models of host-parasite regulation d. gurarie. i. macro-parasites (helminth) regulation macro...
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Models of host-parasite regulation
D. Gurarie
I. Macro-parasites (helminth) regulation
• Macro parasites can regulate the host reproduction and growth (Anderson-May 1978) ;
• Contribute to host birth/death rates (additive or multiplicative)
; or 1 - birth
; or 1 - death
b b b b b
d d d d d
Single host models
• Inf+L system: Burden strata
0 - " " (host) strata; - total host pop.
- parasite pop. in i-strata; - total parasite
- larvae pop.
i ii
i i ii
h i H h
p ih P p
L
00 0 1 0 0
0
1 1 1
; - logistic constraint
...
; 1,2,3...
i ii
ii i i i i i i i
dhbh d H h p L h
dt
dhd H h p p L h h i
dtdL
P H Ldt
Reduced (closure) models
• Fix-k model (k – aggregation index; w=P/H –mean burden)
; - effective rates (mean over distribution)
i
i i
H b w d w H H
w L b w d w w id
L Hw H L
b b d d
20 0 0
1; ; 1
k
i
kb b d d w d id d w d w w
k w b k
For negative binomial with aggregation k
Reduced ‘predator-prey’
• For quasi-equilibrated larvae:
*
0
;
,
,
HwL
H
H b w d w d H H f H w
H dw b w d w w g H w
H k
H1 H0
f=0
g=0
H
Two-host models (schisto)
Life Cycle ofSchistosomiasis
0 0 0 1 0 00
1 1 1
1
- human strata
- cercaria
- susceptible snails;
- infected snails;
i ii
i i i i i i i i i
S C
S S
S
H
h bh d H h p C h
h d H h p p C h h
C Y H C
X N X m X
Y m X Y
M P
- miracidia
- total snail pop.; - miracidium/snail
MM m N
MN X Y m
N
Human ‘burden’ strata + snail ‘prevalence’+ 2 larvae
=> Reduced 6D, 4D, 3D models. Differs from standard Ross-Macdonald
0
w aY w
Y bw N Y Y
-“Mean burden” + “infected prevalence”-(for const N)
Applications
• Regulation and stability of host–parasite populations (Anderson-May 1978)
• Aggregation for single parasite species and multi-strain competition, coexistence, invasion (Dobson-Roberts 1994; Pugliese 2000, Rosa-Pugliese 2002)
• Parasite (schisto) diversity and drug resistance (Feng et al 2001)
• Goal: application to schisto control
II. Micro-parasite regulation (malaria)
• Intra-host P. regulation by host immunity• Community transmission modulated by immune
regulation• Goal: improved SIR (for control, drug resistance,
et al)!
RBC cycle and immune response
• Merozoites: 48 hr replication cycle in RBC with replicating factor:
• Fever suppression at pyrogenic level:
• Immunity (transient, lost in the absence of reinfection). 3 forms: fever control; species transcending (ST); species specific (SS)
12 16r
3 40 10 10 pRBC/x l
Par
asit
emia
(P
aras
ites
/μl)
Tem
p. (
°F)
105 -103 -101 -99 -
96 -
10
100
1,000
10,000
Inoculation by 15 P. falciparum (black line; gametocytes – black dots) + P. vivax (red line) infected A. quadrimaculatus
Day #1 Day #25 Day #50 Day #75
Tx
Day #86
Tx
Zimmerman et al. Figure 1A
Natural infection histories
Zimmerman et al. Figure 1B
Par
as
item
ia (
Pa
ras
ite
s/μ
l)T
emp
. (°
F)
105 -
10
100
1,000
10,000P. vivax
Inoculation by 12 P. falciparum + P. vivax infected A. quadrimaculatus
Day #1
Tx
Day #20 Day #40 Day #60 Day #80 Day #100 Day #120 Day #140
99 -101 -103 -
Continuous regulation models
• Feedback circuits:Stimulation/growth
Inhibition/loss
ugM
b
x
L
fM
b
1
u
b2
x
L
yL
g,N h,
N
c
1
c
2
w
v
gM
b
1
u
b
2
x
L
yL
Single P. with fever and ST
2 species with fever and ST
2 species with fever, ST and SS
DDE models
• Single w. ST:
• 2 species w. ST:
• 2 species w. ST and SS (5D system)
0
1 | ; - parasite
; - ST effector
L
N
dx
dt
du x
dt x
x bu x x
f bx u u
0
1 1
2 2 2 2
21 2 2
1 /
1 /
L
L
M
dx
dt
dy
dt
du x y
dt x
x y b u x
x y b u y
f b x b y u
Results: single species
20 40 60 80 100 120 140
0.25
0.5
0.75
1
1.25
1.5
b 0.2
xtt
ST
20 40 60 80 100 120 140
0.2
0.4
0.6
0.8
1
1.2
1.4
b 0.08
xtt
ST
20 40 60 80 100 120 140
0.2
0.4
0.6
0.8
1
1.2
1.4
b 0.04
xtt
ST
2 species: SS/ST
0 200 400 600 800
02
46
810
u,v,w
0 200 400 600 800
-80
-60
-40
-20
0
goLx,y
Future plans
• Deterministic and stochastic growth/removal models w. discrete time step
• Estimation, validation• Applications to community transmission ???