new immune
response layer the pathogen population is given to dominant order by Eq. (20),
however with constants of integration sr and T∗
r in place of s and T∗
Pr ∼
s2
r
2
sech2

sr(Tr − T∗
r )
2

. (42)
Here we must determine T∗
r by matching Eq. (42) in the limit Tr → −∞, with
Eq. (40). Substituting Tr = tr/ǫ into Eq. (42) and expanding for lim ǫ → 0 with
tr finite and negative, we find, to dominant order, after reexpressing it back in
terms of Tr
Pr ∼ 2s2
r e−srT∗
r esrTr .
We note that this matches Eq. (40) functionally provided that
T∗
r =
[ln(2s2
r ) + B1 − A1e−θr ]
sr
. (43)
Since sr 1 the second pathogen maximum is less than the first.
4 Results and conclusions
In the previous section we obtained results for the evolution curves for the path-
ogen and immune system levels in each region of interest and a uniformly valid
composite over all the regions. A comparison of this uniformly valid approxi-
mate solution with the corresponding numerical solution is shown in Fig. 3. For
the parameter values that were used, the agreement is excellent, although the
second pathogen peak is slightly low and occurs a little early, and the second
immune peak has not been included. Note in particular that the pathogen levels
are in good agreement over four orders of magnitude. Moreover, the agreement

0 1 2 3 4 5 6 7 8
10
–1
10
0
10
1
Immune
level
Time
0 1 2 3 4 5 6 7 8
10
0
Pathogen
load
Time
numerical
asymptotic
numerical
asymptotic
Fig. 3 Comparison of dimensionless immune and pathogen levels from numerical integration with
the uniformly valid solution for ǫ = 0.1( = 10) and p = 1.5
in this region can be improved in two ways: first by making larger, although
its actual value must ultimately be determined by experiment, and second by
carrying out the calculations to higher order in . This latter alternative imposes
a tradeoff, because as the expressions become more accurate, they also become
more cumbersome, and thereby lose their ability to provide insight into what
are the important physical (biological) mechanisms.
The main features of these curves are their peak values and the times at
which they occur. We can easily obtain simple expressions for these quantities
from the general results of Sect. 3.
4.1 First pathogen population peak
Since any local extremum in the pathogen population, P′ = 0, occurs for I = 1
[see Eq. (10)], we get the time of the first peak value, TP, implicitly from the
second Eq. (11) as
1 = I0(TP) + ǫI1(TP) = I0(TP0 + ǫTP1) + ǫI1(TP0) + O(ǫ2
),
where we have made an expansion of TP in powers of ǫ. Taylor expanding I0
and solving, we find TP0 ∼ T∗ and TP1 = −2I1(T∗)/s2. Therefore
TP ∼ T∗
− ǫ
2I1(T∗)
s2
. (44)

The maximum value of the pathogen population is then, using the second equal-
ity in Eq. (23)
PP ∼ P0(TP0 + ǫTP1) + ǫP1(TP0) =
s2
2
+ ǫ

2 − s2
T∗
+ ln

s2
2p

. (45)
Equations (44) and (45) are rather simple approximations to the actual values
(although the second term of Eq. (44) is not simple to evaluate it is a small
correction for small epsilon) yet they are quite accurate even for values of ǫ
that are not very small. Moreover, the first order correction terms are small
enough to be negligible as long as p is not too small, a condition for which the
approximation fails in any event.
On using just the dominant order approximations, and writing them in terms
of dimensional variables, we find, with µ = r − ma0/µ
P̄P ∼ p̄0 exp{θµ} +
(µ)2
2k0m
(46)
and
t̄P − θ ∼
1
2k0mP̄P
ln

2k0mP̄P + µ
2k0mP̄P − µ

. (47)
Note that the first term in Eq. (46) is just the value of the pathogen load at the
onset of immune system activity.
4.2 First immune population peak
After the pathogen has peaked and begun to subside the immune population
itself reaches a maximum. The time that this occurs, TI, is obtained by differen-
tiating the second of Eqs. (11) and setting I′ = 0. This gives
P0(TI) + ǫI′
1(TI) + O(ǫ2
) = 0.
Here TI must be large, so we use Eq. (24) for P0 and the derivative of Eq. (26)
for I′
1. Taking −sTI = ln(aǫ), we find a = (s + 1) exp(−sT∗)/2s2, so to dominant
order
TI ∼ T∗
+
1
s
ln

2s2
(1 + s)ǫ

=
1
s
ln
1
ǫ
+
1
s
ln

2s2
s − 1

. (48)
Using this value back in Eq. (11) along with Eqs. (18) and (26) gives the maxi-
mum immune population
II = 1 + s − (1 + s)
ǫ
s
ln
1
ǫ
+
ǫ
s
C + (1 + s)

1 − ln

2s2
s − 1

. (49)

Equations (48) and (49) are not as accurate as their counterparts for the path-
ogen, because the first neglected term is of a relatively larger order than that
in the former case. Moreover, here both the maximum and the time it occurs
depend on ǫ as well as p, although the variation with ǫ is smaller than that of p.
The accuracy of these approximations can be judged by the results listed
in Table 1. There we have compiled two cases; the first, corresponding to the
values plotted in Figs. 1 and 2, ǫ = 0.1 and p = 1.5, where the approximation is
within its region of validity, and the second, for ǫ = 0.143 and p = 0.312, which
we obtained from using the parameter estimates given in [9]. In this second
case, the value of p is somewhat small [it is approaching O(ǫ)], although the
peak values are still quite accurate.
4.3 Second pathogen population peak
The second time the pathogen peaks, its maximum value is given to dominant
order by Eq. (42) as
PP 2 ∼ s2
r /2 (50)
with sr given by the second Eq. (41), and occurs at a time given by Eq. (39)
tP 2 ∼ θr + ǫT∗
r , (51)
where θr is given by the solution of the first of Eqs. (41), and T∗
r given by Eq.
(43). Accordingly, the maximum does not depend on ǫ. The time however, does
depend on both parameters, although more strongly on p than on ǫ. Results are
listed in Table 2.
Table 1 Numerical and analytical determinations of the first peaks of the pathogen and immune
system levels and times. For the ǫ = 0.1 data, p = 1.5 and in the second case, p = 0.312
ǫ tP PP tI II
Num Asym Num Asym Num Asym Num Asym
0.100 0.06 0.06 2.006 2.007 0.24 0.23 2.60 2.57
0.143 0.25 0.25 0.854 0.854 0.57 0.54 1.79 1.72
Table 2 Numerical and analytical determinations of the second peak of the pathogen level and its
time of occurrence.
ǫ p tP2 PP2
Num Asym Num Asym
0.100 1.500 3.03 2.97 0.366 0.337
0.143 0.312 2.86 2.52 0.273 0.229

4.4 Uniformly valid expansion
Because they match functionally, we can combine our results in each region to
obtain uniformly valid evolution curves. Equations (37) and (38) illustrate how
this is done. In the same way, we can append the result of the reinfection layer,
Eq. (42) to Eq. (38) and subtracting the new common part for Pr. We have
shown a plot of this type in Fig. 2. Results for the peak values and their times,
that are obtained from the uniformly valid approximations are generally less
accurate than those given by the regional solutions.
4.5 Subsequent behavior
The solution described above can be continued by adding a second main region
at the end of the reinfection layer and matching exactly as we did previously,
a second reinfection layer after that, and so on. In this way we obtain the time
evolution of each of these populations as a decaying oscillation about its respec-
tive equilibrium value. Eventually, when I − 1 = O(ǫ) and P − ǫ = O(ǫ3/2), the
nonlinear terms are small and the populations behave like a weakly damped
harmonic oscillator with dominant order natural frequency of ωn = ǫ−1/2. This
means that successive population maxima in this case occur periodically, with
the period given by, to dominant order, tp ∼ 2πǫ1/2.
These results can be obtained from a straightforward, small ǫ, multiple scale
asymptotic analysis, [10], of Eqs. (4) and (5) with the population scalings indi-
cated here, and a time scaling of τ = ωnt.
4.6 Conclusions
We have shown how the method of matched asymptotic expansions provides
a formalism for obtaining an approximate solution to a previously developed
model for the transient dynamics of the interaction between an invading path-
ogen and its host’s immune system in acute infectious diseases. We have shown
that this solution is particularly accurate in predicting the time of occurrence
and amplitude of the maximum pathogenic load and the subsequent remission
periods and reinfection peaks. Not only is this solution quantitatively accurate,
but it is also qualitatively simple and easy to understand.
Due to the similarity of this model to those that characterize infectious dis-
ease, we believe that this formalism could be successfully applied to them as
well. For example, of the various deterministic models presented in [5], the clas-
sic SIR endemic model would seem to be a prime candidate, using the contact
number as the large parameter. This is suggested because the infective-suscep-
tible fraction phase plane resembles a nonlinear oscillator and because of the
similarity of the bifurcation geometry in the two models, as we noted in Sect. 2.2.

References
1. Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases, 2nd edn, pp. 82–88. Hafner,
NY (1975)
2. Barenblatt, G.I.: Scaling, pp. 22–26. Cambridge University Press, Cambridge (2003)
3. Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model
Building, Analysis, and Interpretation, pp. 43–51. Wiley, NY (2000)
4. Endo, Y., Igarashi, T., Nishimura, Y., Buckler, C., Buckler-White, A., Plishka, R., Dimitrov,
D.S., Martin, M.A.: Short- and long-term clinical outcomes in rhesus monkeys inoculated with
a highly pathogenic chimeric simian/human immunodeficiency virus. J. Virol. 74, 6935–6945
(2000)
5. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev 42, 599–653 (2000)
6. Hinch, E.J.: Perturbation Methods, pp. 52–101. Cambridge University Press, Cambridge (1991)
7. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics
Proc. Roy. Soc. London, Ser. A 115, 700–721 (1927)
8. Marsh, K.: Malaria - a neglected disease? Parasitology 104 (Suppl) pp. S53–S69 (1992)
9. Mohtashemi, M., Levins, R.: Transient dynamics and early diagnosis in infectious disease. J.
Math. Biol. 43, 446–470 (2001)
10. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations, pp. 120–121. Wiley-Interscience, New York
(1979)
11. Paunio, M., Peltola, H., Valle, M., Davidkin, I., Virtanen, M., Heinonen, O.P.: Explosive school-
based measles outbreak: intense exposure may have resulted in high risk, even among revacc-
inees. Am. J. Epidemiol. 148, 1103–1110 (1998)
12. Singer, B.: Mathematical Models of infectious diseases: seeking new tools for planning and
evaluating control programs. Popul. Deve. Rev. 10, 347–365 (1984)
13. Van Dyke, M.: Perturbation Methods in Fluid Mechanics. p. 20. Academic, New York (1966)

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