www.biomathforum.org/biomath/index.php/biomath

REVIEW ARTICLE

Coupling Within-Host and Between-Host
Infectious Diseases Models

Maia Martcheva∗, Necibe Tuncer†, Colette M St. Mary‡
∗Department of Mathematics, University of Florida,

Gainesville, maia@ufl.edu
†Department of Mathematics, Florida Atlantic University,

Boca Raton, ntuncer@fau.edu
‡Department of Biology, University of Florida,

Gainesville, stmary@ufl.edu

Received: 6 February 2015, accepted: 9 October 2015, published: 30 October 2015

This article is dedicated to Tanya Kostova’s
anniversary birthday

Abstract—Biological processes occur at distinct
but interlinked scales of organization. Yet, math-
ematical models are often focused on a single
scale. Recently, there has been a significant in-
terest in creating and using models that link the
within-host dynamics and population level dynamics
of infectious diseases. These types of multi-scale
models, called immuno-epidemiological models, fall
in four categories, dependent on the type of the
epidemiological component of the model: network
or individual based models (IBM), “nested” age-
since-infection structured models, ordinary differen-
tial equation (ODE) models, and “size-structured”
models. Immuno-epidemiological multi-scale models
have been used to address a variety of questions,
including what is the impact of within-host dynamics
on population-level quantities such as reproduction
number and prevalence, as well as questions related
to evolution of the pathogen or co-evolution of the
pathogen and the host. Here we review the literature
on immuno-epidemiological modeling as well as the
main insights these models have created.

Keywords-within-host models, between-host mod-

els, immuno-epidemiology, mathematical models,
differential equations, reproduction number, evolu-
tion.

AMS SUBJECT CLASSIFICATION: 92D30,
92D40

I. INTRODUCTION

Biological processes occur at nested scales of
organization. In infectious diseases, the dynamical
interplay between the microparasite and the host
immune system has a strong impact on the epi-
demiological characteristics of the disease, such
as pathogen shedding, population level transmis-
sion, disease-induced host mortality and recovery.
Yet, traditionally, differential equation modeling
of infectious diseases has been strictly separated
by biological scale of organization. Within-host
modeling of infectious diseases has been drawing
significant attention in the last century. Simple
differential equation models, developed to describe
a number of diseases such as HIV, HCV, malaria,

Citation: Maia Martcheva, Necibe Tuncer, Colette M St. Mary, Coupling Within-Host and
Between-Host Infectious Diseases Models, Biomath 4 (2015), 1510091,
http://dx.doi.org/10.11145/j.biomath.2015.10.091

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M. Martcheva et al., Coupling Within-Host and Between-Host Infectious Diseases Models

flu and others, have lead to dramatically improving
our understanding of how microscopic processes
develop and affect the host health. Several books
are devoted to within-host (immunological) mod-
eling of infectious diseases [63], [50], [40] and
multiple articles develop and use such models to
answer an array of biological questions regarding
the pathogen and its interplay with the immune
system and target cells. At the same time numerous
differential equation models have been developed
to model the dynamics of specific diseases or
in general the distribution of pathogens on the
population level. Between-host (epidemiological)
models have addressed a variety of questions re-
lated to public health. Multiple books focus on
the contributions of mathematics to epidemiology
([10], [20], [18], [33] just to mention a few). Some
important public health questions that can be ad-
dressed with epidemic models include the fraction
of the population that needs to be vaccinated to
eradicate a disease, the reproduction number of
various disease outbreaks and what efficacy do
control measures have.

Pathogen reproduction, transmission and evo-
lution are processes that span several scales of
biological organization, i.e. intracellular, within-
host, and population scales. Answering effectively
public health questions at the population level
often requires the understanding and the “lift-
ing” of processes from within-host levels to the
population level. Unfortunately, very rarely do
mathematical models encompass multiple scales
of biological organization. Here we will review
the relatively recent models linking the within-
host scale with the between-host scale. The emer-
gent area of linked data, models and knowl-
edge is called immuno-epidemiology. Hellriegell
defines immuno-epidemiology as the area that

“combines individual- and population-oriented ap-
proaches to create new perspectives” [30]. We
define the mathematical immuno-epidemiology as
the area in which mathematical dynamical models
of within-host disease processes are interlinked
to population-level dynamical models of disease
spread to allow for novel results. The concept of
immuno-epidemiology is not new but in the last
half of the 20th century it was primarily linked to
the interaction of immunology and epidemiology
of macroparasitic diseases and malaria [27], [54],
[28]. Until the 21st century little had been done
linking immunological and epidemiological ideas
in microparasitic diseases, such as viral pathogens.

Why is it important to develop and study multi-
scale infectious disease models? (1) Because im-
munological considerations predict important epi-
demiological determinants, such as disease preva-
lence and reproduction number [19]. Such impor-
tant epidemiological quantities can be explicitly
related (for arbitrary parameter values) to host
pathogen load and immune responses. (2) Data
exist on both scales; linking them is essential
for quantitatively coupling processes across scales.
Furthermore, the biological understanding gained
from more data will be more comprehensive and
accurate. (3) Incorporating explicit immune re-
sponses is important in diseases, such as dengue,
where disease severity depends on the strength of
this response [14]. (4) These models are essential
for elucidating the role of within-host disease
dynamics for pathogen evolution [47], [15].

In this paper we review models linking within-
host and between-host processes and some of the
insights that have resulted from them. First, we
detail some immunological models used in the
coupled frameworks. The variability of immuno-
epidemiological modeling techniques stems from

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the various types of epidemiological models from
which they have arisen. There are four basic types
of epidemiological models being employed: net-
work or individual based epidemiological models,
reviewed in subsection 3.A, ODE epidemiolog-
ical models, reviewed in subsection 3.B, size-
structured epidemiological models, reviewed in
section 3.C, and age-since-infection structured epi-
demiological models, reviewed in subsection 3.D.
Finally, in section 4, we provide further discussion
of the immuno-epidemiological modeling and its
implications to biology.

II. WITHIN-HOST MODELS

Within-host models are dynamical models that
represent, in caricature, the interaction of the
pathogen with the host replication machinery or
immune defenses within a single host individual.
Roughly speaking these models can be classified
into three groups: models that depict the repro-
duction processes of the pathogen within the host;
models that depict the pathogen with the immune
responses; and models that include both the repli-
cation of the pathogen and the immune responses.
The simplest within-host models are of the first
two types. We introduce here an example of each
of the first two types as these types are the ones
typically used in linked models.

A. Within-host model of the pathogen replication
cycle

Within-host models that represent the pathogen
replication cycle assume a typical viral pathogen
that replicates using the machinery of host cells,
called target cells. To introduce such a model,
let x(τ) be the number of pathogen-free (healthy)
target cells (in the blood) and y(τ) be the number
of infected target cells. The amount of pathogen
is denoted by P(τ). The within-host replication

Fig. 1. The dynamics of the pathogen and target cells.
Parameter values are: r = 50000000, b = 0.000000000015,
µ = 0.01, d = 0.5, δ = 3, ν = 250, s = 0.00008.

model has been used for many viral diseases
before [50].

x′ = r − bPx−µx,
y′ = bPx−dy,
P ′ = νdy − (δ + s)P,

(II.1)

where r is the replication rate of target cells, b is
the infection rate, µ is the clearance rate of healthy
cells, d is the clearance rate of infected cells,
ν is the number of pathogen particles released
from lysis of an infected cell, δ is the clearance
rate of pathogen, and s is the shedding rate. The
dynamics of the pathogen and target cells is shown
in Figure 1.

Model (II.1) has been completely analyzed [17].
The reproduction number of the pathogen is given
by

<0 =
rνb

µ(δ + s)
. (II.2)

The model has two equilibria, an infection-free
equilibrium E0 = (r/µ, 0, 0) and an infection
equilibrium E∗ = (x∗,y∗,P∗) where

x∗ =
δ + s

νb
, y∗ =

µ(δ + s)

νbd
(<0 − 1) ,

P∗ =
µ

b
(<0 − 1) .

(II.3)

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B. Models with immune response

The adaptive immune response includes a cel-
lular component which includes various types of
T cells and humoral response which consists of B
cells and antibodies. Simple pathogen-immune re-
sponse models typically include only the pathogen
P(τ) and one type immune response cells, B-
Cells, B(τ). The pathogen replicates according to
the Malthus model or logistic model. B cells kill
the pathogen and B cell production is stimulated
by the pathogen. B cells are cleared at rate d:

P ′ = rP

(
1 −

P

K

)
− �PB,

B′ = aP −dB,
(II.4)

where r is the parasite growth rate, K is its
carrying capacity, � is the killing rate of the
immune response, a is the activation rate of the
immune response and d is the clearance of the
immune response. The dynamics of model (II.4)
is illustrated in Figure 2.

Fig. 2. The dynamics of the pathogen and the immune
response. Parameter values are r = 1, K = 1000, � = 0.1,
a = 0.2.

The model has two equilibria: (0, 0) which is
always unstable and a coexistence equilibrium
(P∗,B∗) where

P∗ =
rdK

�aK + rd
, B∗ =

raK

�aK + rd
. (II.5)

Immunological models vary immensely in com-
plexity and detail. Differences also stem from the
particular processes in the disease being modeled.
For our purposes here these two simple within-host
models are sufficient to illustrate the concepts. In
the next section, we introduce various epidemio-
logical models.

III. IMMUNO-EPIDEMIOLOGICAL MODELS

The epidemiological component of the immuno-
epidemiological models can take several different
forms based on existing modeling frameworks, or
just epidemiologically relevant quantities, such as
the reproduction number, expressed in terms of
the immunological variables. In this section we
consider four types of immuno-epidemiological
models structured by the type of epidemic model.

A. Network epidemic models

In network epidemic models, individuals are
nodes in a network. Each individual or node
can exhibit its own within-host dynamics (see
Figure 3). The models can show the impact of
the individual immune dynamics on population-
level transmission of disease. Tucknell [57] and
Kostova [34] introduced some of the first immuno-
epidemiological models where the epidemiological
component is a network. Kostova linked a network
of n within-host models and showed that even if
the immune response clears the infection in each
individual when isolated, while these individuals
are in a network, the pathogen persists in each
one of them and on a “population level”.

Vickers and Osgood [58] suggest that increased
variance among people’s ability to respond to an
infection, while maintaining the average immune
responsiveness, may worsen the overall impact
of an outbreak within a population. Furthermore,
high values for the network connectivity reduced

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Susceptible Individuals

Infected Individuals

Immune Response

Immuno-Epidemiological Network Model

1

Fig. 3. Network immuno-epidemiological schematic dia-
gram.

the timing between peak viral levels in neigh-
boring individuals. A network based immuno-
epidemiological model was applied by Vickers
and Osgood [59] to study treatment of chlamydia
which suggested that treatment applied up to the
third day post infection has significant chance
of preventing transmission of the disease to the
nearest neighbor. Delivering treatment past the 3rd
day post infection allows for infection of nearest
neighbor as well as reinfection.

Lukens et al. [37] use a simple ODE model of
influenza A and link that model, through infec-
tivity, to large-scale agent-based population-level
model to study influenza A epidemics. The authors
obtain a map of the parameters of the immune
model that characterizes clinical phenotypes of
influenza infection and immune response vari-
ability across the population. At the population-
level, effectively the authors simulate epidemics
in Allegheny County, Pennsylvania and consider
both age-specific and age-independent severity as-
sumptions.

One of the serious drawbacks of network and

agent-based immuno-epidemiological models is
that very few population level quantities that de-
scribe the disease distribution can be computed
analytically in closed form. In particular, in net-
work models there are difficulties for computing
the reproduction number and the prevalence of the
disease. Consequently, little can be learned of the
effect of the immune response on these quantities
outside of extensive simulations [52].

The next three classes of immuno-
epidemiological models remedy this shortcoming
but some of them assume that all infected
individuals undergo the same within-host
dynamics, an assumption that is largely unrealistic.
Still these models have contributed immensely to
our further understanding of the mutual impact of
within-host and between-host processes.

B. ODE immuno-epidemiological models

One way to obtain a simpler ODE immuno-
epidemiological model for chronic diseases is to
consider the immune model at infectious equilib-
rium. In this case one can make the parameters
of a simple ODE epidemic model dependent on
the equilibria values of the pathogen and/or the
immune response. For instance, a simple SI model
becomes:

S′ = Λ −β(P∗)SI −m0S ,
I′ = β(P∗)SI − (m0 + m1(P∗,B∗))I ,

(III.1)
where P∗ and B∗ are given by (II.5) and m0 is
the natural death rate, m1 is the disease-induced
death rate, β is the transmission rate, and Λ is
the recruitment rate. One can then investigate how
within-host parameters, pathogen load and im-
mune response affect the epidemiological quanti-
ties, such as disease-induced mortality, prevalence
and reproduction number. These conclusions are
not necessarily equivalent to conclusions obtained

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Fig. 4. Immuno-epidemiological modeling with environmen-
tal transmission.

from other types of immuno-epidemiological mod-
els such as the nested models considered below.

Another opportunity to connect the within-
host and between host dynamical systems in an
ODE model emerges in environmentally trans-
mitted disease. In this case a chronological-time-
structured within-host ODE system is linked to
a chronological-time-structured ODE epidemio-
logical system through the pathogen load in the
environment (see Figure 4).

Using this novel modeling scenario Feng at
al [12], [21], [22] investigate the transmission of
Toxoplasma gondii. Feng at al [21] link a dynamic
within-host model of the type (II.1), where the in-
fection of target cells depends on population-level
prevalence I, with an SI epidemic model much
like (III.1), where transmission depends on viral
load. Within-host and between-host reproduction
numbers are computed but for the linked model
analysis suggests that long-term behavior of the

infection may depend on the initial conditions.
Articles [12], [22] contain the environment as
an explicit variable and find that infection may
persist on the population level even if the isolated
between-host reproduction number is less than
one; a result that is facilitated by the within-host
dynamics.

C. Size-structured PDE immuno-epidemiological
models

Size-structured PDE immuno-epidemiological
models are perhaps the most complex type of
immuno-epidemiological models. In this case the
epidemiological model consists of physiologically
structured PDEs in which the structural indepen-
dent variables are the dynamical variables of the
ODE immune model. The first such model was
proposed by [45] (see also [44]) where the “size
structure” variable is the immune response. Analy-
sis of this model revealed similarities to age-since-
infection structured models particularly because
the structural variable is strictly increasing in time.
This model was further extended by [23] to model
where the population-level density of infected indi-
viduals is structured by both the viral load and the
immune response. [23] also transforms the size-
structured model to an age-since-infection model
where the independent variables are age-since-
infection and initial pathogen inoculum. Consid-
ering a specific within-host model that allows for
both pathogen extinction and unbounded growth,
the authors investigate the population level impact
of the initial inoculum and of the isolation thresh-
old.

A somewhat different modeling approach to the
same modeling scenario is given by [6], [7], [48].
The authors suggest coupling a classical HIV/HCV
within-host model, given by equations (II.1) with
a size-structured SIR epidemic model. The authors

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establish well-posedness in the special case when
the density of infected individuals is structured
only by the viral load. Furthermore, they develop
numerical methods and discuss the impact of the
(fixed) number of target cells and the burst size
on the epidemic [6], [7]. The results allow them
to determine the distribution of the density of the
infected individuals by their viral load.

In general, the size-structured approach presents
interesting mathematical challenges, such as the
potential for measure-valued solutions, modeling
issues related to creating and linking the models,
computational issues with the large number of
independent variables [6], [7], [48]. Because of
these issues, creating the within-host model is
an important step. Furthermore, the computational
and analytical problems with the large number
of independent variables somewhat restrict the
incorporation of significant realism into the im-
mune system; a problem that can be addressed by
incorporating fewer within-host model dependent
variables as independent variables in the epidemi-
ological model. There are still a lot of interesting
and open questions, related to this approach.

D. Nested models

Nested models are a relatively recent class of
models suggested for the first time by Gilchrist
and Sasaki [25]. The main advantage of nested
models relative to size-structured models is that
these models allow for the use of very realistic
and specific to a given disease models. Establish-
ing well-posedness for these models is generally
not very problematic [51]. Furthermore, since the
number of independent variables in the PDE part
of the model is restricted to two, there are few
difficulties with computation, independent of the
complexity of the immunological or epidemiolog-
ical components.

Because of their advantages, since the Gilchrist
and Sasaki paper, nested models have acquired
significant popularity. The nested models “nest”
a time-post-infection structured immune dynamics
model into a time-post-infection and chronologi-
cal time structured epidemiological model. Nested
models also link the within-host model with the
epidemiological model through the parameters of
the epidemiological models that are expressed in
terms of within-host dependent variables. To in-
troduce a simple nested immuno-epidemiological
model, let S(t) denote the number of susceptible
humans and i(τ,t) be the density of infected
humans. In the simplest case, we can use an SI
epidemiological model. The model takes the form:

S′ = Λ −S
∫ ∞
0

β(τ)i(τ,t)dτ −m0S,

iτ + it = −(m0 + m1(τ))i(τ,t),

i(0, t) = S

∫ ∞
0

β(τ)i(τ,t)dτ ,

(III.2)

where m0 is the natural death rate, m1 is the
disease-induced death rate, Λ is the recruitment
rate and β is the transmission rate. Model (III.2)
can be linked with either of the within-host mod-
els. The transmission rate β(τ) depends on the
pathogen load. Experimental evidence suggests
that the transmission probability does not increase
linearly with the pathogen load but in a saturating
fashion [35]. We will use the following simple
function of saturating growth:

β(τ) =
cP(τ)

Q + P(τ)

where c is the contact rate and Q is the half-
saturating constant. The disease-induced mortality
can be linked in multiple ways to the immune
system. It is thought that two distinct processes
lead to disease-induced mortality in the host. On
one side is the pathogen itself, and on the other is

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the immune response. We take the disease-induced
mortality generated by the pathogen proportional
to the pathogen load. The disease-induced mortal-
ity generated by the immune response has been
taken to be proportional to the growth of the
immune response aBP [25], [53]. We take here
the disease-induced mortality proportional to B2.
The square guarantees that at low values of B,
the immune response almost has no impact on the
disease-induced mortality while at high levels of
B, it has significant impact. This way a trade-
off exists between the necessity of the immune
response to be vigorous enough to clear the virus,
but not too vigorous to kill the host. The disease-
induced mortality is then given by

m1(τ) = ν1rP(τ) + ξ1B
2

where ν and ξ are constants of proportionality.
ξ1 = 0 if we are working with immune model
(II.1).

One of the main disadvantages of the nested
immuno-epidemiological models relative to the
network and size-structured models is that they as-
sume that all individuals exhibit the same immune
dynamics. To remedy this disadvantage one may
consider a multi-group immuno-epidemiological
model where the different groups exhibit differ-
ent immune dynamics. The multi-group model is
somewhat complicated and obtaining analytical
results on it is not easy. This weakens one of
the great advantages of the nested models, namely
that basic epidemiological quantities, such as the
reproduction number and the prevalence, can be
computed in analytical form. A schematic diagram
of nested models is given in Figure 5.

The reproduction number of the immuno-
epidemiological model (III.2) depends on the

Between Host Model

Susceptible Population

S(t)

Infected Population

i(t, ⌧)

Pathogen level in host (P (⌧))

Antibody level in host (B(⌧))
Within Host Model

Natural Deaths

Births

Natural Deaths
+ Disease In-
duced Deaths

Transmission

1

Fig. 5. Nested immuno-epidemiological schematic diagram.

within-host variables and parameters:

R0 =
Λ

m0

∫ ∞
0

β(τ)e−m0τe−
∫

τ

0
m1(σ)dσdτ.

(III.3)
There are two types of questions being ad-

dressed with nested models: How does the within-
host pathogen dynamics affect the population-level
reproduction number and prevalence? What are the
evolutionary and co-evolutionary consequences of
the pathogen and host within-host evolution?

Several articles have suggested that the de-
pendence of the reproduction number R0 on
the pathogen reproduction rate r may be non-
monotone [31], [53], [43]. As the within-host
pathogen reproduction rate r grows, it should
be increasing the population-level reproduction
number R0; however the increased pathogen load
is increasing host mortality, which in turn leads
to decrease in the population-level reproduction
number. This creates a hump-shaped form of R0
as a function of r, which is well-described in the
literature. The dependence of R0 on immunologi-
cal parameters has been further discussed in [32],
[41], [34], [16], [8]. The dependence of prevalence
on some of the within-host parameters may also
be counter-intuitive. For instance, prevalence may
decrease with increase of b in model (II.1) [42].
This in turn implies that within-host medications

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that lower the infection of target cells, that is
decrease b, de facto increase the population-level
prevalence of the disease. This paradox has been
observed in practice in HIV, where medications
lower within-host virus load and increase sur-
vivability of infected individuals which leads to
increasing prevalence. Furthermore, amplification
of the HIV epidemic has been observed through
nested models [55]. More mathematical questions
related to immuno-epidemiological models, such
as well-posedness and optimal control are ad-
dressed in [51].

The importance of multi-scale immuno-
epidemiological modeling is best highlighted
by its role in studying evolution [47]. Gilchrist
and Sasaki were among the first to address
co-evolution [25] using multi-scale approaches
but since then evolution of virulence has been
attracting significant attention. Because evolution
involves multiple interacting strains, a number
of approaches have been developed to handle
the emergent complexity [3], [4], [5], [13],
[15], [24], [26], [36], [38]. One possible way
is to model the strains explicitly on within-host
and/or between-host scales [42], [13]. In the
absence of trade-off mechanisms, the strain that
maximizes its between-host epidemiological
reproduction number dominates; a result first
established rigorously mathematically in [11]. In
the presence of trade-offs there is coexistence
between the strains and invasibility is governed by
population-level invasion numbers. Nested models
further reveal [2] that the optimal virulence in
a co-infection model increases with multiple
infections and that in a linked within-host and
between-host co-infection model, an evolutionary
stable strategy (ESS) can turn into a branching
point [1]. An ESS is a strategy which, if adopted

Fig. 6. Pairwise invasibility plot.

in the population, cannot be invaded by any other
strategy. To see this from a pairwise invasibility
plot (PIP) (see Figure 6), we draw a vertical line
through the singular strategy and confirm that the
vertical line lies entirely in the non-invasibility
region.

Nested models have the potential to link
in a natural way the virus reproduction rate,
population-level fitness, and population level
disease-induced mortality. For HIV, the reproduc-
tion rates of the virus increase at a moderate
rate and the virulence is slightly higher than the
level that maximizes the population-level trans-
missibility of the virus [39]. For Hepatitis C,
slowly replicating strains have a higher fitness and
produce more population-level secondary infec-
tions while strains with higher replication rates
dominate within a host [38]. For Influenza A,
the relative importance of virulence and viral
clearance by the immune system on the viral
fitness and persistence was found to depend on
the temperature [29].

Day et al [15] develop the mathematical theory
that bridges the nested immuno-epidemiological
models to quantitative genetics and evolution of

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traits. Using this framework, the authors show that
the trade-off between transmission and virulence,
studied early on in multi-scale models in [24], is an
interplay of the genetic variation of the pathogen
and the population-level dynamics of the disease.
This framework is the backbone for future research
at the interface of dynamic population modeling
and quantitative genetic modeling.

Conclusions from multi-scale nested models
for chronic diseases can be derived by writing
the epidemiological quantities in term of the
infected equilibrium of the within-host model
[9], [26], [46]. This approach bridges to the
ODE immuno-epidemiological models, discussed
in section III.B. Following this approach article
[26] found that within-host selection favors viral
production rates ν that maximize virulence but
between-host pathogen fitness is maximized at
some intermediate virulence and viral production
rate. Article [9] extends the results in [26] by
incorporating superinfection.

IV. DISCUSSION

These new, more integrative modeling ap-
proaches, each have strengths and weaknesses
mathematically and in terms of their levels of
biological realism, but they have on the whole led
to new and biologically counter intuitive insights.
This is exemplified by even early examples of
these models, such as Kostova [34], which demon-
strated population-level disease persistence even
when individual immune responses are able to
clear infection and result in immunity. A second
example is the population level persistence of
virus, even when the between host reproduction
number is less than 1, as shown by Feng et al.[22].
This result leads to the surprising interpretation
that immune responses do not necessarily tip the
balance of interactions in favor of the host and

decrease disease prevalence. Similarly, drug ad-
ministration may mimic the host immune system
and increase disease prevalence (e.g., references
[42] and [54]).

At another level, when we focus on the evolu-
tion of virulence, immuno-epidemiological models
again lead to counter-intuitive results (e.g., [26]).
The results of these analyses, that selection within
an individual can favor different pathogen traits
than selection among individuals, highlight that
the within-host/among-host model structure char-
acteristic of these models meets the requirements
for trait-group selection to play a role in evolu-
tionary dynamics [60], where virus in individuals
constitutes trait groups from which virus emerge,
intermix and infect susceptible individuals. Wil-
son’s model has been criticized for unrealistic,
or at least uncommon, assumptions of popula-
tion structure (reviewed in [61]), however, infec-
tious disease may represent a common context in
which individual- and group-level selection both
act strongly and at times in conflict.

Ultimately, understanding the evolution of vir-
ulence might be informed by models designed to
understand trait evolution in the context of multi-
level selection such as those discussed in [56].
In general, because there are sometimes conflicts
between within-host and among-host virulence op-
tima, new insights will likely come from relaxing
the biologically-unrealistic, albeit simplifying, as-
sumption that virulence evolves to its within-host
optimum prior to the epidemiological dynamics of
the model. Again, modeling approaches appropri-
ate for relaxing this assumption might be informed
by the body of literature focused in the modeling
of group and individual level selection, especially
those approaches that consider the continuous
nature of the dynamics of within-individual and

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http://dx.doi.org/10.11145/j.biomath.2015.10.091


M. Martcheva et al., Coupling Within-Host and Between-Host Infectious Diseases Models

among-individual processes [56].

ACKNOWLEDGMENTS

The authors acknowledge support from NSF
grants DMS-1515661/DMS-1515442.

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	Introduction
	Within-host Models
	Within-host model of the pathogen replication cycle
	Models with immune response

	Immuno-epidemiological models
	Network epidemic models
	ODE immuno-epidemiological models
	Size-structured PDE immuno-epidemiological models
	Nested models

	Discussion
	References