Bulletin of Mathematical Biology (2007)
DOI 10.1007/s11538-007-9223-z
O R I G I N A L A RT I C L E
Modeling Within-Host Evolution of HIV: Mutation,
Competition and Strain Replacement
Colleen L. Balla , Michael A. Gilchristb , Daniel Coombsa,∗
a
Department of Mathematics and Institute of Applied Mathematics, University of British
Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
b
Department of Ecology and Evolutionary Biology, University of Tennessee, Knoxville,
TN 37996, USA
Received: 25 August 2006 / Accepted: 25 April 2007
© Society for Mathematical Biology 2007
Abstract Virus evolution during infection of a single individual is a well-known feature
of disease progression in chronic viral diseases. However, the simplest models of virus
competition for host resources show the existence of a single dominant strain that grows
most rapidly during the initial period of infection and competitively excludes all other
virus strains. Here, we examine the dynamics of strain replacement in a simple model
that includes a convex trade-off between rapid virus reproduction and long-term host cell
survival. Strains are structured according to their within-cell replication rate. Over the
course of infection, we find a progression in the dominant strain from fast- to moderatelyreplicating virus strains featuring distinct jumps in the replication rate of the dominant
strain over time. We completely analyze the model and provide estimates for the replication rate of the initial dominant strain and its successors. Our model lays the groundwork
for more detailed models of HIV selection and mutation. We outline future directions and
application of related models to other biological situations.
Keywords Virus dynamics · Selection-mutation models · HIV evolution · Virus fitness
1. Introduction
The replication of certain viruses is known to be highly error prone. During long-term
viral infections (caused, for example, in humans by HIV, hepatitis B, hepatitis C, herpes
simplex viruses and human papillomaviruses), where the virus undergoes many cycles
of replication during infection, we therefore expect that significant genotype changes
will occur. In the case of HIV, a point mutation occurs with probability 0.25 during every cycle of replication (Mansky and Temin, 1995) and it is estimated that 109
such cycles occur per day within a single untreated individual (Preston et al., 1988;
∗ Corresponding author.
E-mail address: coombs@math.ubc.ca (Daniel Coombs).
C.L. Ball et al.
Roberts et al., 1988). Shankarappa et al. (1999) tracked viral evolution of nine HIVinfected men over a 6–12 year period. The number of mutations was found to increase
linearly with a slope of approximately 1 point mutation every 2 months (see Kelly et al.,
2003 for further discussion). Further, since HIV is diploid, recombination can occur during reverse transcription. The rate of recombination was estimated to be around 10–15
times higher than that of point mutations (Jetzt et al., 2000). However, for recombination
to be a source of variation, it is necessary that cells be multiply infected. This appears to
be the case for a substantial fraction of cells infected with HIV (Jung et al., 2002) and in
silico work indicates that recombination contributes substantially to the rate of genotypic
variation during infection (Bocharov et al., 2005).
Rapid genotypic variation also appears to lead to substantial variation in phenotype.
In vitro competition assays between HIV samples drawn at different times from the same
patients show a general increase in competitive fitness (Troyer et al., 2005). A particular
phenotype change during the course of infection is the coreceptor switch. Early-stage
disease is usually marked by a predominance of virions that bind the CCR5 coreceptor on
host cells. In about half of all patients, a strain of virus binding CXCR4 on target cells (or
both CCR5 and CXCR4) dominates later on (Regoes and Bonhoeffer, 2005). This switch
impacts on in vitro competitive fitness (Troyer et al., 2005; Arien et al., 2006).
Point mutations in an envelope protein cause the switch to occur (De Jong et al., 1992;
Jensen et al., 2003). Analyses of fitness variations due to point mutations in the gene coding for reverse transcriptase are given in Goudsmit et al., Goudsmit et al. (1996, 1997).
A more recent study examining the effect of random mutations in noncoding, DNAbinding sites of the transcriptional promoter was performed by van Opijnen et al. (2006).
Similarly, point mutations in a single gene have been shown to drastically reduce or eliminate virulence in the case of herpes simplex virus 1 (Brandt, 2005). These few examples demonstrate rapid variation of phenotype in chronic viral infections. Given the long
course of HIV progression, we see that the ecological and evolutionary timescales are not
separable (Kelly et al., 2003).
Chronic viral infections and in particular HIV have been studied using mathematical models for over 20 years (reviewed in Perelson and Nelson, 1999; Nowak and May,
2000). In this paper we will extend the basic model of HIV population dynamics to examine competition between virus strains within a single infected host. We will impose
a biologically natural trade-off between viral replication within host target cells and the
mortality rate of these cells, and allow for rapid generation of genotypic variation. We
show that rapid evolution of viral productivity is a general feature of such models and
discuss its particular application to HIV biology.
2. Competition between HIV strains at the within-host level
We begin with the now-standard model of HIV dynamics, designed to capture the dynamics that take place in an infected individual upon infection with HIV. We track populations
Modeling Within-Host Evolution of HIV: Mutation, Competition
of uninfected target cells T , free virions V , and infected target cells T ∗ :
dT
= λ − dT − kT V ,
dt
dT ∗
= kT V − μT ∗ ,
dt
dV
= pT ∗ − cV .
dt
(1)
Uninfected target cells are produced at a constant rate λ and die at a rate d. Free virions
infect uninfected target cells according to a mass-action law with rate constant k. Infected
target cells then produce new free virions at rate p and are killed with mortality rate μ.
Finally, free virions are cleared at rate c. We omit, as usual, the term for loss of a free
virion when it infects a cell (Perelson and Nelson, 1999; Nowak and May, 2000).
The dynamics of this system are controlled by the within-host basic reproductive number (Nowak and May, 2000)
ρ=
λpk
.
dμc
(2)
ρ gives the expected number of progeny virions produced by a single virion introduced
to a new host. If ρ < 1, then the uninfected equilibrium (T = λ/d, T ∗ = 0, V = 0) is
globally stable while if ρ > 1, then an infected steady state exists and is globally stable.
In an extended model where multiple strains of free virus compete to infect host cells, ρ
is also a useful measure of the relative fitness of competing virus strains in this model.
A virus with a given ρ cannot be invaded by a virus with a lower ρ at the infected steady
state, and always invades any such virus.
2.1. Burst size
A useful quantity implied by Eq. (1) is the viral burst size, which we denote by N . This
is the total number of virions produced during the lifetime of a single infected cell. In this
model, N = p/μ. ρ is proportional to N .
2.2. Reproduction-mortality trade-off
Expression (2) indicates that the most successful viruses reproduce quickly (high p) while
avoiding host cell death (low μ). As previously argued (Coombs et al., 2003; Gilchrist et
al., 2004), this dual optimization is likely to be impossible.
It is known that infected T cells have a much higher death rate than uninfected cells
due to the damaging effects of the virus on the cell. In general terms, the cell’s resources
are co-opted for the benefit of the virus, leading to an increased likelihood of apoptosis.
The HIV gene Tat (Trans-Activator of Transcription) may provide a specific link between
cell death and replication. Tat plays an essential role in viral replication and suppression
of Tat has been shown to significantly inhibit HIV replication in in vitro experiments
(Mhashilkar et al., 1995). Tat has also been shown to directly induce apoptosis and cells
expressing Tat have shown an increased sensitivity to apoptotic signals (e.g., Bartz and
Emerman, 1999; Westendorp et al., 1995).
C.L. Ball et al.
Additionally, as infected cells express viral peptides on their surface they become
targets for the immune system, leaving them open to attack by HIV-specific cytotoxic
CD8+ T cells. (Expression of the gene nef downmodulates MHC molecules on the cell
surface, reducing the visibility of the infected cell to cytotoxic T cells, but presumably allowing increased surveillance by natural killer cells.) The repertoire of peptides presented
on a cell surface has been experimentally shown (in a different system) to represent recently synthesized proteins, not total foreign protein load (Princiotta et al., 2003) and
therefore we will treat the cell death rate as depending on the viral replication rate rather
than the sum of viral production to date. Making this assumption also simplifies the mathematical model (details in Coombs et al., 2003).
To model the relationship between replication rate and cell death, we replace the constant infected cell death rate μ with a function depending on the viral replication rate
μ(p), where we take the possible range of p to be between 0 and some upper bound
pmax . μ(p) is presumably an increasing function with μ(0) = d. Within this model, the
viral fitness ρ is the same as given above, with μ(p) replacing μ (Gilchrist et al., 2004).
We observe that the burst size is now p/μ(p). In Coombs et al. (2003), we analyzed how
different functional forms of μ affect the fitness (in terms of burst size or, equivalently, ρ)
of viral strains structured by their production rates p. In the case of a linearly increasing
function, say μ(p) = d + ηp, we found that the fittest viruses were those with the maximum possible replication rate. The more interesting cases concerned increasing, strictly
concave up mortality functions (Fig. 1). In this case, an interior optimal production rate
can be found by differentiating N = p/μ(p) with respect to p and setting the derivative
to zero, achieving
p∗ =
μ(p ∗ )
.
μ (p ∗ )
(3)
The presence of interior optima given this trade-off between reproduction and target cell
mortality is reminiscent of results concerning the evolution of virulence. When a similar (concave-up) trade-off is imposed between disease transmissibility and disease vir-
Fig. 1 Concave-up trade-off function between viral replication rate p and cell death rate μ(p). p∗ is the
optimal viral replication rate using this function.
Modeling Within-Host Evolution of HIV: Mutation, Competition
ulence, a similar theory at the epidemiological level predicts the most successful disease has an intermediate level of virulence (e.g., Bremermann and Pickering, 1983;
Levin and Pimental, 1981; Anderson and May, 1983).
2.3. Virus mutation
Upon infection of a new cell, HIV releases its genetic information in the form of viral
RNA and then uses the enzyme reverse transcriptase to create a DNA copy of itself. The
DNA copy is integrated into host DNA and is then termed a provirus. The process of
producing provirus is extremely error-prone, with an estimated rate of point mutations
of about 0.25 per HIV genome per replication in addition to variation introduced by recombination. Once a provirus is established, it is transcribed by the more reliable host
machinery, eventually producing new virions.
Until fairly recently, it was thought that most infected T cells are singly infected, that
is, they contain a single provirus. However, it is now believed many infected cells contain multiple proviruses (Jung et al., 2002; Levy et al., 2005). This situation allows for
substantial mutation via recombination between RNA coming from different infecting
virions. Bocharov et al. (2005) showed, using a stochastic in silico model, that multiple
infection of host cells substantially increases the effective mutation rate to well above the
simple rate due to point mutations. The model of virus dynamics given above (1) was
extended by Dixit and Perelson (2005) to allow for multiple infections of target cells.
The model they derived was shown to yield the same population dynamics as model (1)
provided all strains had a constant burst size.
Our major modeling assumptions concerning mutation are as follows. (a) We ignore
multiple infection of target cells. This allows us to present an easily comprehensible model
and interpret our results without complications arising from multiple infection. We plan to
build on this simplified platform to include multiple infection in future work. (b) Because
the process of reverse transcription is responsible for most mutations, we will only consider mutation occurring during the process of reverse transcription, and, therefore, when
a mutation occurs we suppose that the entire infected T cell will change the strain of
viruses it produces. (c) We consider mutation only in one viral parameter, the replication
rate p. We suppose that the parameters λ and d, and the function μ represent mostly host
properties, so viruses are characterized by k, c and p. In the expression for ρ, a trade-off
gives an optimal, bounded value of p while k and c are optimal simply at their highest
possible values. Therefore, we fix k and c at physiological values. This focuses our attention on the effects of the trade-off in p. (d) We do not know in detail how mutations in the
genetic material of the virus will affect its properties from the point of view of model (1).
However, as discussed above, minor changes are known to radically affect behavior. We
therefore explicitly do not assume that single mutations lead exclusively to small changes
in phenotype.
2.4. Competition model
We now modify model (1) to allow for M strains of competing viruses. We suppose that
viral strains differ only in their reproduction rate pi . The mass-action infectivity constant
k and the viral clearance rate c are taken to be the same for each virus strain. Each strain
mutates at a constant rate , and mutates equally to each of the allowed virus strains. Note
C.L. Ball et al.
that actually represents the rate at which viable mutants are produced, rather than the
overall mutation rate. For mathematical convenience, we allow mutation to a virus with
identical characteristics in this model.
M
dT
= λ − dT − kT
Vj ,
dt
j =1
M
kT dTi∗
= k(1 − )T Vi − μ(pi )Ti∗ +
Vj ,
dt
M j =1
dVi
= pi Ti∗ − cVi ,
dt
(4)
i = 1, . . . , M.
If = 0, this system of equations is decoupled and viruses compete independently for
target cells. In this case, the infected steady state consists of viruses only of the strain of
highest ρ (expression (2)). For small , there is a single infected steady state containing
all strains, dominated by the strain(s) with highest ρ.
2.5. Parameters and functions
In the simulations below, the parameters λ, d, k and c are chosen based on estimates from
Stafford et al. (2000). We chose our cell death rate μ(p) to be an increasing concave-up
function. To create this function we considered the measured values from Stafford et al.
(2000). A wide range of production rates were observed and our choice of μ(p) attempts
to capture that. We choose μ(p) = d exp(φp) and choose φ so that the average estimated
cell death rate is achieved by μ(p) when p is at the average estimated production rate
(numerically, φ = 0.0043). Our range of production rates span the range where the virus
experiences positive initial growth in this model. Thus, we only consider virus strains
that would be capable of causing infection when introduced by themselves into a host.
The value of is difficult to estimate for this model; we only suppose 1. Numerical
values of all parameters are given in Table 1.
Table 1 Parameter values for the HIV model
Symbol
Description
Estimate
λ
d
k
μ(p)
φ
c
pmax
pmin
p∗
p∗∗
M
Birth rate of T cells
Death rate of uninfected T cells
Mass-action infectivity of free virions
Death rate of T cells infected with a virus of production rate p
Sensitivity of T cell death rate to viral production rate
Clearance rate of free virions
Maximum viral production rate yielding initial growth
Minimum viral production rate yielding initial growth
Optimal production rate at steady state
Production rate that maximizes growth during initial dynamics
Number of virus strains, varying in production rate
Viral mutation rate within infected T cells
0.1 cells/µL/day
0.01/day
6.5 × 10−4 /cell/day
deφp /day
0.0043 cell day/virion
3/day
1300 virions/cell/day
4 virions/cell/day
230 virions/cell/day
836 virions/cell/day
varies
varies
Modeling Within-Host Evolution of HIV: Mutation, Competition
3. Numerical results
3.1. Two-strain dynamics
Figure 2 shows the numerical solutions of a 2 strain model, with varying initial conditions.
We suppose that one strain has replication rate p ∗ and the other has replication rate 2p ∗ .
We see that the quickly replicating strain dominates initially, before being overtaken by
the optimally replicating strain. When we begin with an initial inoculum containing only
the fast strain we see that after an initial peak, the viral load levels out to a quasi-steady
state where it remains for about 1,000 days, before eventually being overtaken by the
p ∗ strain. When we begin with an initial inoculum containing only the p ∗ strain, the
mutant strain overtakes the V (p ∗ ) for a period of time before the p ∗ strain reinvades.
Because the system begins far from the stable infected equilibrium, the viral ρ defined
by expression (2) is not a useful guide to the initial dynamics (Lenski and May, 1994;
Lipsitch and Nowak, 1995). Also, we stress a limitation of our model: the second strain is
never completely removed from the population. Rather, it remains at a density of order (see below, Eq. (18)).
3.2. Many-strain dynamics
We now allow M to get large, as an approximation to a continuous range of production
rates. Figure 3 shows these results. When we begin with an initial inoculum containing
just one strain, we see this strain dominate initially before observing a large jump in
production rate to an intermediate rate of production that is greater than p ∗ . The dominant
strain then slowly evolves toward p ∗ , each strain being replaced by the strain with the next
largest production rate. These replacements occur more and more slowly the closer p gets
to p ∗ . We observe the jump for any choice of initial production rate, provided we begin
with a single-strain inoculum. The qualitative features of the dynamics do not depend
on .
Fig. 2 Viral strain loads plotted against time. The simulations were performed using two strains (with production rates respectively p∗ = 230 and 2p∗ = 460 virions/cell/day) and a mutation rate = 10−5 . Other
parameters are given in Table 1. (a) Simulation performed using an initial inoculum of 100% 2p∗ -strain.
(b) Simulation performed using an initial inoculum of 100% p∗ -strain. Note that all strains are present at
all times, at levels below resolution of the figures.
C.L. Ball et al.
Fig. 3 Evolution of viral production rate for M = 10 and M = 100 strains. We plot contours of strain
concentration over time followed by the production rate of the most abundant virus strain at each point
in time. Here we take = 10−5 and the single initial strain has p = 873 virions/cell/day. Other parameters: pmin = 4, p∗ = 230, p∗∗ = 836, pmax = 1300 virions/cell/day. (a) M = 10 with contour interval
30 virions/µL; (b) M = 100 with contour interval 5 virions/µL. The contours increase from zero on both
contour plots. Note that all strains are present at all times, at levels below contour resolution of the figures.
3.3. First dominant strain
Figure 4 shows results of simulations performed with an initial inoculum containing just
one strain. We plot the replication rate of the strain that achieves the initial “spike” in concentration. Over a wide range of p-values of that strain, we find that the introduced strain
is the dominant strain during the initial dynamics (the part of the graph corresponding to
a 45◦ line). However, for a range of lower- and higher- replication rate strains, another
strain arises via mutation and dominates the initial dynamics. The p ∗ strain is among this
group, in line with Fig. 2b. The same basic result holds for M = 10 and M = 100.
The key features of the model that we wish to analyze are therefore the early dominance of a single non-optimal (i.e., non p ∗ ) strain, the abrupt jump in the dominant strain
to a considerably lower production rate, and the slow transition to the p ∗ or within-host
optimal state. These features were found in outline in an analogous model (at the epidemic scale, with an imposed trade-off between transmissibility and virulence) by Lenski
and May (1994). The main results of that paper supposed a separation of the ecological
and evolutionary timescales. As indicated above, that assumption does not apply to the
within-host evolution of chronic viruses.
Modeling Within-Host Evolution of HIV: Mutation, Competition
Fig. 4 We plot the production rate of the first strain to dominate when the initial infection is of a single
strain. (A) M = 10 strains. (B) M = 100 strains. Parameters were as Table 1 except /M = 10−6 . Strain
production values are evenly distributed between pmin and pmax in both cases.
4. Model analysis
4.1. Dynamics about the infection-free equilibrium
We would like to understand how the initially dominant strain is selected. To do so, we
perform a linear stability analysis of system (4) about the uninfected equilibrium (T =
λ/d, with all Vi = 0 and Ti∗ = 0) . We expand in powers of the small parameter (see
Appendix A). We find a single potentially unstable eigenvalue δ+ (pi ) defined by
1
pi kλ
−(μ(pi ) + c) + (μ(pi ) + c)2 − 4 μ(pi )c −
.
δ+ (pi ) =
2
d
(5)
Strains for which this eigenvalue is positive will (after a short transient) grow exponentially:
Vi (t) ≈ Vi (0)eδ(pi )t .
(6)
The condition for linear instability of the uninfected steady state is therefore
pi kλ > μ(pi )c d
(7)
for any one of the virus strains present in the host, corresponding to ρ > 1 in Eq. (2).
To leading order and for small times, the strains grow independently and a strain that
was not initially present will not grow. However, because of continuous mutation, every
possible strain will be introduced in small quantities as soon as a host is infected with
any virus strain. While the strain (or strains) that initially infected the host will clearly
have an advantage, it is possible that one of its mutants will out-compete the initial strain,
provided the mutation rate is high enough and the growth rate of the mutant strain is large
enough. In Appendix B, we provide an approximate analysis for the competition between
a single initially-present strain and its mutants, during the early phase of the infection.
C.L. Ball et al.
4.2. Optimal production rate during initial infection
To determine the fastest-growing strain during the initial infection we simply need to find
the production rate, denoted by p ∗∗ , that maximizes Eq. (5). Optimizing (5) yields the
following expression for p ∗∗ :
p ∗∗ =
kλ
μ(p ∗∗ ) − c
+
.
μ (p ∗∗ )
d(μ (p ∗∗ ))2
(8)
The second derivative test shows, after some algebra, that this value of p ∗∗ is a maximum
provided μ (p ∗∗ ) > 0.
It can be shown (Appendix E) that p ∗∗ is unique and always greater than p ∗ . The
biological interpretation of this is that during the initial dynamics uninfected T cells are
abundant and virus strains that quickly kill their host cells can still easily find a new host
to infect. At steady state, however, T cells are much more limiting, and the virus that
optimizes the number of virions an infected T cell produces will have the competitive
advantage. However, we note that p ∗∗ is not, in general, the most quickly replicating
strain. It is dependent upon the exact form of μ(p) as well as the uninfected steady state
level of T cells. As that background level of T cells increases, p ∗∗ also increases. We
also observe (Fig. 4 and Appendix B) that the p ∗∗ strain does not necessarily replace all
possible inoculum strains.
4.3. Graphical interpretation of p ∗ and p ∗∗
It was shown in Coombs et al. (2003) that provided μ (p) is positive and bounded away
from zero, p ∗ can be found easily. On a graph of μ(p), the point (p ∗ , μ(p ∗ )) can be
found by rotating a ruler fixed at the origin counterclockwise until it just touches the
graph (Fig. 1). This is equivalent to finding the contour of N = p/μ(p) that is tangent to
the graph of μ(p). p ∗∗ is found by a slightly more difficult procedure that immediately
shows that p ∗∗ > p ∗ . The contours of δ are straight lines in the μ–p plane defined by
μ = (kλ/(c + δ)d)p − δ.
(9)
Note that δ = 0 corresponds to ρ = 1. p ∗∗ is then found at the contour of δ that is tangent
to the graph of μ(p) (Fig. 5).
4.4. Replacement strain selection
We have established that a unique virus strain will expand most rapidly at the beginning
of the infection. The numerical solutions presented in Fig. 3 indicate that this initially
dominant strain abruptly loses out to another strain, which is not the within-host optimal
p ∗ strain. We now present a simple analysis of these strain replacement dynamics and
show how the second strain is selected among all present strains. Our approximate theory
expands upon the method of Lenski and May (1994).
If = 0, then there are M infected steady states, one corresponding to each virus
strain existing alone (arising from initial conditions where only that virus is present, for
example). For small it can then be shown that the dynamics near these states will be slow.
This is borne out by numerical studies, where we observe that the initially dominant strain
Modeling Within-Host Evolution of HIV: Mutation, Competition
Fig. 5 Graphical interpretation of p∗∗ . See text for details.
(with production rate p ∗∗ above) successfully out-competes its mutant strains, peaks and
then decays to a particular level that it only very slowly decays from. This level is close
to the steady state level of this strain in the model without mutation. We now explore the
rate of decay of the initial strain and the invasion rates of its mutants, and show how the
next successful invader is selected amongst the competitors.
We will call the strain currently established in the host the resident strain, denoted by
VR , and we will denote the invading strain by VI . We set = 0 in system (4) and perform
a linear stability analysis about the steady state where VR only is present. We find that a
small amount of VI introduced to this steady state grows at rate
1
μ(pR )pI
γ (pI ) =
. (10)
−(μ(pI ) + c) + (μ(pI ) + c)2 − 4c μ(pI ) −
2
pR
The condition for the invading strain to grow is thus
pR
pI
>
μ(pI ) μ(pR )
(11)
or equivalently, the reproductive ratio ρ of the invader (Eq. (2)) must exceed that of the
resident. This is expected, since p/μ(p) is the burst size of an infected T cell, and we
know that the virus with the largest burst size will be the fittest at steady state. However,
this is not the whole story.
If we assume that μ(p) c, and for any reasonable choice of parameters it is (Nowak
and May, 2000), then we can expand Eq. (10) in powers of μ(p)/c. This greatly simplifies
the expression for the growth rate, and allows us to see the parameters that dominate the
growth:
γ (pI ) ≈ μ(pR )
pI
− μ(pI ).
pR
(12)
Any strain that satisfies condition (11) will grow. However, the strain that grows the most
quickly will emerge first. One might have expected that the p ∗ strain would invade, since
it is the strain that optimizes the burst size. But Eq. (12) reveals that this is not the case.
C.L. Ball et al.
Fig. 6 Exponential growth rate of free virions. The growth rates of invading strains are plotted on a
log scale when the resident strain is at steady state. The virus that optimizes initial growth rate is in
each case denoted by a star and the initial growth rate of the within-host optimum strain (p∗ strain) is
denoted by a circle. For each curve, the production rate of the resident strain is given by pR . We show the
predicted succession of virus strains if each strain is allowed to reach equilibrium and assuming that the
first established strain is the strain that optimizes the initial growth rate.
Maximizing over pI yields an equation for the viral production rate that can invade most
quickly, p̂I .
μ (p̂I ) −
μ(pR )
= 0.
pR
(13)
Comparison with Eq. (3) indicates that p̂I = p ∗ only when pR = p ∗ and so it is not
true that the virus with the largest burst size necessarily invades. To explain this, we first
note that the concentration of uninfected T cells at steady state is inversely proportional
to ρR . Therefore, the lowest steady state level of uninfected T cells occurs when the p ∗
strain dominates. During initial infection, T cell levels are high, and a strain with high
production rate p (a strain inducing high host cell mortality and with a relatively low
burst size) is most successful. As the dynamics get close to the steady state corresponding
to only this resident strain, the level of T cells is reduced below the initial level, leaving
the resident strain in conditions such that it is no longer the best-adapted strain. However,
the concentration of T cells is still higher than it will be at steady state. Therefore, a strain
that replicates more quickly, but also kills T cells more quickly, can out-compete a strain
that better optimizes burst size, but replicates more slowly. Strains with a replication rate
too close to the resident strain have a burst size nearly as small as the resident strain and,
thus, are not as fit as strains further away. Therefore, since it is the concentration of T cells
that determines the optimal rate of replication and since as soon as a new strain invades
it lowers the concentration of T cells by a discrete amount, in this approximate theory,
we predict discrete jumps in production rate. The same phenomena were observed at the
epidemiological scale by Lenski and May (1994).
Figure 6 shows the predicted succession of virus strains, if we assume that each strain
is allowed to reach a steady state that excludes all other strains. In full simulations, we
only observe a single discrete jump from the initially invading (p ∗∗ ) strain to its successor.
The reason for this is that as that first strain is being invaded, several strains with similar
Modeling Within-Host Evolution of HIV: Mutation, Competition
fitness begin to invade it. Although one of these strains invades more quickly than the
rest and will obtain a higher concentration than the others, other strains are still present in
measurable quantities and this blurs the next transition.
4.5. Replacement dynamics
In our simulations we take μ(p) to be an exponential function, μ(p) = d exp(φp). In this
case, we can solve Eq. (3) to find p ∗ = 1/φ and then simplify (13) to obtain
pˆI ≈ p ∗ ln
p∗
pR
+ pR .
(14)
Substituting into (12) we get
∗ ∗ μ(pR )
p
p
∗
γ (pˆI ) ≈
(pˆI − p ) = μ(pR )
t −1 +1 .
ln
pR
pR
pR
(15)
We can make several observations from Eqs. (14) and (15). First, as pI gets close to
p ∗ , γ (pR ) gets small. The closer pI is to p ∗ , the longer this analysis predicts it will take
for a secondary virus strain to invade. Further, γ (p) is inversely proportional to the burst
size (or basic reproductive number) of the resident strain. Thus, as the burst size of the
resident strain increases, the invasion rate of its mutants will grow slower. And, since each
successive virus strain has a larger burst size than the previous strain, the replacement rate
of the strains should decrease with each successive strain.
4.6. Infected steady state
Finally we present approximate formula for the density of each virus strain at the infected
steady state of the system. We denote the infected steady state by (T̂ , Tˆi∗ , V̂i ) and consider
this steady state as a perturbation away from the infected steady state of the no-mutation
( = 0) model where only the p ∗ strain is present. We write the density of the optimal
within-host strain as V̂∗ and then expand the system of Eq. (4) in powers of to obtain
T̂ ≈
μ(p ∗ )c
+ O(),
kp ∗
d ∗
(ρ − 1) + O(),
k
d ∗
ρi
(ρ − 1) ∗
V̂i ≈
+ O 2
i
Mk
ρ −ρ
(16)
V̂∗ ≈
(17)
for all other strains.
(18)
(ρ is defined by Eq. (2)). We observe that, to first order in , all strains are present at
steady state. While the overall magnitude of the subdominant strains is determined by ρ ∗ ,
their relative frequencies are controlled by the comparison between their individual ρ i
with ρ ∗ .
C.L. Ball et al.
5. Discussion
We have presented a general framework for competition between virus strains for host
resources in the presence of a simple trade-off defined by the function μ(p). We showed
that, under a wide range of conditions, there is an initial advantage to strains that replicate more quickly than the long-term dominant strain. Although the viral population will
always eventually be dominated by a slower-replicating strain, in our simulations that
“optimal” strain was widely out-competed during initial infection.
5.1. Implications for strain competition at the epidemiological scale
This result has important implications in our understanding of HIV strain fitness. If the
within-host optimal strain was not at such a disadvantage during the initial dynamics,
we would expect this strain to essentially exclude all others within a population of hosts,
since this strain could not be outcompeted within any single host, at any time, and therefore all transmissions to new hosts would be dominated by that strain. In our model, the
dominant strain within a host changes over time, at least allowing the possibility of strain
coexistence at the inter-host level.
Very quickly replicating strains do not persist for long within the host before being
replaced with a strain of intermediate virulence. It is the strains of intermediate replication
rate that seem to have best chance of surviving within a host and successfully establishing
themselves within new hosts. They have a high enough fitness during the initial dynamics
to out-compete their mutants and they are able to utilize host resources effectively enough
that they are replaced by fitter strains only very slowly.
An ongoing debate in HIV epidemiology concerns the importance of early versus late
transmissions from an infected individual (Rapatski et al., 2005, for example). Our results emphasize another dimension to this discussion, in that strains transmitted over the
infected host’s lifespan will change according to the within-host dynamics of strain replacement (Li et al., 2004).
Blattner et al. (2004) studied 22 untreated HIV patients from Trinidad. They recorded
their symptoms during the acute phase of infection and took frequent viral load measurements. They found that a rapid early clearance of HIV was associated with a significantly
lower viral load during steady state and a slower progression to AIDS. Interestingly, they
also found that a greater number of symptoms during initial infection was associated with
a lower steady state viral load. A greater number of symptoms would likely indicate a
stronger immune response that could (among many other possibilities) be due to a higher
viral peak or to a greater expression of viral proteins on the cell surface. In our model
quickly replicating strains had higher initial peaks due to their increased initial fitness,
but declined more quickly because they had depleted their supply of T cells. These strains
then had lower steady state levels, with viral loads slowly increasing as the viral population evolved toward more fit strains. By contrast, more slowly replicating strains show
a lower initial peak which declines more gradually and finally reaches a higher steady
state level. Of course, this discussion is speculative, since our model is missing a lot of
potentially important details and we also should note that Blattner et al. (2004) did not
measure peak viremia.
Modeling Within-Host Evolution of HIV: Mutation, Competition
5.2. Understanding in vitro assays to determine viral fitness
These results demonstrate the need to use caution when defining viral fitness (see Wu et
al., 2006 for a discussion of viral fitness in vitro). Recently, Arien et al. (2005) published
their results of dual virus competition assays between HIV-1 isolates from 1986–1989 and
more recent isolates from 2002–2003. They paired historical isolates with recent isolates
in a complete medium and measured the concentration of each virus strain upon peak
virus production. They then gave each strain a relative fitness score, based on the ratio
between the final viral load and the initial inoculum. To compare the two stains, they
looked at the ratio of relative fitness between the strains. They found that historical strains
out-competed recent strains in 74% of cases and concluded that this provides evidence
that historical strains are fitter than recent strains. They followed the method described
in Quinones-Mateu et al. (2000), whereby the virus was incubated with cells which were
replenished weekly. However, they amplified and measured the viral ratio upon peak virus
production. In other words, they did not wait for the system to equilibriate. We have shown
that the strain that is fittest initially always has a greater production rate than the fittest
strain at steady state if there exists a trade-off between production rate and cell death and if
μ(p) is a concave up function. Because they did not wait for the system to equilibriate, it
is not clear which strain would have been the fittest at steady state. It would be interesting
to let this experiment run for longer and take measurements at different time intervals to
see if the fittest strain differed throughout time.
5.3. Limitations of the presented model
The model we have presented contains major oversimplifications. We were concerned
with understanding how rapid mutation would lead to dynamic variation in the dominant
viral strain, in the presence of a simple trade-off. Major assumptions that we hope to
change in future work include the following.
Stochasticity of mutation events: We have presented a completely continuous and deterministic model. An obvious criticism is that mutation events occurs randomly during
reverse transcription. In our model, all considered strains arise simultaneously via mutation. When the number of infected cells is small, this is a serious oversimplification that
should be addressed in future work. As the number of infected cells grows, this simplification is probably of less concern. Also, less fit strains are never completely eliminated
from the population in our model (Eq. (18)).
Cell tropism: as reviewed by Regoes and Bonhoeffer (2005), many HIV-infected patients exhibit a switch from a CCR5-tropic to a CXCR4-tropic dominant viral strain over
time. We cannot model this switch in the model presented here. See Ribeiro et al. (2006)
for a more detailed discussion of modeling the coreceptor switch. Bjorndal et al. (1997)
demonstrated that CCR5 strains replicated more slowly in a particular in vitro situation
(cultured cells expressing both CCR5 and CXCR4 coreceptors) than CXCR4 strains.
Within the constraints of the experimental system, this indicates a general increase in
the replicative ability of the predominant strain over time within a patient.
Evolution of cell infectivity: Due to our focus on the production-cell death trade-off, we
have neglected mutation in the parameter k (governing host cell infectivity). For HIV, the
glycoproteins gp120 and gp41 are most important for binding to CD4 and are encoded
on the env gene (actually, a precursor glycoprotein, gp160, is encoded, but this is later
C.L. Ball et al.
cleaved by a host cell enzyme to form gp120 and gp411 ). Unsurprisingly, these proteins
are highly conserved; much variation would lead to a major decline in fitness. This partly
justifies our neglect of k. Nonetheless, a more precise model featuring multiple cell types
(for example, to look at coreceptor switching) would require mutation in cell infectivity
parameters.
Multiple infection of host cells: The multiple infection model of Dixit and Perelson
(2005) may be modified to include our framework of mutation between viral replication
rates. Modifying our assumption of constant mutation to more accurately reflect distinct
processes of recombination and point mutation should be possible within such a model.
5.4. Evolution at ecological timescales
A standard way of looking at the competitive evolution of a trait is provided by adaptive
dynamics. The adaptive dynamics framework, as described, for example, by Dieckmann
(2002, 2004) allows ecological models to be viewed from an evolutionary perspective,
under the assumption that mutations cause only small changes in phenotype and that
mutations are rare. Therefore, evolutionary dynamics play out over a longer time scale
than that required for the existent strains to reach an ecological steady state. Questions of
invasion are then naturally treated using an established mathematical framework.
Here we have studied within-host evolution of a system where it is not possible to separate the evolutionary and ecological time-scales, and mutations appear to be extremely
frequent and possibly cause large changes in phenotype. In Appendix C, we show how the
system behaves differently if one supposes that minor changes in phenotype are the norm.
Evolution on ecological timescales has also been observed in biological systems operating
on larger scales: in a predator-prey system consisting of a rotifer and an algae (Yoshida
et al., 2003); in the size of flies under spatial habitat variation (Huey et al., 2000) and in
the evolution of reproductive isolation in fish during colonization of new environments
(Hendry et al., 2000). Analysis of such situations in general terms represents a challenge
for theoretical evolutionary biology. Here, we have been able to characterize the majority
of the dynamic effects in a very simple model by using elementary arguments. We propose
that our methods of analysis will be applicable to other systems where the assumptions of
adaptive dynamics are violated.
Finally, the virus dynamics model presented here is very close in mathematical form
to certain population-scale epidemic models (Lenski and May, 1994). In Appendix D, we
outline how our elements of our theory can be extended to model the spread of disease
strains varying in transmissibility and virulence within a population of susceptible hosts.
Acknowledgements
This work was supported by funding from NSERC and MITACS NCE. We thank Eric
Cytrynbaum and Vitaly Ganusov for useful discussions and two anonymous reviewers for
very helpful comments and suggestions.
1 120 + 41 160.
Modeling Within-Host Evolution of HIV: Mutation, Competition
Appendix A Linear stability analysis of the uninfected steady state
To explore the dynamics about the uninfected steady state, (T = λ/d, Vi = 0, Ti∗ = 0),
we expand the state variables in powers of the small parameter , defining x = T − λ/d,
yi = Ti∗ and zi = Vi where x, y and z are of order . Substituting these into (4), we obtain
to first order in :
M
dx
= −dx − (kλ/d)
zj ,
dt
j =1
dyi
= (kλ/d)zi − μ(pi )yi ,
dt
dzi
= pyi − czi .
dt
(A.1)
We assume a solution of the form x(t) = x0 eδt , zi = zi0 (p)eδt and yi = yi0 (p)eδt , obtaining
δx0 eδt = −dx0 eδt − (kλ/d)
M
zj 0 eδt ,
j =1
δyi0 e = (kλ/d)zi0 e − μ(pi )yi0 eδt ,
δt
δt
(A.2)
δzi0 eδt = pyi0 eδt − czi0 eδt .
The eigenvalues are δ = −d and
1
pi kλ
2
δ± (pi ) =
−(μ(pi ) + c) ± (μ(pi ) + c) − 4 μ(pi )c −
.
2
d
(A.3)
We can therefore write zi0 = ξ1 eδ+ (pi ) + ξ2 eδ− (pi ) + ξ3 e−dt where the ξk are arbitrary constants that can be found by applying the initial conditions. By applying the initial conditions zi0 = Vi (0), x0 = 0 and yi0 = 0 we find that
μ − c + (μ(p ) + c)2 − 4μ(p )c − pi kλ i
i
d
ξ1 =
Vi (0),
p
kλ
i
2 (μ(pi ) + c)2 − 4 μ(pi )c − d
c − μ + (μ(p ) + c)2 − 4μ(p )c − pi kλ i
i
d
ξ2 =
Vi (0),
p
kλ
2 (μ(pi ) + c)2 − 4 μ(pi )c − id
ξ3 = 0.
(A.4)
(A.5)
(A.6)
C.L. Ball et al.
ξ1,2 can be rewritten as
δ+ (pi ) + μ(pi )
ξ1 =
Vi (0),
2δ+ (pi ) + μ(pi ) + c
δ+ (pi ) + c
ξ2 =
Vi (0) and
2δ+ (pi ) + μ(pi ) + c
Appendix B
ξ3 = 0.
(A.7)
Early-stage strain replacement
A common simplification of model (1) is to suppose that the V equation is in steady
state so T ∗ = (c/p)V (Nowak and May, 2000). This is a reasonable approximation in the
study of HIV infection because the V dynamics are relatively fast compared to the T ∗
dynamics. We make this approximation within our model, in this appendix only, to obtain
the simplified system
M
dT
= λ − dT − kT
Vj ,
dt
j =1
M
ck
ckT dVi
= (1 − )T Vi − μ(pi )Vi +
Vj ,
dt
p
pi M j =1
i = 1, . . . , M.
(B.1)
We now expand about the uninfected steady state (of this model) in powers of : T =
(λ/d) + x (1) + 2 x (2) , Vi = zi(1) + 2 zi(2) , to obtain to O()
kλ (1)
dx (1)
= −dx (1) −
z ,
dt
d j =1 j
M
dzi(1)
=
dt
ckλ
− μ(pi ) zi(1) .
pi d
(B.2)
(B.3)
Solving and applying the initial conditions T (0) = λ/d, VI (0) = V0 and Vi (0) = 0 for
i = I , we find that
zI(1) = V0 exp (ckλ/pI d) − μ(pI ) t ,
(B.4)
zi(1) = 0,
i = I.
(B.5)
The growth rate in Eq. (B.4) is a close relative of the growth rate δ+ defined in Appendix A, Eq. (A.3). Now going to O( 2 ),
dzi(2)
=
dt
ckλ
ckλ (1)
− μ(pi ) zi(2) +
z .
pi d
dpi M I
(B.6)
The first term in this equation represents the exponential growth of strain i. The second
term shows the seeding of this strain by mutation from the initial strain I . We now write
Modeling Within-Host Evolution of HIV: Mutation, Competition
the approximate growth rates for this approximate model as ri = (ckλ/pi d) − μ(pi ) to
obtain the simple expression
ckλ
dzi(2)
= ri zi(2) +
V0 exp(rI t)
dt
dpi M
zi(2) =
with solution
ckλ
e rI t − e ri t
V0
.
dpi M
rI − ri
(B.7)
(B.8)
To approximate the time tˆ that a mutant strain, Vk , k = I , surpasses the initial strain, VI ,
we set VI = Vk and solve for time:
ckλ
erI tˆ − eri tˆ
V0
+ O 3 ,
V0 erI tˆ + O 2 = 2
dpi M
rI − ri
dpi M
1
ˆt =
(ri − rI ) .
ln 1 +
ri − rI
ckλ
(B.9)
(B.10)
This expression is valid only at early times (when all the strains are still at low levels). This
is acceptable since we are only interested in examining the possibility that the initial strain
is overtaken by its mutant at short times. For small , this expression can be approximated
by
tˆ ∼ ln(1/)/(ri − rI ),
(B.11)
indicating that faster-growing mutants may invade the initial strain early in the infection
provided is sufficiently large. If tˆ is found to be large then the linear analysis is no longer
valid, but we should suspect that the initial strain out-competes its mutants. As one might
expect, a high mutation rate and a small initial inoculum of a weak initial competitor lead
to the initial strain being overtaken by its faster-expanding mutant while a low mutation
rate and large initial inoculum lead to the initial strain establishing itself. Also, as the
initial number of T cells (λ/d) becomes smaller, the longer it takes for a mutant strain to
invade. We also note that a single strain (maximizing ri ) is the fastest to invade. This strain
corresponds to the p ∗∗ strain in the full model. Figure 4 shows that all initially present
strains that are out-competed by a mutant at early times are indeed outcompeted by the
same strain. This strain is the one closest to the p ∗∗ strain.
Appendix C Nearest neighbor mutation
We have assumed an equal probability of mutation to any virus strain, but mutation is
complicated and we cannot predict how changes in the HIV genotype will affect production rate. The theory of adaptive dynamics takes the opposite point of view—that mutation
can only produce small changes. While it appears that small genotypic changes can produce large phenotype changes, the frequency with which this occurs is unknown and it
may be the case that most mutations have small effects. We present here for comparison
the extreme case where mutations can only occur to the strain closest to the strain being
mutated. To consider this ‘nearest neighbor’ approach, we modify Eq. (4) to allow each
C.L. Ball et al.
Fig. C.1 Evolution of the viral production rate when only nearest neighbor mutation is allowed. We
use M = 50 virus strains, mutation rate of = 10−5 and initial condition of 100% p∗∗ -strain. (a) Each
contour line corresponds to a 15 virion/µL increase in viral concentration. (b) The production rate of the
most abundant strain plotted through time.
strain to mutate only to strains of neighboring production rate. The equations describing
infected T cells become
kT
dT1∗
=k 1−
V2 ,
(C.1)
T Vi − μ(p1 )T1∗ +
dt
2
2
kT
dTi∗
= k(1 − )T Vi − μ(pi )Ti∗ +
(Vj −1 + Vj +1 ) for i = 1, M,
dt
2
dTM∗
kT
=k 1−
VM−1 ,
T VM − μ(pM )TM∗ +
dt
2
2
(C.2)
(C.3)
and the rest of the model follows as before.
When we examine this model numerically (see Fig. C.1) we no longer observe jumps in
production rate. Instead, if we start with a quickly replicating strain, each strain is replaced
by the strain that has the next closest production rate to p ∗ . The evolution toward steady
state occurs much more slowly than it did when we allowed equal rates of mutation,
because we no longer observe the initial sudden jump to a lower production rate. This
pattern of strain replacement is very similar to that of other models of adaptive dynamics
in other contexts (Dieckmann, 2002).
We might expect the rate of strain replacement to eventually match the rate observed
in our first model since after the first jump we observe next strain replacement in our first
model as well. However, we observe a much slower rate of strain replacement with the
nearest neighbor model. This is explained by noting that the pattern of strain replacement
occurs by different mechanisms in each model. In the first model each strain is created by
mutation at equal rates. The fittest strain at that concentration of T cells will grow and peak
first, but there are a number of strains surrounding it that have similar levels of fitness and
grow at a similar rate. Once the first strain has peaked, the next successive strain quickly
peaks. In the second model, a strain must wait for its nearest neighbor to appear before
it is created by mutation. As the fitness difference between the 2 strains decreases, the
rate of strain replacement also decreases. Thus, the succession of strains is not dependent
Modeling Within-Host Evolution of HIV: Mutation, Competition
upon the fitness of each strain within the host, but on the order of appearance and the size
of the mutational jumps.
Appendix D Strain selection in epidemics
We extend our theory by considering the spread of an infectious disease through a population, assuming that the disease generally does not persist for long within its host, and,
therefore, the selection pressures placed on the parasite to infect new hosts outweigh the
within host selection pressures. We consider susceptible (S) and infected (Ii ) individuals
and allow new infections to occur by contact between infected and susceptible individuals,
and structure disease strains by some measure of within-host replication rate of disease
propagules (which we intentionally leave vague), p. We take α(pi ) to be the death rate
of individuals infected with strain i, r(pi ) as the recovery rate and β(pi ) as the infectivity. pi is the production rate of pathogen strain i within its host. We assume that an
increased pathogen production rate will increase both the infectivity and the virulence of
the pathogen. Therefore, both β(p) and α(p) are increasing functions of p. We ignore
co- and super-infection in our model; this is justifiable at the beginning of an epidemic in
many situations even when multiple infection can play a role later on.
D.1 Initial dynamics
By assuming that the number of susceptibles changes slowly compared to the growth
of infected individuals during the initial infection, we can model the initial spread of
disease. Taking the number of susceptibles to be approximately constant (S0 ) during the
initial epidemic, we can write a dynamic equation for the number of infected individuals
of class i as
dIi
≈ β(pi )S0 Ii − α(pi ) + r(pi ) Ii .
dt
(D.1)
Solving this we find the approximate initial growth rate of each virus strain.
Ii (t) ≈ Ii0 e[β(pi )S0 −(α(pi )+r(pi ))]t
(D.2)
where Ii0 is the number of individuals infected with strain i initially introduced into the
population. We see that the production rate p ∗∗ that optimizes initial growth depends on
the initial concentration of susceptibles:
β (p ∗∗ )
α (p ∗∗ ) + r (p ∗∗ )
=
1
.
S0
(D.3)
Following the method described in Alizon and van Baalen (2005), in order that p ∗∗ is
a maximum we require that d 2 β/d(α + r)2 < 0|p=p∗∗ < 0. We can gain some intuition
about equation (D.3) by plotting α(p) + r(p) against β(p) and considering the contours
of fitness, given by β = (α + r)/S0 . p ∗∗ occurs when the slope of the tangent line to
the parametric curve is equal to 1 over the initial number of susceptibles S0 . Because of
condition (D.3), the larger the initial population of susceptibles, the larger the virulence
of the initial strain. If there is a small initial population of susceptibles then less virulent
strains will be favored. Figure D.1a shows the graphical interpretation of this.
C.L. Ball et al.
Fig. D.1 The parametric curve α(p) + r(p) plotted against β(p). (a) Fittest strain during initial dynamics
is found where the curve has slope 1/S0 (shown for two values of S0 ). (b) The fittest strain at steady
state (denoted by a circle). The virulence that optimizes fitness is found when the tangent line to the curve
passes through the origin.
D.2 Steady state behavior
The fittest strain at steady state maximizes the number of infections produced by a single
host, represented by N (p). This is equivalent to the usual R0 formulation.
∞
β(p)
N (p) = Ŝ
Ŝ,
(D.4)
e−(α(p)+r(p))t β(p)dt =
α(p) + r(p)
0
where Ŝ is the number of susceptibles at steady state. We find the optimal rate of replication, p ∗ , by solving dN/dp = 0:
α(p ∗ ) + r(p ∗ )
β(p ∗ )
=
.
β (p ∗ ) α (p ∗ ) + r (p ∗ )
(D.5)
We require that d 2 N/dt 2 |p=p∗ < 0 and, thus, we impose the following restrictions:
β (p ∗ ) α (p ∗ )
<
β(p ∗ )
α(p ∗ )
and
β (p ∗ ) α (p ∗ )
< ∗ .
β (p ∗ )
α (p )
(D.6)
Figure D.1b shows a graphical interpretation analogous to Fig. 3. We see that p = p ∗
when the tangent line to β(α(p)) passes through the origin (see Lenski and May, 1994
and Alizon and van Baalen, 2005).
Appendix E
Proving p∗∗ > p∗ and uniqueness of p∗∗
From the burst size N = p/μ(p),
N (p) =
μ(p) − pμ (p)
.
(μ(p))2
(E.1)
Modeling Within-Host Evolution of HIV: Mutation, Competition
We will assume p ∗ exists and maximizes N (p) so that for p < p ∗ , N (p) > 0 and for
p > p ∗ , N (p) < 0. Now let
1
kλ
f (p) = −
pμ
(p)
.
(E.2)
μ(p) − c +
μ (p)
dμ (p)
Note that f (p ∗∗ ) = 0. We will first show that p ∗∗ > p ∗ by showing that roots of f (p) are
greater than p ∗ . Consider f (p) for p ≤ p ∗ . For concave up μ(p), μ (p) < μ (p ∗ ) on this
range so
1
kλ
− pμ (p)
f (p) ≥ μ(p) − c +
μ (p)
dμ (p ∗ )
1
kλp ∗
= −
pμ
(p)
from Eq. (3)
μ(p) − c +
∗
μ (p)
dμ(p )
>
1
μ (p)
≥0
μ(p) − pμ (p)
from Eq. (7)
because N (p) > 0 on this range Eq. (E.1) .
We have shown that if p ∗∗ exists, then p ∗∗ > p ∗ . Now consider p > p ∗ . To prove that a
root exists, consider limp→∞ f (p):
μ(p) − c − pμ (p)
kλ
+ lim
p→∞
p→∞ (μ (p))2 d
μ (p)
lim f (p) = lim
p→∞
μ(p) − c − pμ (p)
p→∞
μ (p)
= lim
since μ (p) → ∞ as p → ∞
μ(p) − pμ (p)
p→∞
μ (p)
< lim
< 0.
Since f (p ∗ ) > 0 and limp→∞ f (p) < 0, by the intermediate value theorem there exists
a root, p ∗∗ , of f (p) = 0 that is greater than p ∗ . We have proved that p ∗∗ always exists
provided p ∗ exists, μ (p ∗∗ ) > 0, and we have shown that p ∗∗ > p ∗ .
To show that p ∗∗ is unique we consider turning points of f (p), located at
μ (p)
2kλ
f (p) = − μ(p)
−
c
+
=0
(μ (p))2
dμ (p)
yielding μ(p̂) − c = (−2kλ)/(dμ(p̂)) at the turning points. Substituting p̂ into f (p) we
get
f (p̂) =
kλ
−kλ
−2kλ
+
− p̂ =
− p̂ < 0.
d(μ (p̂))2 d(μ (p̂))2
d(μ (p̂))2
Since f (p) = 0 only when f (p) < 0, and f (p) < 0 as p → ∞, it is impossible for f (p)
to pass through 0 more than once. Therefore, p ∗∗ is unique.
C.L. Ball et al.
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