modelling two host strains with an indirectly transmitted pathogen angela giafis 20 th april 2005
Post on 19-Dec-2015
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TRANSCRIPT
Motivation
1. Disease can be spread by contact with infectious materials (free stages) in the environment.
2. Interested in what happens when 2 different host types, one more susceptible to infection the other more resistant, are subjected to such an infection.
Structure
• Discuss the differences between two
models.
• Equilibrium solutions, feasibility and
stability.
• Show parameter plots.
• Look at some dynamical illustrations.
Models
22222
22222
2222222
11111
11111
1111111
1
1
WIdtdW
IWSdtdI
IWSKHSr
dtdS
WIdtdW
IWSdtdI
IWSKHSr
dtdS
WIdt
dW
IWSWSdt
dI
WSHSqSrdt
dS
WSHSqSrdt
dS
2211
222222
1111111
2
Model 1:
Model 2:
Both models have demography (births and deaths) and infection but as we see there are details that differ and this turns out to matter.
Much of the behaviour of model 1 is governed by the term D0 which is the basic depression ratio, where
.)(
0W
ISKD
Equilibrium Solutions and Feasibility: Model 1
1. Total extinction (0,0,0,0,0,0)
2. Uninfected coexistence (S1
*,0,0,K-S1*,0,0)
with S1* є [0,K]
3. Strain 1 alone with the pathogen (Ŝ1,Î1,Ŵ1,0,0,0)
4. Strain 2 alone with the pathogen (0,0,0,Ŝ2,Î2,Ŵ2)
1. Feasible
2. Feasible
3. Feasible if K > HT,1
4. Feasible if K > HT,2
Equilibrium Solutions and Feasibility: Model 2
1. Total extinction (0,0,0,0)
2. Strain 1 alone at its carrying
capacity (K1,0,0,0)
3. Strain 2 alone at its carrying
capacity (0,K2,0,0)
4. Strain 1 alone with the pathogen
(Ŝ1,0,Î1,Ŵ1)
5. Strain 2 alone with the pathogen
(0,Ŝ2,Î2,Ŵ2)
6. Coexistence of the strains and
the pathogen (S1*,S2
*,I*,W*)
1. Feasible
2. Feasible
3. Feasible
4. Feasible if K1 > HT,1
5. Feasible if K2 > HT,2
6. Feasible if q1υ2-q2υ1 < 0 and HT,1<K12<HT,2
Stability: Model 1
22
222222
22222
222
2222222
11
111111
111111
11111
111
1
0000
00
1
0000
00
1
SWS
SK
SrW
K
Sr
K
HrS
K
Sr
K
Sr
SSW
SK
Sr
K
SrS
K
SrW
K
Sr
K
Hr
1. (0,0,0,0,0,0) is unstable
2. (S1*,0,0,K-S1
*,0,0) is neutrally stable iff (K-S1*)/HT,2+S1
*/HT,1<1 and given feasibility. For point stability we need ABC-C2-A2D>0, if ABC-C2-A2D<0 we expect limit cycles.
3. (Ŝ1,Î1,Ŵ1,0,0,0) is stable iff D0,1<D0,2 and given feasibility. For point stability we need A1B1-C1>0, if A1B1-C1<0 we expect limit cycles.
4. (0,0,0,Ŝ2,Î2,Ŵ2) is stable iff D0,2<D0,1 and given feasibility. For point stability we need A2B2-C2>0, if A2B2-C2<0 we expect limit cycles.
Stability: Model 2
00
221121
22222222222
11111111111
SSWW
SSqWSqHqrSq
SSqSqWSqHqr
1. (0,0,0,0) is unstable
2. (K1,0,0,0) is stable iff K1<HT,1
3. (0,K2,0,0) is unstable
4. (Ŝ1,0,Î1,Ŵ1) is stable iff HT,1>K12 and given feasibility. For point stability we need A1B1-C1>0, if A1B1-C1<0 we expect limit cycles.
5. (0,Ŝ2,Î2,Ŵ2) is stable iff K12>HT,2 and given feasibility. For point stability we need A2B2-C2>0, if A2B2-C2<0 we expect limit cycles.
6. (S1*,S2
*,I*,W*) is stable given feasibility. For point stability we need ABC-C2-A2D>0, if ABC-C2-A2D<0 we expect limit cycles.
Parameter Plots
• Trade-off : individual hosts pay for their increased resistance to the pathogen by a reduction in the contribution to the overall fitness.
• Our parameter plots are representative of our stability conditions.
• The susceptible strain (strain 1) values are fixed and we will vary two of the resistant strain (strain 2) parameters.
Summary
• In both models we considered two cases, one where the more susceptible strain is point stable (A1B1-C1>0) and one where we expect to see limit cycles (A1B1-C1<0).
• Model 1: coexistence not possible. • Model 2: coexistence possible. Indeed cyclic coexistence
of all the populations is possible.
• Outcome depends on balance between costs and benefits.