Transient dynamics and early diagnostics in infectious disease

Digital Object Identifier (DOI):10.1007/s002850100103

J. Math. Biol. 43,446–470 (2001) Mathematical Biology

Mojdeh Mohtashemi · Richard Levins

Transient dynamics and early diagnosticsin infectious disease

Received: 25 January 2000 / Revised version: 30 November 2000 /Published online: 12 October 2001 – c© Springer-Verlag 2001

Abstract. To date, mathematical models of the dynamics of infectious disease have con-sistently focused on understanding the long-term behavior of the interacting components,where the steady state solutions are paramount. However for most acute infections, the long-term behavior of the pathogen population is of little importance to the host and populationhealth. We introduce the notion of transient pathology, where the short-term dynamics ofinteraction between the immune system and pathogens is the principal focus. We identifythe amplifying effect of the absence of a fully operative immune system on the pathogenesisof the initial inoculum, and its implication for the acute severity of the infection. We thenformalize the underlying dynamics, and derive two measures of transient pathogenicity: thepeak of infection (maximum pathogenic load) and the time to peak of infection, both crucialto understanding the early dynamics of infection and its consequences for early interven-tion.

1. Introduction

Microorganisms or pathogens are often characterized by their small size and theirshort generation time during which they replicate at very high rates. The short in-fection period, relative to the life span of the host, is another important feature ofmicroorganisms that cause acute pathology in their hosts. Once recovered frominfection, the host acquires life-long or transient immunity against re-infection.Such a characterization of pathogens broadly includes viruses, bacteria, fungi, andprotozoa [2,10,16,18].

When an invading pathogen finds its way through the epithelial surfaces andestablishes a site of infection, components of cellular immunity are able to respondto infection rather quickly. However, it takes several days for the humoral immunityto be induced. But the immune system may be inactive for some initial period formany different reasons. For instance, a pulse of 500 grams of sugar can inhibit

M. Mohtashemi: Massachusetts Institute of Technology, Department of ComputerScience, 545 Technology Square, Room 421, Cambridge, MA 02139, USA.e-mail: [email protected]

R. Levins: Department of Population and International Health, Harvard School of PublicHealth, 677 Huntington Avenue, Boston, MA 02115, USA.e-mail: [email protected]

Key words or phrases: Transient pathogenicity – Acute infection – Amplified initial inoculum– Maximum pathogenic load – Time to peak of infection

Transient dynamics and diagnostics in infectious disease 447

immune activity for up to 5 hours; an immunization can tie it up for a week or two;a major emotional trauma can reduce immune activity for months; malnutrition candelay the immune response indefinitely.

When an infection is naturally acquired, the infecting dose generally consists ofa small number of microorganisms. This, on its own, is quite insufficient to causesignificant pathology or even stimulate an immune response. But the pathogen thenmultiplies, and if the immune system has to be induced, and this may indeed takeseveral days, then such an “initial delay period” on the part of the host can becritical. For most acute infections, the pathogen replicates rapidly and at very highrates and thus can manage to stay ahead of the host’s defenses and take advantageof such a delay period.

If the immune system is readily activated, it may surpass the reproduction ofthe pathogen from the start, so that the pathogenic load declines. But if the immuneresponse has to be induced, during this window of opportunity for the pathogen,there can be extensive increase in antigenic mass which may lead to the death ofthe host. It is during this initial delay period that the pathogen replicates freely untilthe host’s defenses are activated. Hence, the initial inoculum that only consisted ofa few microorganisms at the time of inoculation, in the absence of a fully effectiveimmunity grows almost without bound. The newly activated immunity is now toface an exponentially “amplified initial inoculum”. Therefore we ask: When doesthe size of the initial inoculum and the initial period of immune inactivity affect theoutcome of infection?

It is reported that for Shigella, oral ingestion of 10 bacteria is sufficient to pro-duce disease in 1–3 days [3,18]. For TB, the inhalation of 1–10 mycobacteriumtuberculosis can start an infection in 4–12 weeks [3,18]. For Salmonella, a largenumber of bacteria, 106–109, must be ingested to produce disease in healthy hosts;however, studies performed on volunteers have demonstrated that the larger the in-oculum size the greater the attack rate [3,10,16]. For malaria, Marsh reports that forPlasmodium falciparum “in a small child a sporoziote dose in the hundreds couldresult in a clinically significant patent parasitaemia on the first round of blood-stagecycle” [17]; it is claimed by the same author that death occurs within a few days ofinoculation with >105 sporozoites. In experimental influenza in volunteers, inocu-lation with 0.6-3 tissue culture infectious dose (TCID50) via the aerosol route wasrequired to cause disease in 18–72 hours [10,16]. For measles, it has been reportedthat the second case in a household is always more severe than the first, presumablybecause more intense contact results in a greater inoculum [1,24,35]. For HIV, giv-en the similarities between the in vivo pathogenesis, as well as the genetic basis, ofHIV-1 and simian immunodeficiency virus (SIV), experimental animal studies withSIV has now been established as a model for studying the AIDS pathogenesis [15].Studies with rhesus macaques, intravenously inoculated with a highly pathogenicsimian/human immunodeficiency virus, have revealed that a larger inoculum re-sults in a more severe infection [7]. Infants are the best group of human subjectsfor studying the role of initial inoculum in the pathogenesis of HIV, because ifthey are not HIV-positive at birth but test positive shortly thereafter, then we knowwith high certainty that they must have been initially infected at birth. In a studywith 254 HIV-1 positive children, Mofenson et al. demonstrated that if the baseline

448 M. Mohtashemi, R. Levins

RNA level at 1 year of age were >105 copies/mL, it was prognostic of a worseoutcome [19].

Hence, it is of great importance to have a mathematical framework for mod-eling the early dynamics of the immuno-patho system and for making inferencesabout the transient outcomes of infection namely, the “time of crisis” and “extentof crisis”, based on the size of the initial inoculum, the period during which theimmune system is not fully effective, and the infection-specific parameters.

Mathematical models of the immuno-patho systems are increasingly common.Much work has been devoted to modeling the viral dynamics of HIV infection,mainly due to greater availability of quantitative information for HIV [5,8,12,21,22,25,26,31,34]. The major themes of such studies are to investigate the chronicevolution of the disease, the viral diversity at equilibrium, or to estimate the rel-evant and vital infection-specific parameters. However, the short term dynamicsof interaction between the immune system and pathogens for acute infections, orchronic infections exhibiting interesting transient behavior shortly after the startof infection, have seldom been studied. Despite the numerous literature on math-ematical modeling of the HIV dynamics, and although we have known for sometime that the virus concentration transiently increases to a peak before it declinesto a quasi steady state during the early stages of infection [4], only a handful ofinvestigators have attempted to address transients in the early stages of the HIVinfection [20,23,28,33].

Phillips [28] develops a mathematical model of interaction between the free vi-rions and the CD4+ T cells, divided into uninfected, latently infected and activelyinfected cells, and uses the model to argue that the decline in the virus concentrationimmediately after the peak is the result of target cell limitation and not due to aspecific immune response. Nowak et al. [23] use the standard mathematical modelof viral dynamics consisting of three components: uninfected cells, infected cells,and free virus particles, applied to SIV RNA in longitudinal specimens that werecollected from SIV-infected macaques to estimate R0, the basic reproductive ratio,during primary SIV infection. Murray et al. [20] constructs a mathematical modelof the interaction of HIV-1 with the CD4+ T cells at primary infection to demon-strate that the initial viral containment is the result of an effective immune response.More recently, Stafford et al. [33] used the standard viral dynamics model, appliedto data from ten anti-retroviral, durg-naive, infected patients, to demonstrate thatthe target-cell-limited model of primary infection is in accordance with the ob-served kinetics of the time of initial infection until shortly after the initial peak ofinfection.

In this paper we introduce a simple, yet sufficiently descriptive model of inter-action between the immune system and a pathogen to address the importance andimplications of transients in the dynamics of infectious disease. Our goal is to studythe qualitative behavior of the pathogen population growth during the early stagesof the development of infection and to formulate measures that are descriptive oftransient pathology, in terms of the initial inoculum, the initial period of immuneinactivity, and the system parameters. This model may readily be applied to infec-tions that exhibit similar dynamics, or will have to be further extended or modifiedto be representative of the dynamics of the interacting components.


Finally, we note that understanding the early dynamics of infection has directconsequences for clinical medicine and public health intervention [3,11,13,14,16,29,32]. Anticipating the maximum pathogenic load and the time of its occurrencehas immediate implication for the choice of effective intervention scheme(s) andassessing the duration of the critical period for intervention, beyond which any in-tervention strategy may prove ineffective. However, such quantitative assessmentsrequire more precise modeling of the dynamics of the infection at hand.

2. A framework for analysis of transients

2.1. Interaction model

Our model of the dynamic interaction between the immune system and a pathogenis a time-dependent, two-variable, nonlinear system of ordinary differential equa-tions. The variables are I (t), the immune level of an infected host at time t ; andP(t), the pathogenic load at time t . The following system of ordinary differentialequations models the dynamics of the immuno-patho system

dI

dt= a0 − µI (t) + kP (t) (1)

dP

dt= rP (t) − mI (t)P (t) (2)

where a0 represents the innate immunity; µ is the rate of decay of the immune sys-tem; k is the rate of induction of the immune system; r is the reproductive rate ofthe pathogen; and m is the rate of removal of the pathogen by the immune system.Furthermore, for the dynamics to be biologically meaningful, we assume that allsystem parameters are positive. It is interesting to note the dual effects of some ofthe parameters: a0 and µ are both properties of the immune system, yet µ favorsthe pathogen; m and k, on the other hand, are properties of interaction between theimmune system and the pathogen, yet they both favor the immune system.

Now suppose that the immune system is not fully responsive for some initialperiod, θ . During this initial delay period, the pathogen will replicate more freelysince the only control element is due to the innate immunity, a0, partially offset byµ: a0

µ. Hence, in the absence of fully responsive immune system the dynamics of

the patho-system is as follows

P(0) = p0 (3)dP

dt= P(t)(r − m

a0

µ) for 0 ≤ t ≤ θ (4)

where p0 is the arbitrary initial pathogenic inoculum. Equations 3–4 have the sim-

ple solution: P(t) = p0e(r−m

a0µ

)t . This initial dynamics at t = θ , when the immunesystem becomes fully responsive, sets the initial conditions for the system of equa-

tions 1–2 so that P(0) = p0e(r−m

a0µ

)θ becomes the initial pathogenic load for


the new dynamics. We will refer to this quantity as the amplified initial inoculum.Therefore, we have

I (0) = a0

µ, P (0) = p0e

(r−ma0µ

)θ (5)

dI

dt= a0 − µI (t) + kP (t) (6)

dP

dt= rP (t) − mI (t)P (t) (7)

2.1.1. Regions of pathogenicityOur dynamic model of interaction between the immune elements and a patho-gen consists of three qualitatively distinct regions of pathogenicity dependingon the steady state values and the relative magnitude of the parameters of thesystem of equations 5–7. There are two equilibrium points, namely (I ∗

1 , P ∗1 ) =

(a0

µ, 0) and (I ∗

2 , P ∗2 ) = (

r

m,rµ − ma0

mk), corresponding to equilibrium solutions

of the system of equations 5–7. The qualitative behavior of the solutions of thesystem of equations 5-7 and the conditions for local stability in each region can bedetermined by linearizing the system in the neighborhood of each critical point:

1. Linearizing the system in the neighborhood of (a0

µ, 0), the Jacobian matrix is

J1 =−µ k

0µr − ma0

µ

(8)

whose eigenvalues are: λ1 = −µ and λ2 = µr − ma0

µ. For the system to be

locally stable, both eigenvalues must be negative real numbers. Hence, we musthave the following conditions on the system parameters:

1-(a) µ > 01-(b) ma0 > µr

2. Linearizing the system in the neighborhood of (r

m,rµ − ma0

mk), the Jacobian

matrix is

J2 =( −µ k

ma0 − µr

k0

)(9)

whose eigenvalues are:λ3 = −µ+√

µ2−4(rµ−ma0)

2 andλ4 = −µ−√

µ2−4(rµ−ma0)

2 .

For the system to be locally stable around (r

m,rµ − ma0

mk), the trace of J2 must

be negative and the determinant of J2 must be positive [6,9], which implies:

2-(a) µ > 02-(b) µr > ma0


If the parameters of the system are such that condition 1-(b) is satisfied, then clearlycondition 2-(b) is violated, in which case the eigenvalues λ3, λ4 will be real andof opposite signs since

√µ2 + 4(ma0 + rµ) > µ. Consequently, the critical point

(r

m,rµ − ma0

mk) will be an unstable saddle point. On the other hand, if condition

2-(b) is satisfied, then λ1, λ2 are real and of opposite signs, and the critical point

(a0

µ, 0) will be an unstable saddle point. We distinguish two different regions un-

der this condition. If µ2 < 4(rµ − ma0), then λ3, λ4 are complex conjugates andthe dynamics will be a damped oscillation to the steady state, P ∗

2 . Otherwise, ifµ2 > 4(rµ−ma0), then λ3, λ4 are negative real, in which case, due to the relative-ly small reproduction rate, r , the pathogen is not capable of producing extensivedamage and is quickly regulated by the immune elements. In this case, a small peakmay be observed before the steady decline to a steady state. Putting it all together,we distinguish three regions of pathogenicity as follows:

Region I: If (a0

µ, 0) is a stable steady state, then by condition 1-(b), it must be

that the offensive parameters of the immune system, i.e., a0 and m are much largercompared to r and µ. Hence, the immune system can exceed the reproduction ofthe pathogen from the start and the pathogenic load declines. Figure 1(a) illustratesthe dynamics of this region.

Region II: If (r

m,rµ − ma0

mk) is a stable steady state and µ2 < 4(rµ − ma0),

then the pathogenic load will take off until the immune system is able to react andreverse the pathogen population growth and regulate it to some equilibrium valuethrough damped oscillations. Figure 1(b) illustrates the dynamics of region II.

Region III: If (r

m,rµ − ma0

mk) is a stable steady state and µ2 > 4(rµ − ma0),

then it must be that the pathogen reproduction rate, r is relatively small but thatcondition 2-(b) is still satisfied. In this case, a small peak may be observed, butinfection is promptly regulated by the immune elements. Figure 1(c) illustrates thedynamics of this region.

It is worth noting that in region II, where the steady state is associated withcomplex eigenvalues, we can use Hopf bifurcation theorem to eliminate the pos-sibility of limit cycles. For limit cycles to exist, by Hopf bifurcation theorem, thereal part of the eigenvalues must change sign as some of the system parametersare tuned [6,30]. However, since µ > 0 and it does not explicitly depend on anyother parameter, there is no bifurcation value and therefore the real part, −µ

2 < 0.Having eliminated the possibility of limit cycles, once the system is in region II,all transients lead to the same behavior asymptotically.

Measures of transient pathogenicity. Clearly, we are interested in the analytic prop-erties of regions II and III since these are the regions in which the dynamics ofinfection may exhibit interesting transient properties. In the remainder of this pa-per, we will focus on the analytical properties of region II, since region III can be



Fig. 1. Regions of pathogenesis. (a) Region I: Pathogen population growth cannot pick upfrom the start; immune elements are strong. (b) Region II: Pathogen produces sufficientdamage before the immune elements are able to regulate its population growth. (c) RegionIII: Pathogen may exhibit small growth, but is not capable of producing extensive damagedue to its relatively small reproduction rate; immune elements regulate its growth to anequilibrium value quite rapidly.

characterized as a trivial case of region II. We define two measures of transientpathogenicity corresponding to the dynamics of region II as follows:

Definition: Region II, wherein the pathogenic overload can cause severe diseaseor death in a short time period, best characterizes the dynamics of acute infections.We define two measures of pathogenesis pertaining to the transient dynamics ofthis region:

1. Maximum pathogenic load, or hpeak (for height of the peak), defined as thepathogen population at the peak of infection. If this is large enough, the hostmay not recover.

2. Time to peak of infection, or tpeak (for time-to-peak), defined as the time from thefull activation of the immune system to the peak of infection. If the interventionis applied past this point, the host may not recover.


In later sections, we develop the analytic tools needed to explore the dynamics ofthe measures of transient pathogenicity.

2.1.2. A quadratic model of interactionTo date, we do not know of any exact analytic solutions for the nonlinear system ofequations 5–7; however, we do know how to approximate these solutions. We areinterested in functional solutions of I (t) and P(t) in terms of the initial conditionsand various parameters of the system. To obtain such solutions, we first integrateequation 6 to get

I (t) − I (0) = a0t − µ

∫ t

0I (τ )dτ + k

∫ t

0P(τ)dτ

or equivalently

I (t) = a0

µ+ a0t − µ

∫ t

0I (τ )dτ + k

∫ t

0P(τ)dτ since I (0) = a0

µ

Def= a0

µ+ a0t − µtE[I ] + ktE[P ] (10)

where E[x]Def= 1

t

∫ t

0x(τ)dτ is the expected or average value of a variable x(t)(see

[27]). Adapting the notation x(t) for E[x], we rewrite equation 10 as follows

I (t) = a0

µ+ a0t − µtI(t) + ktP (t) (11)

where I (t) and P(t) are the average values of the immune level and the pathogenpopulation respectively, for the period [0, t]. Clearly, I (t) and P(t) are time variantentities; furthermore, all our derivations have been exact thus far. To approximatethe “transient” solutions of I (t) and P(t), we need to define a period during whichthese solutions would closely model the behavior of the immuno-patho system atthe peak. This period is of particular importance for the upcoming transient analysis,and will be discussed in detail in section 2.1.3. Therefore, if we further approximateI (t) and P(t) with constant terms I and P , derived for the period of interest, thenI (t) can be approximated as follows

I (t) = a0

µ+ a0t − µIt + kP t (12)

where I (t) represents an approximate solution of I (t) for the period of interest.Similarly, we can find an approximate functional form for P(t) by integratingequation 7 for the period of interest∫ t

0

dP

dτdτ = r

∫ t

0P(τ)dτ − m

∫ t

0I (τ )P (τ)dτ

Def= rtP (t) − m

∫ t

0I (τ )P (τ)dτ (13)


In equation 13, if we replace I (t) with the value of I (t) from equation 12, and P(t)

with the constant term P , we will then have

Pq(t) − P(0) = rP t − m

∫ t

0(a0

µ+ a0τ − µIτ + kP τ)P (τ)dτ

= rP t − ma0

µ

∫ t

0P(τ)dτ − m(a0 − µI + kP )

∫ t

0τP (τ)dτ

= rP t − ma0

µP t − m(a0 − µI + kP )P

t2

2

or, equivalently

Pq(t) = P(0) + P t(r − ma0

µ) − mP

t2

2(a0 − µI + kP ) (14)

where the quadratic function, Pq(t) (“q” for quadratic), represents an approxima-tion of P(t). As will be demonstrated in later sections, having approximated P(t)

and I (t) in terms of the initial inoculum, various parameters of the system, as wellas short term average measures of immunity and pathogenesis, namely I and P , weare now equipped with a powerful analytic tool for studying the transient behaviorof infection, as well as predicting the severity and time of the peak of infection.

2.1.3. Measures of transient pathogenicity: tpeak and hpeak

Equipped with the qualitative solutions of section 2.1.2, we derive two measures oftransient pathology, tpeak and hpeak , pertaining to region II. Differentiating Pq(t)

from equation 14 with respect to t we get

˙Pq = P(r − m

a0

µ) − mP t(a0 − µI + kP ) (15)

since I and P are constant. Equation 15 can be equated to zero and solved for thevalue of t for which the pathogenic load is maximum. That is

tpeak = rµ − ma0

µm(a0 − µI + kP )(16)

For tpeak > 0, both the numerator and denominator must have the same sign. Insection 2.1.1, we showed that in order for the interaction model of the system ofequations 5–7 to be locally stable in region II, we must have the condition

rµ − ma0 > 0 (17)

Hence for tpeak > 0, since its numerator is always positive, we must also have

a0 − µI + kP > 0 (18)


since µ > 0 and m > 0. Therefore both the numerator and denominator of tpeak

are positive. To derive hpeak , we evaluate Pq(t) at tpeak as follows

hpeak = Pq(tpeak )

= P(0) + P(rµ − ma0)

2

2mµ2(a0 − µI + kP )

where P(0) is the amplified initial inoculum as in equation 5. What are the conse-quences of such transient outcomes of an acute infection for the host? Can thesemeasures be utilized to device effective early intervention strategies? How does theduration of inactivity of the immune system, θ , and the size of the amplified initialinoculum, P(0), affect these measures and the critical timing for intervention? θ

not only influences the early dynamics of infection, it may also be influenced byfactors such as the state of nutrition or the stress level of the host. This is unlikesome of the other parameters of the system, such as r or µ, which may be inherentphysiological characteristics of a pathogen and the immune system, and that wemay or may not be able to influence them. This has significant consequences forintervention. A boost to the immune system, by way of good nutrition or throughintervention, may shorten θ . θ manifests its effect on the transient dynamics throughP(0); an increase in θ causes an exponential increase in P(0) (see equation 5 andcondition 17). In this paper, we will examine the effect of θ on the early dynamicsof infection through examining that of P(0). Hence, we ask: How does the size ofthe amplified initial inoculum affect the early dynamics of infection? Our transientoutcomes, tpeak and hpeak , also depend on the average measures, I and P . There-fore, in order to determine the impact of P(0) on tpeak and hpeak , we must firstexamine the dependency of I and P on P(0).

Approximating I and P. In section 2.1.2, we derived the approximations I (t)

and Pq(t) describing the transient dynamics of the immuno-patho system, basedon the assumption that there exists a period during which Pq(t) closely modelsthe early dynamics of the pathogen population growth, in turn implying that thisshould lead to close approximations of tpeak and hpeak . Let T denote this period.We further define T to be the time at which P(t) regains its initial value, i.e. whenP(T ) = P(0). The constant terms, I and P , are then derived in a way that theywould approximate the average values of I (t) and P(t) for the period of interest,T . This reasonable assumption that T is defined in a way that P(T ) = P(0), notonly greatly simplifies the analysis involved in the derivations of constant terms, italso results in close approximation of the measures of transience, tpeak and hpeak ,since it enforces symmetry in the shape of Pq(t). This can be realized from the dy-namics of Figure 2, where Pq(t) from equation 14 is plotted against the numericalsolution of P(t) obtained by solving the system of equations 5–7 using a fourthorder Runge-Kutta method. The constant terms, I and P , are then derived in a waythat they would approximate the average values of I (t) and P(t) for the period ofinterest, T .


Fig. 2. Pathogenic load in the first few days of a hypothetical infection after the full activa-tion of the immune system. The dashed parabola is the plot of the quadratic function, Pq(t);the solid plot is the graph of P(t) based on the numerical solution of the system of equa-tions 5–7 using a fourth order Runge-Kutta method. The constants of the system are: a0 =0.66; µ = 0.35; k = 0.005; r = 2.86; m = 0.22; the initial immunity, I (0) = a0

µ= 1.886;

the initial inoculum, p0 = 10; the amplified initial inoculum, P(0) = 1, 698; and θ = 2.1days. At T = 1.5296, when P(t) ≈ P(0), then I = r

m= 13, P ≈ 3684, tpeak ≈ 0.765

days, and hpeak ≈ 5143.

To find I , divide both sides of equation 7 by P and take expected value fromboth sides, for 0 ≤ t ≤ T , to get

E (1

P

dP

dt)

Def= 1

T

∫ T

0

1

P

dP

dtdt

Def= 1

T(ln P(T ) − ln P(0))

= E (r − mI (t)) by equation 7

= r − mI (19)


Equation 19 together with the assumption that P(T ) = P(0) implies that

I = r

m(20)

To find P , note that at t = T equation 14 can be written as

Pq(T ) = P(0) + PT (r − ma0

µ) − mP

T 2

2(a0 − µI + kP )

= P(0) since P(T ) = P(0)

which can then be solved for P

P = µr − ma0

mk+ 2(rµ − ma0)

µmkTfor 0 ≤ t ≤ T (21)

which implies that given T , P is constant for the duration t ∈ [0, T ]. Equations 20and 21 are the constant approximations of the average values I (t) and P(t) respec-tively, for the duration t ∈ [0, T ]. Substituting the right hand side of equation 21for P in equation 16, we get

tpeak = T

2

as should be expected from the symmetry assumption. Figure 2 is illustrative ofhow closely our quadratic model approximates tpeak and hpeak .

P and T are inversely associated. As T get smaller, P increases, which in turncauses tpeak to decrease. Clearly, T is an important measure; it provides a boundon the critical period for intervention beyond which any intervention scheme mayprove ineffective. Such a critical period must be less than tpeak . The relationshipsbetween T , P , tpeak , and therefore hpeak are clear. But how do we determine T

and what is its relationship with the initial inoculum P(0)? In other words, it is notclear how P depends on P(0) since such a dependency is masked by T .

2.1.4. A shortcoming of the quadratic modelOur interest in studying the transient behavior of the immuno-patho system is moti-vated by the observation that while components of the immune system are inactive,there is potential for exponential increase in the antigenic mass. This means thatupon full activation, the immune system is to encounter a massive pathogenic load.Although our quadratic model of pathogenesis closely models the transient mea-sures tpeak and hpeak , the effect of P(0) on these measures remains obscure since itis not clear how P(0) is affecting P or T . To understand why our quadratic modellacks an explicit dependency on P(0) in the equation for P , note that Pq is concavedown if the initial slope is increasing and concave up otherwise. Assuming thatin the absence of a fully effective immunity the initial slope is always increasing,and therefore Pq is concave down, then it must be that the second derivative of

the quadratic function is negative for the period of interest, that is, ¨Pq(t) < 0 for

0 ≤ t ≤ T . If Pq(t) is indeed a good model of P(t) at the end points 0 and T , then


Fig. 3. Pathogenic load in the first two days of hypothetical infections for different valuesof P(0) after the full activation of the immune system. (a) As in the dynamics of Figure 2,with P(0) = 1, 698; and the threshold ≈ 5435 (b) A hypothetical infection with k = 0.007;θ = 2.41 days; T = 0.70078; P(0) = 3624; the threshold ≈ 3882; the peak measures areestimated as tpeak ≈ 0.35; and hpeak ≈ 5803; all other parameters are as in figure 2. Noticethe closer approximation of the peak measures as P(0) approaches its threshold.

it must be that P < 0 as well, for 0 ≤ t ≤ T . Consider the second derivative of theoriginal equation at t = 0

P (0) = P (0)(r − mI (0)) − mP(0)I (0) by equation 7

= P(0)(r − ma0

µ)2 − mk[P(0)]2 by equations 5–7

< 0

which implies

P(0) >(r − ma0)

2

µ2mk(22)


Fig. 3. Continued.

This means that our model is a good approximation to the actual curve at t = 0 ifcondition 22 is satisfied, that is, if P(0) is large enough. This is because as P(0)

takes on larger values and becomes closer to the threshold of condition 22, P(t) be-comes more parabolic in shape and matches the dynamics of the quadratic functionPq(t). In fact, this phenomenon can be observed from the dynamics of Figure 3.In Figure 3(a), where P(0) = 1, 698 and is well below its threshold (≈ 5435),notice how the curve of P(t) is “sigmoid” near the end points, 0 and T . Comparethat to the dynamics of Figure 3(b), where P(0) = 3624 and is much closer toits threshold (≈ 3882); notice how tight the peak approximations become as P(0)

approaches its threshold and P(t) becomes more parabolic. This indicates that ourquadratic model is not as good for modeling the end points as it is for modeling thepeak measures, unless P(0) is greater than some threshold.

Our goal is to model the behavior of the pathogen population growth at the peakin a way that the role of the amplified initial inoculum in the dynamics is explicitlyclear. Under the quadratic model, if P is as in equation 21, and if P(0) is larger


than the threshold of condition 22, then Pq(t) closely follows the behavior of P(t)

both at the peak and at the end points. However, in equation 21 the nature of thedependency on P(0) is not clear. Furthermore, there is no reason to assume thatP(0) is almost always larger than the threshold of condition 22. Hence, we need toexpand upon our quadratic model of interaction.

2.2. Toward a hybrid model of interaction

Our quadratic model of the immuno-patho system nicely models the peak mea-sures, tpeak and hpeak , but it fails to express theses measures solely in terms ofthe amplified initial inoculum and various parameters of the system, unless P(0)

is greater than some threshold. Although this condition may well be realistic formany acute infections, it narrows the domain of possibilities. By now, it should beclear why the quadratic model fails to closely model the end points. Looking at thedynamics of P (t) from Figure 4, it is easy to see that a good model of P(t) mustbe a fourth order polynomial and not a second order. Hence, it is no surprise thatthe peak measures can be traced so closely but not the end points, resulting in anobscure relationship between these measures and P(0), which is masked by P andtherefore T .

To arrive at the quadratic model, we approximated the equation for P(t) twiceby replacing I (t) with I (t) and assuming the average measures as constants. Inwhat follows, we will use only the approximation for I (t) and will end up withan exponential function. We will then combine the results from the two models,thus making a “hybrid” to derive a new value of P with explicit dependency onP(0). This means that the peak measures, tpeak and hpeak , can then be explicitlyexpressed in terms of P(0) as well.

2.2.1. An exponential model of interactionIn section 2.1.2, we defined Pq to denote the quadratic model. By the same analo-gy, let Pe denote the exponential model to be derived below. The first and secondderivatives are defined accordingly. To derive Pe, divide both sides of equation 7by P and integrate both sides to get∫ t

0

1

P

dP

dτdτ

Def= ln P(t) − ln P(0)

=∫ t

0(r − mI (t)) by equation 7

≈∫ t

0(r − mI(t))

= rt − ma0

µt − ma0

t2

2+ mµI

t2

2− mkP

t2

2by equation 11

Let �(t) = rt − ma0µ

t − ma0t2

2 + mµI t2

2 − mkP t2

2 . Then we have

Pe = P(0)e�(t)


Fig. 4. Pathogen population in the first two days of a hypothetical infection after the fullactivation of the immune system. The dashed curve is the plot of the exponential func-tion, Pe(t); the solid curve is the graph of P(t) based on the numerical solution of thesystem of equations 5-7 using a fourth order Runge-Kutta method. All parameters are asin Figure 2. Notice how closely Pe(t) approximates the end points, P(0) and P(T ) =P(1.5296).

Note that in the new model, the approximation comes from replacing I (t) withI (t) from section 2.1.2. This results in a higher order function, or an exponentialone. Figure 4 demonstrates how closely Pe models the end points. The new modelprovides a tight upper bound on tpeak , and a lower bound on hpeak . In fact, it canbe shown that both the quadratic and the exponential functions result in the sameapproximation for tpeak , if fed by the same value of P . To demonstrate this, evaluate˙

Pe at tpeak from equation 16 as follows


Fig. 5. The slope of Pq(t) versus the slope of Pe(t) for the dynamics of Figure 2. The dashedline is the slope of the quadratic function Pq(t); the solid curve is the slope of the exponentialfunction Pe(t). Notice that the two models coincide exactly at and around the peak and awayfrom the end points.

˙Pe(tpeak) = P(0)(r − m

a0

µ− ma0tpeak + mµItpeak − mkP tpeak)e

�(tpeak)

= P(0)

[(r − m

a0

µ) −

r − ma0µ

m(a0 − µI + kP )m(a0 − µI + kP )

]e�(tpeak)

= 0

This proves the claim that both functions approximate tpeak the same way, sincethe equation for tpeak was derived from the quadratic model (see Figure 5).

2.2.2. Predicting transience from P(0)

To derive a relationship between P and P(0), we will combine our quadratic andexponential models in such a way that we will be able to capture the strengths ofeach model namely, close approximation of the peak measures by the quadraticmodel, and close approximation of the end points by the exponential model. DefineP new to be the new average measure of the pathogen population resulting from


combining the quadratic and exponential models of interaction. Now, compute thesecond derivative of the exponential model at tpeak to get

¨Pe(tpeak) = P(0)[(−ma0 + mµI − mkP new)e�(tpeak) +

(r − ma0

µ− ma0tpeak + mµItpeak − mkP newtpeak)

2e�(tpeak)]

= −mP(0)(a0 − µI + kP new) exp

((rµ − ma0)

2

2µ2m(a0 − µI + kP new)

)(23)

since tpeak = rµ−ma0

µm(a0−µI+kP new)by equation 16, and�(tpeak)= (rµ−ma0)

2

2µ2m(a0−µI +kP new).

On the other hand, ¨Pq at tpeak is

¨Pq(tpeak) = −mP new(a0 − µI + kP new) by equation 15 (24)

Both Pe and Pq model the behavior of the system at the peak reasonably well, andwe have shown that in fact both models approximate tpeak exactly the same. Fig-ure 5 demonstrates this effect by comparing the slope of the quadratic function, Pq ,versus the slope of the exponential function, Pe; the two models coincide exactly

at and around the peak. Therefore, it would be reasonable to assume that ¨Pq and

¨Pe evaluate to the same value at tpeak . Equating equations 23 and 24, we get

P(0) = P new exp

(− (rµ − ma0)

2

2µ2m(a0 − µI + kP new)

)(25)

Equation 25 is essential to predicting the early behavior of infection from P(0) andthe parameters of the system. By combining the quadratic and exponential modelsof interaction we were able to derive P(0) as a function of P new. Although it maynot be easy to solve for P new analytically, given P(0), we can always solve for itnumerically. Notice that equation 25 defines an explicit relationship between P(0)

and P new, and there is no longer any dependency on T . Substituting the numericalvalue of P new, obtained from equation 25, into the quadratic function Pq fromequation 14, or equivalently into tpeak and hpeak from equations 16 and 19, we canderive numerical estimates of the peak measures solely in terms of the initial inoc-ulum and parameters of the system. Figure 6 illustrates the result of the combinedmethod of approximation. Compare that to the dynamics of Figure 2: the hybridmethod provides a closer approximation of hpeak and a very tight lower bound ontpeak .

But more importantly, having a functional relationship between P(0) and P new

is the key to understanding the underlying dynamics. We may ask: How does vari-ation in the initial inoculum affect the transient outcome of infection? With equa-tion 25, as will be demonstrated in the next section, it is possible to determine thequalitative effect of change in the initial inoculum on the measures of transience.It is imperative to note that this effect can never be realized through numericalsimulations only. With numerical methods, the above question can be answered fordifferent values of the initial inoculum. But a set of numerical answers are only


Fig. 6. Pathogen population in the first few days of a hypothetical infection after the fullactivation of the immune system. The dashed parabola is the plot of the combined method ofapproximation using the quadratic and the exponential functions, where Pq(t) is evaluatedat P new; the solid plot is the graph of P(t) based on the numerical solution of the system ofequations 5–7 using a fourth order Runge-Kutta method. All parameters are as in Figure 2.P new ≈ 3974; tpeak ≈ 0.695 days; and hpeak ≈ 5077.

valid answers for the domain in which they are tested, and therefore cannot begeneralized to unexamined situations. Hence, our understanding of the underlyingdynamics will not be enhanced and will be limited to special cases.

2.3. Qualitative effects of P(0) on the measures of transience

Before proceeding to the peak analysis, we need to clarify how a change in P(0)

impacts the average measure P new since both tpeak and hpeak depend on this term.As established before, we assume that a change in the amplified initial inoculumis the result of a change in θ only. Furthermore, an increase in θ causes P(0) to


increase exponentially since the exponent of P(0) in equation 5 is always positiveby condition 17.

2.3.1. Effect of P(0) on P new

By equation 25, it is clear that an increase in P new causes an increase in P(0)

since both the numerator and the denominator in the exponent of equation 25 arepositive (see conditions 17 and 18). On the other hand, although it is not easy toobtain an analytic solution for P new in terms of P(0) from equation 25, we canstill show that an increase in P(0) should cause an increase in P new as well. Tosee this, note that if P(0) is increasing, it can only be associated with a change inP new since the increase in P(0) is assumed to be associated with an increase in θ

and all other parameters in equation 25 are assumed to be constant. Suppose forcontradiction that an increase in θ causes P new to decrease. This would imply thatthe exponent in equation 25 is increasing or the exponential term is decreasing. Butif both terms in the product are decreasing, then P(0) must be decreasing as well,which contradicts the original assumption. Therefore, if P(0) is increasing, P new

must be increasing as well, implying that P(0) and P new are positively correlated.

2.3.2. Effect of P(0) on tpeak

An increase in P(0) causes P new to increase, which in turn causes tpeak to decrease(see equation 16). Hence, as the amplified initial inoculum takes on larger valuesas θ increases, the time of “crisis” will occur earlier, requiring faster intervention.

2.3.3. Effect of P(0) on hpeak

The effect of P(0) on hpeak is less obvious since P new appears in both the numeratorand the denominator of the second term in the equation for hpeak (see equation 19).Differentiating hpeak with respect to P(0) or P new will not help. Instead, we willinfer the direction of change in hpeak by deriving an equation for hpeak in terms ofthe second derivative of P(t) from the original equations. Evaluation of equation 6at tpeak produces

I (tpeak) = a0 − µI (tpeak) + kP (tpeak)

= a0 − µr

m+ khpeak (26)

since P(tpeak) = hpeak , and since

I (tpeak) = − P (tpeak)

mP (tpeak)+ rP (tpeak)

mP (tpeak)by equation 7

= r

msince P (tpeak) = 0

Differentiation of equation 7 at tpeak to get an explicit relationship between I (tpeak),hpeak , and P (tpeak) produces

P (tpeak) = rP (tpeak) − m[I (tpeak)P (tpeak) + I (tpeak)P (tpeak)

]= −mI(tpeak)hpeak since P (tpeak) = 0


= −m(a0 − µr

m+ khpeak)hpeak by equation 26

= (µr − ma0)hpeak − mk(hpeak)2

Solving the above quadratic equation for hpeak we get

hpeak = 1

2km

[(rµ − ma0) ±

√(rµ − ma0)2 − 4mkP (tpeak)

](27)

Clearly, hpeak > 0, so we will only consider the positive solution of equation 27.Replacing P (tpeak) in equation 27 with either its quadratic or its exponential ap-proximation is the final step needed in order to infer the direction of change in hpeak

with respect to P(0). We will do both. If we replace P (tpeak) with its quadraticapproximation from equation 24 we get

hpeak = 1

2km

[(rµ − ma0)

+√

(rµ − ma0)2 + 4m2kP new(a0 − µI + kP new)

](28)

Now, an increase in P(0) causes P new to increase, which in turn causes hpeak toincrease. Let β = a0 − µI + kP new. Then replacing P (tpeak) with its exponentialapproximation from equation 23 gets

hpeak = 1

2km

[(rµ − ma0)

+√

(rµ − ma0)2 + 4m2kP (0)β exp

((rµ − ma0)2

2µ2mβ

)]

>1

2km

[(rµ − ma0) +

√(rµ − ma0)2 + 4m2kP (0)β

](29)

since 0 <(rµ−ma0)

2

2µ2mβ< ∞ by conditions 17 and 18 from section 2.1.3. An increase

in P(0) also causes an increase in P new, which together cause inequality 29 andhpeak to increase.

The intuition behind the argument that hpeak and P(0) change in the samedirection is much more straightforward than the preceding analysis. Note that anincrease in P(0) causes tpeak , and similarly T , to decrease. On the other hand, anincrease in P(0) causes P new to increase. The only way to compensate for the effectof an increase in P new in a shorter time period is for hpeak to increase, resulting ina “thinner” and “taller” parabolic shape.


3. Discussion

We defined the notion of “transient pathogenicity” to encompass the early dynam-ics of the within-host pathogenesis of acute infectious disease. We recognized theamplifying effect of the initial period of immune inactivity, θ , on the pathogen-ic growth of the initial inoculum. We further demonstrated that in the absence ofa fully effective immunity, the initial pathogenic population consisting of a fewmicroorganisms will grow freely and without bound until the immune system isfully activated. This “amplified initial inoculum”, P(0), is therefore the new initialpathogenic load at the time of full activation of the immune system, when the new-ly activated immunity is to face an extensive antigenic mass. We then developed aquadratic model and an exponential model of the pathogen population growth tostudy transients in infectious disease. We further derived the transient measures,tpeak (the time from the full activation of the immune system to the peak of in-fection) and hpeak (maximum pathogenic load of infection), under the quadraticmodel. The two models were then combined in order to express the outcomes ofthe early dynamics of infection in terms of P(0) and the parameters of the systemonly. The qualitative analysis of this combined method of approximation enabledus to derive explicit relationships between P(0) and and the peak measures, tpeak

and hpeak . Therefore, we have established qualitatively that

1. As the duration of immune inactivity, θ , increases, the initial antigenic massincreases exponentially with θ . Upon activation, the immune system must there-fore face and combat a massive pathogenic load.

2. As the duration of immune inactivity, θ , increases, tpeak decreases, causing the“crisis” to occur earlier, making the critical window for intervention smaller.

3. As the duration of immune inactivity, θ , increases, hpeak increases, making theintensity of “crisis” greater.

4. Putting 2 and 3 together, as the duration of immune inactivity, θ , increases, thedamage to the host will be much more extensive in a shorter time period.

Although some of the parameters of the immuno-patho dynamics may be inherentphysiological properties of the interacting components that we may not be able toinfluence, θ may very well be influenced by the nutrition state or the stress levelof the host. A boost to the immune system may be equivalent to reducing θ ; thusreducing the maximum pathogenic load and delaying the time of its occurrence.

To obtain a more quantitative assessment of the measures of early dynamics,tpeak and hpeak , for specific infections, the methodology outlined in this paper willhave to be extended or modified to encompass the particular dynamics under study.If it is indeed possible to predict the within-host early behavior of acute infectionsfrom the initial inoculum and a few related immunological and pathological param-eters, and if there is sufficient data on average parameter values for a cohort, then itshould be possible to classify infectious disease based on their peak measures andthe critical period to intervene.

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Transient dynamics and early diagnostics in infectious disease