Dynamic regulation of single- and mixed-species malaria infections:
specific and non-specific mechanisms of control
David E. Gurarie1, Peter A. Zimmerman2, and Charles H. King2
Department; 2Center for Global Health and Diseases, Case University, Cleveland, OH
ABSTRACT
ANALYSIS OF EQUILIBRIA
Intra-host immune regulation of parasite species
involves a complex web of different cell types, cytokines
and pathways, and exhibits complex dynamics (Ref. 1-2).
Here we develop simplified mathematical models of single
and mixed-species infections based on the immune
regulatory mechanisms (stimulation - clearing). Our
approach exploit differential-delay equations and dynamical
systems (cf. Ref. 3-4). It explains species interaction and its
outcomes in terms of ‘predation/ competition’. In many cases
it allows a detailed analysis of the long-term behavior
(equilibria, limit cycles, stability, bifurcations). Our
analysis has several applications:
• it explains ‘sequential patterns’ of parasitemia observed
in many clinical and field studies
• It exhibits possible clinical outcomes (species ‘coexistence’
or ‘domination’)
• The latter has potential implications for chemotherapy
control (drug resistance), and community transmission.
Our models highlight the role of the adoptive speciesspecific immunity as the primary regulating mechanism of
parasite growth.
DE system (1) is a version of ‘2-prey + 3-predator’
model. It has 7 essential parameters:
(i) NS and SS efficiencies (“clearing”x”stimulation”/”decay”): eN,
eF,eV
(ii) relative growth rate: a = a2/a1;
(iii) cross-reactivities: 0<,<1; (iv) time-lag L.
System (1) will typically equilibrate in a damped-oscillating
pattern (Fig. 6-7), and its equilibrium (endemic) states are
approximately described by the classical Volterra-Lotka
competition
V
III
2)
3)
4)
ss
IV
II
II
IV
x1
x2
F
x1
IV
x2 F
II
x2
x1
ss
1
0.8
0.6
0.4
0.2
Days
50
100
Remark: Changing relative growth a in Fig. 3 is relevant to
analysis of drug treatment. Indeed, having ‘susceptible’ and
‘resistant’ strains, their growth rates are modified by persistent
drug, that creates an effective ‘selection /domination’.
Next plot (Fig.4) shows regions of coexistence and Fastdomination in in the full (5D) model for a range of crossreactivities: 0<<.8, and NS/SS efficiencies: 0< en;es <2.5.
Conclusion: A hypothetical host population distributed over
parameter region (0< en,s <2.5) has predominant coexistence
equilibrium for low  (different species) , but it changes to a
‘dominant fast species’ for strongly cross-reactive
(homologous) strains, high .
NS
efficiency en
Fig. 4
cs
150
variation of the surface protein of Pf-species, or random inoculation with a
heterologous strain, whereby cs ‘relaxes’ to its low to high value, via ‘affinity
maturation’ (development of specific antibodies).
cn
L
0.3 ; c s
1.8
6 days
L
1
1
0.1
0.1
0.01
0.01
0.001
0.001
Fig.6: Time series
9 days
0.0001
0
50
100
150
L
200
250
300
0
50
100
12 days
150
L
1
1
0.1
0.1
0.01
0.01
0.001
0.001
0.0001
200
250
300
200
250
300
15 days
(parasitemia - solid
black, SS-effector solid gray, NSeffector - dashed)
show transition from
stable equilibrium to
limit cycle (Hopf)
past L=12.
0.0001
0
50
100
150
200
250
300
en
es
0
1.2; L
0.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
150
es
200
250
300
50
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
150
es
200
250
300
50
1
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
100
150
200
250
300
50
100
200
250
300
200
250
300
200
250
300
6.8
150
es
0.8
50
100
9.
Fig.7: Two-spp
2.4
150
es
1
100
100
4.6
0.8
50
150
es
1
100
100
7
0.8
50
50
model (1) undergoes
2 bifurcation as SSefficiency changes
from a low value .2
to high 11.2
11.2
150
Fig.8: Randomly
a2
2
x
0.8
x
0.7
x
0.6
x
0.5
V
I
1.5
sn
cn
200
Fig.5: Randomly varying SS-clearing rate. Random drops represent either antigenic
0.0001
2.5
F
cs t
F
Depending on parameters we get either species domination (F right, or V - left), or
coexistence (middle).
K
F
III
I
m
cs
y2
We incorporate 2 elements of adaptive SS-immunity in our
system (1): (i) time-lag L (several days) in stimulating SSeffectors; (ii) variable SS-clearing rate cs(t) (Fig. 5).
Adaptive immunity brings about new dynamic features –
transition from stable equilibrium to limit cycles
(recrudescent parasitemia), or more complicated chaotic
dynamics (Hopf bifurcation). We demonstrate it for a
single spp model, by looking at the effect of time-lag L
(Fig.6), and for mixed spp by varying SS-efficiency (Fig.7)
y1
Fig. 3: Phase-plane portraits of reduced system (1), stable equilibrium marked in black.
m
J
<<
y1
y2
‘Parasite + immunity’ regulatory circuit
growth (solid arrows) at rates
a1,2
Clearing (dashed arrows) by
(i) fever F; (ii) non-specific
(NS) immune effector I; (iii) a
1
specific (SS) immune
effectors J,K
Immune effectors: I,J,K
degrade (deactivate at
certain rates m)
Species F,V stimulate
production of immune
effectors and trigger ‘fever
switch’
a<a2
a2<a< a1
III
y1
We stress basic patterns and processes rather than
detailed structure and components on the ‘immuneparasite’ system. Those are
• Parasite replication
• Stimulation of immune effectors (by parasite Ag)
• Parasite clearing by immune effectors
Mathematical models can be either
Deterministic (continuous -‘Differential Equations’ or
discrete -‘finite-difference’), or stochastic, or combine both.
1)
V
a1<a
y2
MODELING PRINCIPLES
Two species: F,V
V
DYNAMIC PATTERNS FOR ADAPTIVE
SS-IMMUNITY
SS efficiency es
1Math.
1
cn
I
0.5
m
II
0
0.5
1
1.5
2
2.5
SUMMARY AND CONCLUSIONS
varying SS-clearing
creates ‘chaotic’
sequential patterns,
and could bring about
the demise of a ‘slow
spp’ (dashed). Upper
and lower panels
show ‘linear’ and ‘log’
plots of the process,
left and right columns
compare ‘fixed SS’ vs.
‘random SS’ (Fig.5).
Fig.1: Two-species ‘parasite-effector’ circuit
(1)
dF
dt
•
5 variables: F,V-spp,I,J,K- effectors, obey a coupled
differential (Ref.4), or differential-delay system (Ref.5)
•
 a1F  b1F  cn I  cs  J   K   F ;
•
 a2V  b1F  cn I  cs  J  K  V ;
dt
growth
dV
clearing
dI
dt
dJ
dt
 s n  F  V   mn I ;
printed by
NS stimulation

s sF
|L  ms J ;
dK
www.postersession.com
SS-stimulation
dt
 s sV |L  ms K ; Fig.2 : Fever controller F(F+V), turns
SS
on and off past ‘threshold’ levels
1.
2.
3.
4.
5.
Intra-host immune regulation of Plasmodium species (single or mixed) is modeled by systems of differential, or differential-delay
equations, that resemble some familiar patterns of population Biology (‘predation competition' ).
We combine analytic and numeric tools to explore possible outcomes: ‘coexistence’, ‘domination’, ‘cycles’, sequential patterns, in
terms of the basic parameters: NS and SS efficiencies, cross-reactivities, fitness (relative grwoth rates), time lags (in SS
induction).
Further applications (in progress) include: (i) Individual-based models of community transmission; (ii) Development of drug resistance;
(iii) Prediction of disease outcome and control strategies (treatment/vaccination) on individual and community level
References:
Boyd, M.F., Kitchen, S.F. (1937) Am J Trop Med 17, 855-861; 18, 505-514.
Bruce, M.C., Day, K.P. (2002), Curr Opin Microbiol 5, 431-7; 2003, Trends Parasitol. 19,271-7.
Anderson, R.M., May, R.M., Gupta, S., (1989), Parasitology, 99 Suppl, S59-79.
Mason, D.P., McKenzie, F.E., (1999), Am J Trop Med Hyg 61, 367-74; J Theor Biol 198, 549-66.
Gurarie,D., P.A. Zimmerman, C.H. King, (2005), J Theor Biol (to appear)