Page 1

A simple model of pathogen–immune dynamics including specific

and non-specific immunity

Andrea Pugliesea,*,1, Alberto Gandolfib

aDepartment of Mathematics, University of Trento, Via Sommarive 14, Trento 38050, Italy

bIstituto di Analisi dei Sistemi ed Informatica A. Ruberti CNR, Rome, Italy

a r t i c l ei n f o

Article history:

Received 10 December 2007

Accepted 20 April 2008

Available online 1 May 2008

Keywords:

Mathematical model

Immune response

Pathogen dynamics

Vaccination

Bifurcations

a b s t r a c t

We present and analyze a model for the dynamics of the interactions between a pathogen and its host’s

immune response. The model consists of two differential equations, one for pathogen load, the other one

for an index of specific immunity. Differently from other simple models in the literature, this model

exhibits, according to the hosts’ or pathogen’s parameter values, or to the initial infection size, a rich rep-

ertoire of behaviours: immediate clearing of the pathogen through aspecific immune response; or acute

infection followed by clearing of the pathogen through specific immune response; or uncontrolled infec-

tions; or acute infection followed by convergence to a stable state of chronic infection; or periodic solu-

tions with intermittent acute infections. The model can also mimic some features of immune response

after vaccination. This model could be a basis on which to build epidemic models including immunolog-

ical features.

? 2008 Elsevier Inc. All rights reserved.

1. Introduction

Several recent papers [1–3] have started to bring immunologi-

cal considerations into models for epidemic spread, especially con-

cerning the evolution of virulence [4,5]. Gilchrist and Sasaki [4]

have developed the so-called ‘nested’ approach, in which an expli-

cit model of pathogen–immune response dynamics within each

individual host is coupled to a model of epidemic spread between

hosts; an approach that has been followed in [5–7]. The idea of

integrating host immune response in population models had arisen

also in studies of the dynamics of macroparasites [8], especially

when the impact of phenomena like waning immunity is

investigated.

While an accurate modelling of the dynamics of the interactions

between pathogens and the several types of immune cells is a fas-

cinating subject (see for instance [9–11]), in a ‘nested’ approach the

immunological model needs be reasonably simple, but able to

qualitatively reproduce the basic behaviour of disease dynamics

caused by different pathogen agents.

Models that include different types of immune cells have been

used as an ingredient of epidemic models by Kostova [7] and have

been fitted to data on CD8 T cell responses in HIV [12] and chori-

omeningitis [13] infections.

Since we do not aim at an accurate model of immune response,

but simply to a phenomenological description of the time course of

infections, we follow instead the approach used in [14,4,5] and in

several models in [9] of describing the immune response through

a single variable, the level of pathogen-specific immunity, that

may represent some precise quantity, like the density of specific

B-cells or T-cells or antibodies, or a more generic index related to

the different types of immune cells specific for that pathogen

agent. Analogously, the pathogen load is described through a single

variable, giving thus rise to a two-dimensional dynamical system.

Most dynamical description of pathogen–immune interactions

give rise to a stereotyped behaviour. Some models [4,5] are tailored

for a short-term description of successful immune response: thus,

for all parameter values, an infection gives rise to a strong and per-

sistent immunity, that completely clears the pathogen. Other mod-

els [14,9] consider long-term effects, such as immunity decay, and

share some properties of epidemic models; one may define a num-

ber R representing pathogen’s reproduction ratio: when R > 1,

from any initial inoculum a sizeable infection will occur, followed

by a growth in immune response reducing the infection to a posi-

tive equilibrium, where the pathogen persists at a positive (possi-

bly low) level contrasted by host’s immune response; on the other

hand, when R < 1, the infection cannot start and the immune sys-

tem eventually completely clears the pathogen.

We build here a slightly more complex model, that considers

also the effect of aspecific immune response, such as mediated

by macrophages, and Holling-type functional responses of immune

cells to pathogen level. We show that this slight increase in

0025-5564/$ - see front matter ? 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.mbs.2008.04.004

* Corresponding author.

E-mail addresses: pugliese@science.unitn.it (A. Pugliese), alberto.gandolfi@

iasi.cnr.it (A. Gandolfi).

1Supported in part by the project EPICO of the Province of Trento, Italy.

Mathematical Biosciences 214 (2008) 73–80

Contents lists available at ScienceDirect

Mathematical Biosciences

journal homepage: www.elsevier.com/locate/mbs

Page 2

complexity allows for a much more diverse behaviour of the sys-

tem, according to parameter values and initial conditions. Hence,

we believe that the resulting may be a satisfactory flexible descrip-

tion of the qualitative features of the dynamical interactions be-

tween pathogens and immune system.

2. The model

We describe an infected individual through its pathogen load P

and its level of specific immunity B. The variable B may represent

some precise quantity, like the density of specific B-cells or anti-

bodies, or a more generic index related to the different types of im-

mune cells specific for that pathogen agent.

As discussed in Section 1, several authors have described the

interactions of a pathogen and its host’s immune response through

a dynamical system involving these two variables. Gilchrist and

Sasaki [4] used the following model, where r is the pathogen’s rep-

lication rate, the immune cells proliferate proportionally (with a

proportionality constant a) to the pathogen load, and the pathogen

is killed by immune cells with a Lotka-Volterra-type predator–prey

relationship with constant c:

?

Thesystemhastobecompleted

Pðt0Þ ¼ P0> 0, Bðt0Þ ¼ B0> 0. It is very easy to see (actually sys-

tem (1) can be transformed, changing t into ?t, into the classical

Kermack–McKendrick epidemic model) that PðtÞ initially in-

creases (if r > cB0) to a maximum and then declines to 0, while

BðtÞ increases to an asymptotic level (depending on initial condi-

tions) B1. Hence, every infection is, after an acute phase, com-

pletely cleared.

André and Gandon [5] simplified (1), by assuming that the pro-

liferation rate of immune cells is independent of pathogen load, as

long as it is positive [15], obtaining the system

?

The behaviour of PðtÞ is similar to the previous case, but has the

advantage of being expressed through a closed formula, while BðtÞ

grows to infinity, so that the solution can be adequate only for t

not too large.

Mohtashemi and Levins [14] considered instead two other fea-

tures of the immune system, the spontaneous production (at a very

low rate) of specific cells, as well as their decay (these aspects were

not considered in [4,5] that are limited to acute infections). More-

over, they assumed instead that immune cells are produced by an-

other compartment (constant in size) proportionally to pathogen

load. Thus they studied the system

?

If we take PðtÞ ? 0, i.e. we consider an uninfected individual, it is

easy to see that BðtÞ will approach the equilibrium value h=d, that

can then be considered the typical value for uninfected individual.

If we consider the dynamics within an individual infected at time

t0, it is then natural to add to (3) the initial conditions Bðt0Þ ¼ h=d,

Pðt0Þ ¼ P0> 0, where P0is the inoculum size.

If the threshold condition r > ch=d is satisfied, the pathogen-

free equilibrium ð0;h=dÞ is unstable, and pathogen load will

initially increase; then ðPðtÞ;BðtÞÞ will converge to the positive

equilibrium ðdr

r 6 ch=d, the pathogen load immediately starts decreasing and

the infection is completely cleared. In summary, there are only

two possible behaviours: either an infection fails immediately,

P0¼ rP ? cBP;

B0¼ aBP:

ð1Þ

withinitialconditions

P0¼ rP ? cBP;

B0¼ bB:

ð2Þ

P0¼ rP ? cBP;

B0¼ kP ? dB þ h:

ð3Þ

c? hÞ1

kwhich is globally stable [16]. Otherwise, if

or, if it initially succeeds, it leads to an equilibrium where

the pathogen is maintained at low density.

Several other models are proposed in Nowak–May [9] for

virus–immune dynamics, but all share this feature: if an infec-

tion is possible, it is never cleared completely. It must be re-

marked that, if the deterministic model is viewed as an

approximation of a more realistic stochastic model, it is possible

(or perhaps likely) that stochastic extinction of pathogen agents

occurs when the pathogen load, as predicted by the determinis-

tic model, is low.

Kostova [7] extends these models, by considering two types

of immune cells: effector T-cells and memory T-cells. Assuming

that the latter do not decay at all, the typical behaviour of the

system is an acute infection, controlled first by effector T-cells,

then by memory T-cells, until complete clearance of

pathogen.

Here we propose and analyze another extension of model (3),

sharing part of the structure with the model proposed by

d’Onofrio [17] for tumour–immune interactions, but with the

inclusion of aspecific immune response. The resulting model

exhibits a rich dynamical repertoire, allowing for acute infection,

possibly dose-dependent, followed by deterministic pathogen

clearance, or for chronic infection, possibly with periodic fluctu-

ations in pathogen load and immune response. We assume that

the predation of immune cells on pathogens follows a Holling

function of the 2nd type, and moreover we postulate that a sim-

ilar action is performed by aspecific cells, whose density is a

constant M. Thus, the first equation of (1), (2), or (3) is modified

to

the

P0¼ rP ?

csP

1 þ asPB ?

cuP

1 þ auPM:

ð4Þ

As for the second equation, we keep the spontaneous production

and the decay of specific immune cells as in (3), but assume, as in

(1), auto-replication of specific immune cells, stimulated by patho-

gen load, but with a maximum replication rate. The resulting equa-

tion is

B0¼

kP

1 þ kmPB ? dB þ h:

It is convenient to rescale the variables, so as to obtain a non-

dimensional system. We choose the following changes of variables:

ð5Þ

x ¼kP

Letting ¼:

_ x ¼ ax ?

_ y ¼

where

d;

y ¼csB

d

;

s ¼ dt:

d

ds, we obtain the system

xy

1þbsx?

1þcx? y þ g;

mx

1þbux;

xy

(

ð6Þ

a ¼r

d;

bs¼asd

k;

bu¼aud

k

;

c ¼kmd

k

;

g ¼csh

d2;

m ¼cuM

d

:

Before proceeding with the analysis, we make the following mini-

mal assumptions on the parameters of system (6):

(1) The replication rate of the specific immune cells is higher

than their decay rate, at least when the pathogen load is very

high. Looking at (5), this translates into

(2) The specific immune response responds better to high path-

ogen loads than the aspecific response. This translates into

as< au, i.e. bu> bsP 0.

k

km> d, i.e. c < 1.

There may be other natural assumptions on the order of magni-

tude of the parameters, but they are not needed in a preliminary

analysis.

74

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

Page 3

3. Equilibria and null-clines

System (6) has clearly the pathogen-free equilibrium ð0;gÞ.

In order to look for other equilibria, consider the null-clines.

Letting _ y ¼ 0, we obtain

y ¼ fðxÞ :¼

1 ?

The function f is positive for x < ximm¼

asymptote. ximm> 0, because of Assumption 1. f is increasing and

convex in ½0;ximmÞ.

Letting _ x ¼ 0, we obtain

m

1 þ bux

Computing the derivatives, one sees that g is increasing and con-

cave, because of Assumption 2.

Summarizing, the signs of the time derivatives of x and y are as

follows:

g

x

1þcx

:

ð7Þ

1

1?c, which is a vertical

y ¼ gðxÞ :¼ ð1 þ bsxÞ a ?

??

:

ð8Þ

? _ y > 0 for all x P ximmand, when x 2 ½0;ximmÞ, _ y < 0 for y > fðxÞ.

? _ x > ½<?0 for y < ½>?gðxÞ.

From this information about f and g one obtains (see Fig. 1) the

following conclusion:

? If gð0Þ > fð0Þ, i.e. a ? m > g, there exists a unique x?2 ð0;ximmÞ

such that fðx?Þ ¼ gðx?Þ. This corresponds to a positive equilib-

rium ðx?;y?¼ fðx?ÞÞ of (6);

? If gð0Þ < fð0Þ, there may be 0 or 2 solutions (1 in non-generic

cases) of fðxÞ ¼ gðxÞ. Some conditions that guarantee that there

are no solutions (hence, no positive equilibria of (6)) are

g0ð0Þ 6 f0ð0Þ or gðximmÞ 6 fð0Þ.

Recalling the definitions of f and g, and checking also the Jaco-

bian of (6) at the equilibria, one obtains

Proposition 1.

? If a > g þ m, the pathogen-free equilibrium is unstable and there

exists a unique positive equilibrium ðx?;y?Þ. ðx?;y?Þ may be locally

asymptotically stable or unstable; in the latter case, if the solutions

are bounded (see Proposition 2 below), Poincaré–Bendixson theory

implies the existence of a periodic solution surrounding the

equilibrium.

? If a < g þ m, the pathogen-free equilibrium is locally asymptotically

stable, and there exist either 0, 1 and 2 positive equilibria. If there

are 2 positive equilibria ðx?

ðx?

or unstable; in the latter case, it may be an unstable focus, or an

unstable node.

1;y?

1Þ and ðx?;y?Þ with x?

1< x?, then

1;y?

1Þ is a saddle point, while ðx?;y?Þ may be asymptotically stable

Proof. Most statements follow immediately from the above argu-

ments. We need only to check the Jacobian of (6) at a positive equi-

librium ðx?;y?Þ. Through some computations, we obtain

a ?

y?

ð1þcx?Þ2

J?¼

y?

ð1þbsx?Þ2?

m

ð1þbux?Þ2

?

x?

1þcx?? 1

x?

1þbsx?

0

@

1

A¼

x?

1þbsx?g0ðx?Þ

gf0ðx?Þ

fðx?Þ

?

?

x?

1þbsx?

g

fðx?Þ

!

:

ð9Þ

It follows

detðJ?Þ ¼

x?

1 þ bsx?

g

fðx?Þðf0ðx?Þ ? g0ðx?ÞÞ:

ð10Þ

If there is only 1 equilibrium, we have (generically) f0ðx?Þ > g0ðx?Þ

(see Fig. 1); if there are 2 equilibria, we have f0ðx?

f0ðx?Þ > g0ðx?Þ. We then have from (10) that ðx?

values of opposite sign, so that it is a saddle point; on the other

hand, to ascertain the stability of ðx?;y?Þ, we need to study the sign

of trðJ?Þ ¼

negative.

h

1Þ < g0ðx?

1Þ has 2 eigen-

1Þ and

1;y?

x?

1þbsx?g0ðx?Þ ?

g

fðx?Þ, which may be either positive or

The different possibilities are outlined in Figs. 3 and 4, where

the phase plane and some typical solutions are plotted for different

values of the parameter a.

Sincetheparametersaandg þ mrepresentthereplicationrateof

the pathogen and, respectively, the overall efficacy of the immune

system (see Eq. (4)), we note that the condition a > g þ m corre-

sponds to a case of very high virulence capable of overcoming the

non-specific immunity, whatever the extent of the initial infection.

From the phase plane, one may note that in principle there may

exist solutions diverging to þ1 with both xðtÞ and yðtÞ increasing,

and with yðtÞ < gðxðtÞÞ. In Appendix A, we show the following

Proposition 2. If bs¼ 0 or c ¼ 0 or a <1

starting from ðx0;gÞ with x0> 0 are bounded and converge either to

an equilibrium or a periodic solution of (6).

If bs> 0 and a >1

ðx0;gÞ with x0> 0 such that xðtÞ and yðtÞ are monotonically increasing

and limt!1xðtÞ ¼ limt!1yðtÞ ¼ þ1.

c? 1, all solution of (6)

c? 1, there exist solutions of (6) starting from

Remark 1. It is clear from the structure of (6) that diverging solu-

tions are possible, because it is assumed that pathogens grow

exponentially, in absence of specific immune response, and that

growth may be faster than the maximal growth of immune

response. One could change the assumption of pathogen exponen-

tial growth, since pathogen growth will eventually be limited by

the total host resources. We instead retain the assumption of expo-

nential growth, interpreting diverging solutions as instances in

which host defenses are not able to respond to pathogen replica-

tion, and infection ends with host death.

Moreover, we recall that the parameter 1=c represents the maxi-

mal replication rate of the immune system. Thus is clear and intui-

tive the meaning of the condition a > 1=c ? 1, which implies that

for sufficiently high initial infections (of a naive individual, i.e. at

the basal level of specific immunity) the infection evolves un-

bounded. Models satisfying this conditions appear to be the most

realistic.

Fig. 1. The function f (solid line) and four instances of the function g (dotted and

dashed lines) for different values of a ¼ 7:5; 7:8; 8; 8:4. Other parameter values are

g ¼ 0:05, c ¼ 0:05, m ¼ 8, bs¼ 10?10, bu¼ 0:1.

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

75

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Note that bsor c ¼ 0 imply instead that the immune system can

always control the infection whatever the initial infection be.

Therefore, the Holling-type functional form of the pathogen re-

moval and of the response of the immune system to pathogen level

is, in this model, necessary for the possibility that large infections

escape immune control.

4. Bifurcation diagrams and examples

The qualitative behaviour of solutions may be summarized as a

bifurcation diagram, choosing as parameter, for instance, a (see

Fig. 2). At a ¼ a0:¼ g þ m, the equilibrium ð0;gÞ undergoes a trans-

critical bifurcation, which, under the condition mbuþ gbs> g, is

subcritical, meaning that a positive equilibrium exists unstable

for a < a0, a close to a0. In this case, there will exists a1< a0 in

whichthepositiveequilibrium

bifurcation.

For a < a1 there are no positive equilibria, and the pathogen-

free equilibrium ð0;gÞ is asymptotically stable; actually, if all

solutions are bounded (Proposition 2), Poincaré–Bendixson theory

implies that it is globally attractive from positive initial points.

Still, the system is excitable, in the sense that if the initial pathogen

dose x0is sufficiently large, then, starting from ðx0;gÞ, xðtÞ grows to

high values before the solution eventually approaches ð0;gÞ.

For a1< a < a0, the pathogen-free equilibrium ð0;gÞ is still

asymptotically stable, but there also two positive equilibria

ðx?

sitive equilibrium ðx?;y?Þ, and no solution with x0> 0 will ap-

proach the pathogen-free equilibrium.

Concerning the stability of the positive equilibria, we can say

that, when there are two equilibria ða1< a < a0Þ, the lower one

ðx?

ough, the equilibrium ðx?;y?Þ is always unstable.

There are two possible paths to instability: in one case (see

Fig. 2(a)), ðx?;y?Þ is asymptotically stable for a close to (and larger

than) a1; there exists then a value aHat which the positive equilib-

rium ðx?;y?Þ undergoes a Hopf bifurcation: for a < aH, ðx?;y?Þ is

asymptotically stable, while it is unstable for a > aH. In Fig. 3, we

show the phase plane of solutions for a < aH and for a > aH. It

may also happen that there are several Hopf bifurcation values,

undergoesa saddle-node

1;y?

1Þ and ðx?;y?Þ with x?

1< x?. For a > a0, there exists a unique po-

1;y?

1Þ is an unstable saddle point. Moreover, when a is large en-

with the positive equilibrium alternately losing and acquiring

stability.

In the other case (Fig. 2(b)), the equilibrium ðx?;y?Þ is unstable

for all a. The limit cycle existing for a > a0 because of Poincaré–

Bendixson theory, emerges through a homoclinic bifurcation at

a ¼ a0. Numerically, it appears that, for a < a0, all solutions (except

for the exceptional ones lying on the stable manifold of ðx?

converge to ðg;0Þ, while for a > a0they converge to a periodic solu-

tion (see Fig. 4).

There exists moreover a value a1¼1

all solutions are bounded, and converge to one of the previous

alternatives; for a > a1, there exist diverging solutions of (6).

Under the condition mbuþ gbs< g (which seems less likely,

since entails a large spontaneous production of effective immune

cells), the bifurcation diagram is simpler with a supercritical bifur-

cation of the pathogen-free equilibrium at a0and a unique positive

equilibrium for a > a0.

1;y?

1Þ)

c? 1, such that for a < a1

5. Examples of behaviour of the model

First of all, we show how different behaviours of the system

may occur by changing the inoculum size or parameters of the

model, reflecting individual variation.

In Fig. 5 we show some examples of the time course of

infections.

The solid lines correspond to three different initial sizes of the

inoculum: it can be seen that, all other things beings the same, this

may lead to a sub-threshold infection, or to a normal infection later

controlled by immunity, or to a catastrophic infection correspond-

ing, from the mathematical point of view, to a diverging solution,

and, from the biological point of view, to a potentially lethal event.

The dotted lines corresponds to the same initial inocula in an

individual whose immune cells replicate faster; this corresponds

to a higher k in terms of the original parameters, hence smaller

bs, buand c for adimensional parameters. In this case, the infection

is always controlled.

The dashed lines simulate the same initial inocula in an individ-

ual with a lower level of aspecific response ðMÞ; now also the smal-

ler inoculum size leads to a normal infection, while, in the other

Fig. 2. The bifurcation diagram of equilibria with respect to a. The circle with H corresponds to the value (at a ¼ aH) of Hopf bifurcation; the circle with T to the tangent

(saddle-node) bifurcation at a ¼ a1. Parameter values are g ¼ 0:05, c ¼ 0:05, bs¼ 10?10, bu¼ 0:1. In the left panel (a), m ¼ 1; in the right panel (b) m ¼ 8.

76

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

Page 5

cases, the simulations are very similar to the reference ones (solid

lines).

We then consider what could be the effect of vaccination, mod-

elled in an extremely simple way. We assume that vaccination is

realized by inoculating an attenuated pathogen in the sense that

its replication rate a is lower than the wild type; in all other re-

spects, the attenuated pathogen is identical to the wild type, so

that the immune response is the same.

In Fig. 6, we compare the time courses of a normal infection

with that of a vaccination. In the example, when the vaccine inoc-

ulum is too small (in the figure, a same size inoculum of the wild

type leads to an infection), the pathogen content decreases imme-

diately and no effective immune response mounts. With a larger

inoculum, one obtains an attenuated infection (the peak value of

xðtÞ is several orders of magnitude smaller than in a normal infec-

tion) and an immune response that, while lower than in the case of

a normal infection, is still protective. Thus the model can repro-

duce both vaccination failures (the threshold for the initial inocu-

lum depends on the other parameters m, bu;... that may be

differentamong individuals),

vaccination.

The protective effect can be seen in Fig. 7. There we show the

results of some simulations in which after an initial vaccination

or normal infection, the individual is reinfected, at fixed times after

the initial infection, with the normal pathogen. It can be seen that

when the second infection occurs not too long after the first one (in

the figure at t ¼ 1), whether it is vaccination or natural infection,

the new infection is immediately cleared and the immunity is

andthe protectiveeffect of

ab

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

......

.

..

....

Fig. 3. Examples of the phase plane of the system (6). Parameter values are g ¼ 0:05, c ¼ 0:05, bs¼ 10?10, bu¼ 0:1, m ¼ 1:5. In the left panel (a), a ¼ 1:75; the solution shown

is slowly converging to the equilibrium at the centre of the spiral. In the right panel (b) a ¼ 1:9; two solutions are shown, one starting from ð0:1;gÞ and spiraling inwards, the

other one starting from (1,0.8) and spiraling outwards; both are converging to a limit cycle in between.

ab

Fig. 4. Examples of the phase plane of the system (6). Parameter values are g ¼ 0:05, c ¼ 0:05, bs¼ 10?10, bu¼ 0:1, m ¼ 8. In the left panel (a), a ¼ 7:6215; three solutions are

shown: the left one starting from ð0:5;0:05Þ converges to ð0;gÞ with a monotone decrease of xðtÞ, while yðtÞ remains very close to g; the two other solutions, starting from

ð0:7;gÞ and ð0:8;gÞ converge to ð0;gÞ after an initial increase (rather different between them) of xðtÞ. In the right panel (b), a ¼ 8:5; two solutions are shown, one (dotted line)

starting from ð0:8;gÞ, the other one (solid line) starting from ð0:5;gÞ: both appear to converge to a limit cycle (basically what appears to be the middle curve) which, for a part,

lies very close to the y-axis.

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

77

Page 6

not boosted. Waiting a longer time for the second infection (t ¼ 2

or t ¼ 3), still there are no effects in the case of a naturally infected

individuals; however, after a vaccination, a mild infection occurs

(similar to that occurring immediately after vaccination) with a

boosting of the immune response; in this case, immunity level

actually becomes higher than in the case of a naturally immunized

individual. If the second infection occurs long after initial vaccina-

tion (not shown in the figures), without any exposure to the path-

ogen in the mean time, then immunity has dropped so low that the

new infection is similar to that occurring in a naive individual. A

similar reinfection in a naturally infected individual would instead

lead to mild infection, and a boosting of the infection.

It seems that this simple model is able to mimic several of the

phenomena occurring in vaccinations, from failures, to waning

immunity with time, and boosting of immunity through reexpo-

sures at the correct schedules. It may also occur that while moder-

ate reexposures lead to very mild reinfections and boosting of

immunity, extreme reexposures lead to significant reinfections,

as documented for measles [18]. Models of this type could then

be useful for an appropriate design of revaccination schedules, that

would depend on how much the pathogen is circulating in the pop-

ulation. In fact, the model, at least for the parameter values used in

this simulation, predicts that an effective immunity can be main-

tained as long as an individual is confronted, from time to time,

with the pathogen. As natural exposures become rare, then revac-

cinations need be more frequent.

There is a long-standing debate on the mechanisms main-

taining immunity in individuals, whether it is due to very

long-livedmemorycells,or

[19,20]. Varying the parameter values, this model can support

residual antigenicstimulation

ab

Fig. 5. Examples of the time course of solutions of the system (6). In the left panel (a), pathogen load xðtÞ in logarithmic scale; in the right panel (b), immunity level yðtÞ. The

solid lines are obtained, for different initial conditions ð0:4;gÞ, ð1;gÞ and ð100;gÞ, with parameter values g ¼ 0:05, c ¼ 0:02, bs¼ 10?8, bu¼ 2, m ¼ 200, a ¼ 100. The dotted

lines are obtained, from the same initial conditions, with bu¼ 1:5, c ¼ 0:015, bs¼ 7:5 ? 10?9; the one starting from ð0:4;gÞ is identical to (and hidden from) the solid line. The

dashed lines are obtained, from the same initial conditions, with m ¼ 150; the one starting form ð100;gÞ is identical to (and hidden from) the solid line.

ab

Fig. 6. Simulation of vaccination with system (6). In the left panel (a), pathogen load xðtÞ in logarithmic scale; in the right panel (b), immunity level yðtÞ. The solid line is

obtained with initial condition ð1;gÞ and parameter values g ¼ 0:05, c ¼ 0:02, bs¼ 10?8, bu¼ 2, m ¼ 200, a ¼ 100. The dotted lines simulate vaccination and are obtained with

a ¼ 20 and initial conditions ð1;gÞ and ð5;gÞ.

78

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

Page 7

any of these. In terms of the original parameters, 1=d is the

average duration of a B-particle; if this time is of the order

of magnitude of human life, then no other mechanisms are

needed. On the other hand, with a larger d, maintenance of

immunity is compatible either with convergence of the system

to a stable positive equilibrium (phenomenon that seems to be

possible only when a is moderately large, i.e. with slowly rep-

licating pathogens) or with not too infrequent reexposures to

the pathogen.

Appendix A. Boundedness of solutions

We prove here Proposition 2.

Let us prove the boundedness of trajectories when a <1

c ¼ 0 or bs¼ 0]; we show that for each x0> 0, the solution of (6)

starting from ðx0;gÞ is trapped in a bounded invariant set; hence

the conclusion follows.

Choose d such a <1

such that gðMÞ > g. Hence, for each x P M

x

1 þ cxP1

Take the solution of (6) starting from ðM;gÞ and assume that it re-

mains for all t > 0 in the set fy < gðxÞg. We would then have

_ x > 0 and _ y > 0 for all t > 0, and we could write yðtÞ ¼ hðxðtÞÞ for

an increasing function h.

Then

?

x a ?

using (A.1). By a comparison principle, we obtain

c? 1 [or

c? 1 ? d and take M P max x0;

1

c?d

cd

no

and

c? d

and

gðxÞ > 0:

ðA:1Þ

h0ðxÞ ¼

hðxÞ

?

x

1þcx? 1

hðxÞ

1þbsx?

?

1þbux

þ g

m

? P

hðxÞ

1

c? d ? 1

ax

??

ðA:2Þ

hðxÞ P g

Since, if bs> 0, gðxÞ grows linearly, while hðxÞ grows superlinearly,

necessarily there must exist xM> x such that hðxMÞ ¼ gðxMÞ and

hðxÞ < gðxÞ for M < x < xM. This means that the solution of (6) start-

ing from ðx0;gÞ enters the region fy > gðxÞg where _ x < 0. This will be

the first part of the boundary of the invariant region.

x

M

??

1c?d?1

a

8 x P M:

ðA:3Þ

If bs¼ 0, one needs only choose d such that1

same argument would then show that hðxÞ grows with a positive

exponent, while gðxÞ tends to a constant, again proving the exis-

tence of xM> x such that hðxMÞ ¼ gðxMÞ.

Finally, if c ¼ 0, one has to choose M P maxfx0;a þ 1 þ dg with

d > 0 and repeat the same argument.

Consider now the solution of (6) starting from ðxM;yM¼ gðxMÞÞ

for any xM>

prove that eventually the trajectory enters the region C ¼

x <

where _ y < 0.

Assume the opposite. Then, since yðtÞ is monotone increasing,

and no equilibriaexist with

limt!s?yðtÞ ¼ þ1, where ½0;sÞ is the maximal interval of existence

of solutions. Since xðtÞ is

limt!s?xðtÞ ¼ xm. Since we have assumed that ðxðtÞ;yðtÞÞ never en-

ters the region C, necessarily (see Fig. 1) we have xmP

First of all, we show that s ¼ þ1. In fact, if c > 0, we have

_ y 6

y; this implies an exponential bound for yðtÞ, thus glo-

bal existence. If c ¼ 0, we have _ x 6 ax ?

d

dtðð1 þ bsxMÞxðtÞ þ yðtÞÞ 6 að1 þ bsxMÞxðtÞ ? yðtÞ þ g

6 aðð1 þ bsxMÞxðtÞ þ yðtÞÞ þ g;

which again yields an exponential bound for ðð1 þ bsxMÞxðtÞ þ yðtÞÞ,

thus global existence.

Now, let T be such that yðTÞ > K with K P yMto be chosen later.

From (6), we then obtain, for t > T,

c? 1 ? d > 0; the

1

1?c. Initially we have _ x < 0 and _ y > 0, but we

o

y > yM,

1

1?c;y > fðxÞ

n

necessarily wehave

monotone decreasing, we have

1

1?c.

1

c? 1

??

xy

1þbsxMso that

_ x 6 ax ?

K

1 ? c þ bs

?

m

1 ? c þ bu

:

The comparison principle then implies

?

K

1 ? c þ bs

If K has been chosen large enough that the coefficient of eatin (A.4)

is negative, we have limt!1xðtÞ ¼ ?1, reaching a contradiction.

There exists then a point ðxC;yCÞ with xC<

where the trajectory enters the region C.

xðtÞ 6 eat

xM?

K

1 ? c þ bs

þ

þ

m

1 ? c þ bu

?1

?1

a

?

?

?

þ

m

1 ? c þ bu

a:

ðA:4Þ

1

1?cand yC¼ fðxCÞ

ab

Fig. 7. Reinfections in naturally infected and vaccinated individuals. In the left panel (a), pathogen load xðtÞ in logarithmic scale; in the right panel (b), immunity level yðtÞ.

The solid line corresponds to a natural infection and is the same as in Fig. 6. The dotted line corresponds to a vaccinated individual and is the same as in Fig. 6 with initial

condition ð5;gÞ. The dashed lines correspond to reinfections (at t ¼ 1; 2 or 3) of the naturally infected individual (these cannot be seen in (b), since almost no change occurs in

immunity level). The dot-and-dash lines correspond to reinfections (at t ¼ 1; 2 or 3) of the vaccinated individual. In the reinfections a ¼ 100, like in the natural infection.

A. Pugliese, A. Gandolfi/Mathematical Biosciences 214 (2008) 73–80

79

Page 8

The boundaries of the bounded invariant set B are then the two

arcs of trajectory from ðM;gÞ to ðxM;gðxMÞÞ, and then from

ðxM;gðxMÞÞ to ðxC;fðxCÞÞ, and the segments fðx;fðxCÞÞ; x 2 ½0;xC?g,

fð0;yÞ; y 2 ½g;fðxCÞ?g, fðx;gÞ; x 2 ½0;M?g.

The solution of (6) starting from ðx0;gÞ [where x0> 0 is arbi-

trary] remains inside B, hence is bounded.

Assume now a >1

ing from ðx0;gÞ with x0> 0 to be chosen later. From the second of

(6), we obtain

c? 1 and bs> 0. Take the solution of (6) start-

yðtÞ 6

g

1 ? ce

1

c?1

ðÞ:

ðA:5Þ

Then we have

_ x P ax ?

g

bsð1 ? cÞe

1

c?1

ðÞ?m

bu

;

ðA:6Þ

which implies

0

xðtÞ P

x0?

g

ð1 ? cÞbsa ?1

cþ 1

???m

abu

@

1

Aeat:

ðA:7Þ

If x0is chosen large enough that the coefficient of eatin (A.7) is po-

sitive, (A.5) and (A.7) together imply

yðtÞ 6

g

1 ? c

xðtÞ

g

ð

x0?

ð1?cÞbsa?1

cþ1

Þ?m

abu

0

@

1

A

1c?1

a

:

ðA:8Þ

Since1

ough, yðtÞ < gðxðtÞÞ so that we obtain a solution lying at all times

in the set fy < gðxÞg so that _ x > 0 and _ y > 0. Furthermore, the

inequality _ x 6 ax implies global existence, and consequently from

(A.7) xðtÞ ! þ1 and

c? 1 < a and bs> 0, this inequality implies, for x0 large en-

lim

t!1xðtÞ ¼ lim

t!1yðtÞ ¼ þ1:

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