Evolution of malaria virulence in cross-generation
transmission through selective immune pressure
David Gurarie
Mathematics Department & Center for Global Health and Diseases,
Case Western Reserve University,
Cleveland, OH 44106
Abstract
Theoretical arguments and some mathematical models of host-parasite
coevolution (e.g. [1- 6]) suggest host immunity as the driving source for the
evolution of parasite virulence. Imperfect vaccines in particular, can play the role
and recent work [7] sets to test these ideas experimentally, using the mouse
malaria model, Plasmodium chabaudi. To this end the authors evolve parasite
lines in immunized and nonimmunized ("naïve") mice using serial passage of
infected blood samples. They find parasite lines evolved in immunized mice
become more virulent than those evolved in naive mice. Furthermore, this feature
persisted even when the evolved strains were transmitted through mosquitoes.
Here we develop a mathematical model of parasite dynamics that
qualitatively reproduces the experimental results of [7]. Our model accounts for
the basic in-host processes: (i) production and depletion of red blood cells (RBC);
(ii) immune-modulated parasite growth/ replication, (iii) immune stimulation and
clearing of parasite. Besides we introduce multiple parasite strains with variable
levels of virulence, and allow random mutations during replication process. The
virulence is represented by a single parameter immune stimulation threshold. So
more virulent strains have higher "tolerance levels", hence increased RBC
depletion (anemia).
Numeric simulations with our model exhibit, as in [7] the overall
evolution of virulence in serial passage of parasite strains, and its enhancement
through partial (imperfect) immunization.
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
I. Introduction
Malaria is a mosquito-borne infection transmitted by Plasmodium protozoa. Four species of
parasite (P. falciparum, P. vivax, P. ovale and P malariae) are known to infect humans, and there
are similar types infecting other mammalian hosts. The parasites are transmitted by infected
Anopheles mosquitoes that acquire infection by feeding on blood from infected hosts. The clinical
symptoms of malaria include fever, chills, pain, and sweats, which develop 7-14 days after the bite
of an infectious mosquito. The duration and severity of disease may vary, depending on many
factors such as intensity of infection, biology of the host, its immune status et al.
The parasite has two distinct phases of reproduction: a sexual (gametocyte-mating) stage in
the mosquito, and an asexual phase in mammalian hosts, where it undergoes several stages and
patterns of replication. Initially, in human/ mammalian infection, a sporozoite carried in the insect
saliva enters the blood stream, migrates to liver, and invades liver cells. After a 7-day period, it
releases large number (10,000-30,000) of short-lived but extracellular blood stages, called
merozoites. The merozoites rapidly invade red blood cells (RBC) and turn into trophozoites. Inside
the RBC they undergo asexual replication over a 48-72 hr period (depending on species), then
bursts the cell, releasing a dozen of new merozoites to continue the cycle. This exponential parasite
growth, associated with cyclical episodes of hemolysis, is mostly responsible for the clinical
symptoms of malaria.
Concurrent with parasite growth the host develops immunity (innate and specific responses)
that allows it to control parasitaemia, and under suitable conditions eliminate it altogether. As many
other parasites Plasmodium has adopted various mechanism of immune evasion, including genetic
mutations on each replication cycle. The resulting diversification into multiple parasite strains, and
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
selective pressures (e.g. immunity or drug treatment) can gradual evolve such features as drug
resistance or virulence.
Here we set up a mathematic model of in-host RBC-parasite dynamics for single and multi
strains to explore the evolution of virulence through selective immune pressure, suggested by
theoretical studies [1-6] and experimental work [7]. Our model utilizes some features of continuous
in-host models (e.g. [8-11]), as well as discrete and stochastic ones [12-13]. We implanted it on
Wolfram Mathematica 5 package, and conducted a series of numeric experiments that corroborate
the basic conclusions of earlier theoretical and experimental work.
II. Methods: Stochastic model of in-host dynamics for single and multiple P.
strains.
The state of the system at discrete time
1,2,...
t
=
is described by (non-infected) RBC
density
t
N (per
l
of blood), parasite densities:
t
Y - young stage (newly infected iRBC),
t
X - old
stage/ schizont, and the immune effector variable
t
J . The system below incorporates the following
basic processes:
·
RBC production and removal through natural death and infection
·
Merozoite release and invasion of un-infected RBC
·
iRBC survival in a given immune environment
J
·
immune stimulations by the young and old stages
Y, X
Besides for multiple strains, designated by
{
}
,
;
1,2,...
i
i
Y X
i
=
of different virulence types we
allow random (genetic) transitions on each time step = ½ replication cycle, from type j to i with
prescribed probabilities
0;
1
ij
ij
i
a
a
=
.
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
Production
. For RBC production/loss we take a finite-difference equation
1
production
loss
infectious removal
t
t
t
N
N
Y
+
=
+
-
(1)
with constant production rate
4
5 10 / day
=
, natural loss
.99
=
(for 100 day life span for RBC),
based on equilibrium RBC level
6
0
5 10 /
1
N
l
=
-
. The production term could be made to
depend on RBC-deficit (relative to `norm')
(
)
0
1
/
N N
-
, - enhanced RBC production.
Invasion
. Released merozoites,
t
rX (with replication number
8
r
=
per schizont) compete
for the available RBC population
N. We assume the maximal number of potential invasions
depends on
rX and N as
(
)
,
t
t
t
t
t
rX N
C
rX N
rX
N
=
=
+
, where
(
)
,
xy
x y
x y
=
+
(2)
Such function
has the property that
C rX
- for small rX, but saturates at value
C
N
, as rX
increases (as the number of successful invasions should not exceed the total available RBCs). Each
potential invading merozoite can succeed with a (density-dependent) probability
(
)
(
)
0
1
,
1
/
t
t
t
t
t
p
p rX N
rX
N x
=
=
+
(3)
Such p drops to 1/2, when the relative density
/
rX N
(`merozoites' over `RBCs') reaches threshold
0
x . The combined effect of C potential merozoites having probability of invasion p is given by the
binomial distribution
(
)
1
Bi
|
t
t
t
Y
C p
+
=
(4)
which determines the number of young stages at the next time-step.
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
Survival
. Once invaded, the infected RBC will survive through the old stage (next
replication cycle) with a probability
( )
q J
that depends on the immune environment - the level of J
effector. We take survival probability
( )
1
1
q J
J
=
+
(5)
similar to
(
)
/
p rX N
in (3). So the surviving (old-stage) population is given by another binomial
equation
(
)
(
)
1
Bi
|
/
t
t
t
X
Y q J V
+
=
(6)
where V designates the strain-specific sensitivity threshold. Higher V means higher likelihood of
survival in a given J- environment. Threshold V will serve as a marker of virulence. Indeed a
`comparison plot' (Figure 2) shows larger V having higher parasite densities X Y
+ , and higher
anemia rates (RBC-loss).
Immune stimulation and loss
. The effector variable is stimulated by the combined iRBC
density, weighted by its immunogenic factors and relative duration of 2 (young-old) stages of
development,
1
loss
stimulation
2
t
t
t
t
t
X
Y
J
J
N
+
+
=
+
(7)
Here we assume both stages of equal duration, and `stimulation function'
depending on relative
density `iRBC/RBC'. The loss factor
.99
, corresponds to .01/day loss-rate of immune effector.
The result is a discrete time step (1 day) coupled stochastic system (1) -(7)
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
(
)
(
)
(
)
1
1
1
1
;
Bi
|
;
Bi
|
/
;
2
t
t
t
t
t
t
t
t
t
t
t
t
t
t
N
N
Y
Y
C p
X
Y q J V
X
Y
J
J
N
+
+
+
+
= +
-
=
=
+
=
+
(8)
for variables
{
}
, , ,
N Y X J
(single strain), with functions
(
)
,
C rX N
of (2), invasion probability p
(3), survival probability q (5), and simulation
( )
0
x
s x
=
, taken as a simple linear function. The
essential parameters of the model are
RBC production/loss
4
5 10 ;
.99
=
=
Merozoite replication factor
8
r
=
Invasion probability threshold
0
.6
x
=
Survival probability (virulence) threshold
.1
.3
V
Immune stimulation coefficient
0
8
s
=
Immune loss rate
.99
=
Table 1
: Basic in-host parameters
Some
parameters
e.g.
; ;r
are fixed (known), while others
0
0
; ; ;
x s
V
- can be estimated
from the experimental lab data. Some of them could be strain and host specific.
Multi-strain model
. A single-strain system (8) can be extended to a multi-strain one, with
strains labeled by the genotype index
1,2,...
i
n
=
. We assume the released merozoites (on each time
step) can change their genotypes according to `switching (stochastic) matrix'
ij
A
a
= . The
`invading crop' of type i is then given by
;
i
i
ij j
j
C
a r X N
=
(9)
with the (density dependent) invasion probability
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
0
1
1
j
j
j
i
r X
x N
p
p
= =
+
(10)
and J-dependent survival probability
1
1
/
i
i
q
J V
=
+
, each strain having its specific virulence
threshold
i
V . All strains stimulate production of immune effector J, either through their individual
contributions,
2
j
j
j
X
Y
N
+
(with possibly `strain-specific' functions
{
}
j
=
). Alternatively
we can take their combined density
2
j
j
j
X
Y
N
+
, assuming homologous strains. The resulting
2n+2 system of equations takes the form
(
)
(
)
(
)
1
1
1
1
;
Bi
|
;
Bi
|
/
;
2
j
t
t
t
j
i
i
t
t
t
i
i
t
t
t
i
j
j
t
t
j
t
t
t
N
N
Y
Y
C
p
X
Y
q J V
X
Y
J
J
N
+
+
+
+
= +
-
=
=
+
=
+
(11)
Random mutations:
In simulations below we use the parameters of Table 1, and tri-
diagonal transition matrix A with entries
.99
ii
a
=
, and
, 1
.005
i i
a
±
=
(off-diagonal). So each
genotype generates 99% of the same type (on each replication cycle), while .5% would `up' or
`down' the `geno-scale'. There are many ways to modify such matrix A.
III. Results: numeric simulations and analysis
Single strain
in a single host over 80-day period. We take system (8) with virulence
threshold
.2
V
=
. Typically we observe parasitemia to reached its peak by day 15, and subside by
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
day 30. The follow-up repeated (damped) cycles are approximately
35
-day long, and anemia (RBC
depletion) can reach up to 40% of the normal RBC-level.
0
20
40
60
80
0
1
×
10
6
2
×
10
6
3
×
10
6
4
×
10
6
5
×
10
6
Day
C
B
R;
C
B
R
i
0
20
40
60
80
0
0.5
1
1.5
2
2.5
Day
J;
l
a
v
i
v
r
u
Sb
o
r
p.
0
20
40
60
80
0
0.5
1
1.5
2
2.5
Day
J;
l
a
v
i
v
r
u
Sb
o
r
p.
20
40
60
80
100000
200000
300000
400000
500000
600000
Figure 1
: Upper left plot: RBC-level N (thin) and combined Y+X density (thick). Upper right
plot: effector J (thick curve) and survival probability q(J) shaded. Bottom plot: iRBC,
X+Y (black curve), Y (gray)
Inter-strain comparison.
Here we consider 6 virulent strains (in separate hosts), with
threshold parameter V varying over the range
0.1
0.3
V
. Figure 2 shows several effects of
higher virulence: increases anemia (from 30% maximal depletion at
0.1
V
=
, to 60% reduction at
0.3
V
=
), and enhanced (but delayed) production of the immune effector (right plot).
10
20
30
40
0
1
×
10
6
2
×
10
6
3
×
10
6
4
×
10
6
5
×
10
6
Day
C
B
R;
C
B
R
i
0
10
20
30
40
0
1
2
3
4
Day
J;
l
a
v
i
v
r
u
Sb
o
r
p.
Figure 2
: Comparison of 6 virulent strains (fading shades of gray): left plot shows levels of
RBC (top) and iRBC (bottom); right plot - the corresponding immune effector curves.
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
Multi-strain in-host competition
. Now we allow 8 strains (
1, 2,...8
i
=
) of increased
virulence 0.1
0.3
i
V
described by system (11) to compete within a host over 40-day period. We
initialize the system with the utmost benign strain (
1
i
=
), the mutations matrix A (for each
replication cycle) will gradually shift it more virulent types. Four panels of Figure 3 show: i) RBC
t
N and combined parasitemia
(
)
1,...8
i
i
t
t
i
Y
X
=
+
(top left), ii) the immune effector (top right); iii)
Log-parasitemia
{
}
:
1, 2,...
i
i
t
t
Y
X
i
+
=
spreading to higher-virulence types (bottom left), and
distribution of 20-day average parasitemia over strains
1, 2,...8
i
=
(bottom right). Only #1 and #2
maintain substantial levels over 20-day period. Note that the cumulative RBC, parasitemia and
effector J behave similar to a single-strain case of Figure 1.
0
10
20
30
40
0
1
×
10
6
2
×
10
6
3
×
10
6
4
×
10
6
5
×
10
6
Day
C
B
R;
C
B
R
i
0
10
20
30
40
0
0.25
0.5
0.75
1
1.25
1.5
1.75
Day
e
n
u
m
m
IJ
10
20
30
40
1
10
100
1000
10000
100000.
1
2
3
4
5
6
7
8
20000
40000
60000
80000
100000
120000
Figure 3
: Evolution of multi-strain infection over 40-day period.
Serial passage experiments
. Our next example attempts to replicate the mice experiments
of [7]. Here infected (multi-strain) blood sample taken at the end of a 20-day run, is transmitted
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
from infected to the next-generation immuno naïve host, and let to run for another 20 days (past the
upsurge peak). We run a sequence of 16 transmission cycles and follow the evolution of RBC,
parasitemia (iRBC) and immune effector J (Figure 4). Note that anemia (RBC depletion level)
increases with each transmission cycle from about 40% to 55%, as more virulent strains take over.
Figure 5 corroborates this conclusion by showing period average (over 20 days) distribution of 8
parasite densities, from the first host (top left) to the last (bottom right). One sees clear transition
from benign strains (host #1) to virulent (#16). The system seems to stabilize by 12th cycle.
0
50
100
150
200
250
300
0
0.5
1
1.5
2
2.5
Day
e
n
u
m
m
IJ
0
50
100
150
200
250
300
0
1
×
10
6
2
×
10
6
3
×
10
6
4
×
10
6
5
×
10
6
Day
C
B
R;
C
B
R
i
Figure 4
: RBC and combined parasitemia (top); immune effector (bottom) of a sequence of
16 immuno-naïve mice, each one subjected to 20-day infectious period
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
Figure 5
: Parasitemia profiles on the 8-strain virulence scale (each sample averaged over its
20-day period), for 16 transmission cycles. Cycle ## proceed in rows from "top-left" to
"bottom-right".
Serial passage run with 16 generations of partially vaccinated mice
. We repeat the
above experiment, but now each mouse is given an initial level of immune effector
0
.5
J
= (about
1/5 of the maximal attainable level of J in Figure 5) due to vaccination. In the previous case `naïve
mice' had
0
0
J
= . A comparison between 2 cases (Figure 6), shows the gradual increase of anemia
due to virulence, which reach comparable levels (along with maximal J-levels) by the 9th cycle.
Although the bottom level anemia (RBC) of vaccinated mice seem to overshoot somewhat the
`naïve' ones at late cycles. More interesting and instructive view of the spread of virulence
however, is the period-average distribution of parasitemia among 8 strains in two cases (Figure 7).
For each transmission cycle the `naïve distribution' (red) looks more benign (shifted to the left)
compared to the `vaccinated distribution' (blue). By late cycle (#11-16) the most virulent strain #8
takes dominance, furthermore #8 (blue) exceeds the comparable level of #8 (red).
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
0
50
100
150
200
250
300
0
0.5
1
1.5
2
2.5
Day
e
n
u
m
m
IJ
0
50
100
150
200
250
300
0
1
×
10
6
2
×
10
6
3
×
10
6
4
×
10
6
5
×
10
6
Day
C
B
R;
C
B
R
i
Figure 6
: Same as Figure 4 for comparison of 16 generations of partially vaccinated mice
(black) vs. 16 naïve generations (green).
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
Figure 7
: Comparison of time-average strain distribution for two cases: naïve (red) vs.
partially vaccinated (blue). The 16 transmission cycles are ordered as in Figure 5
Nature Precedings : hdl:10101/npre.2007.203.1 : Posted 23 Jun 2007
IV. Conclusions
We develop a mathematical model of multiple competing parasite strains of varying
virulence subjected to selective immune pressure. Its numeric simulations show how in-host
competition followed by the serial passage of infection over several generations can shift the strain
profile towards more virulent pattern. Furthermore, we observe the enhanced evolution of virulence
when subsequent inocula are given a partial immune protection (immune boost). Our model
corroborates the earlier experimental conclusions of [7], and pinpoints a plausible mechanism of
immune selection of virulent strains.
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