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)
b  b  b; or b 1  b  - birth
d  d  d ; or d 1  d  - death
Single host models
•
Inf+L system: Burden strata
hi - "i" (host) strata; H   i 0 hi - total host pop.

pi  ihi - parasite pop. in i-strata; P   i pi - total parasite
L - larvae pop.

dh0
  bi hi   d 0   H  h0   p1   L 0h0 ;  - logistic constraint
dt
i 0
...
dhi
   d i   H  hi    pi 1  pi    L i 1hi 1  i hi  ; i  1, 2,3...
dt
dL
 P   H  L
dt
Reduced (closure) models
• Fix-k model (k – aggregation index; w=P/H –mean burden)
 H  b  w   d  w    H  H

 w   L    b  w   d  w   w  id i

 L   Hw      H  L
b  bi ; d  d i - effective rates (mean over distribution)
For negative binomial with aggregation k
k
k
 1  2



b  b0 
;
d

d

w

d
;
id

d
w


d

1
w

w
0
i
0


 k

 k  wb 




Reduced ‘predator-prey’
• For quasi-equilibrated larvae:
L* 
 Hw
;
 H
 H  b  w    d 0  wd    H  H  f  H , w 


 H
d 
w




b
w


d

w w  g  H , w 



  H
k 


g=0
f=0
H
H1
H0
Two-host models (schisto)
Life Cycle of
Schistosomiasis
Human ‘burden’ strata + snail ‘prevalence’+ 2 larvae


h0   bi hi   d 0   H  h0   p1  C 0h0
i 0

h    d   H  h    p  p   C  h   h  - human strata
i
i
i 1
i
i 1 i 1
i i
 i
C  SY   C   H  C - cercaria
 X        S N  X   S   m  X - susceptible snails;

Y   S   m  X  1Y - infected snails;
M  H P   M M    m  N - miracidia
N  X  Y - total snail pop.; m 
M
- miracidium/snail
N
=> Reduced 6D, 4D, 3D models. Differs from standard Ross-Macdonald
w  aY   w
Y  bw  N 0  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, RosaPugliese 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)!
Life cycle
More details
RBC cycle and immune response
• Merozoites: 48 hr replication cycle in RBC with
replicating factor: r  12  16
• Fever suppression at pyrogenic level:
x0  103  104 pRBC/l
• Immunity (transient, lost in the absence of
reinfection). 3 forms: fever control; species
transcending (ST); species specific (SS)
Natural infection histories
Temp. (°F)
Day #1
Day #25
Day #75
Day #50
Tx
Tx
105 103 101 99 96 -
Day #86
Parasitemia (Parasites/μl)
10,000
1,000
100
10
Inoculation by 15 P. falciparum (black line; gametocytes – black dots) + P. vivax (red line) infected A. quadrimaculatus
Zimmerman et al. Figure 1A
Parasitemia (Parasites/μl)
Temp. (°F)
Day #1 Day #20
Day #40
Day #60
Day #80
Day #100
Day #120
Day #140
Tx
105 103 101 99 -
10,000
P. vivax
1,000
100
10
Inoculation by 12 P. falciparum + P. vivax infected A. quadrimaculatus
Zimmerman et al. Figure 1B
Continuous regulation models
• Feedback circuits:

gM
x
L
Stimulation/growth

u
Single P. with fever and ST
Inhibition/loss
b
2
1
x
y
L
g

b
L
b
1
2
u

1
2
f
M
x

L
b
g,N
2 species with fever and ST
M
c
1
1
L
b2
1
h,
c
N
2
u

v
y
w
2
2 species with fever, ST and SS
DDE models
dx
• Single w. ST:
• 2 species w. ST:
dt
 1    x |L   bu  x; x - parasite
 x
 f        bx  u; u - ST effector
dt
 x0  N
du
dx
dt
dy
dt
 1  1   x  y /   L  b1u  x
  2 1   2   x  y /   L   2b2u  y
x y
2


 f




b
x




1
2 b2 y   u

dt
 x0  M
du
• 2 species w. ST and SS (5D system)
Results: single species
b
0.04
1.4
x t
1.2
1
0.8
t
0.6
0.4
0.2
ST
20
40
60
b
80
100
120
140
80
100
120
140
0.08
1.4
x t
1.2
1
0.8
t
0.6
ST
0.4
0.2
20
40
60
b
1.5
1.25
0.2
ST
x t
1
0.75
t
0.5
0.25
20
40
60
80
100
120
140
2 species: SS/ST
0
Log x,y
-20
-40
-60
-80
0
200
400
600
800
10
u,v,w
8
6
4
2
0
0
200
400
600
800
Future plans
• Deterministic and stochastic growth/removal
models w. discrete time step
• Estimation, validation
• Applications to community transmission ???