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quancol . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Spatial Representations and Analysis Techniques Part II: Analysis Vashti Galpin University of Edinburgh Bertinoro 22 June 2016 SFM-16 1 / 60 quancol .
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Vashti Galpin University of Edinburgh Bertinoro 22 June 2016
SFM-16 1 / 60
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1
Large discrete space models
2
SIR in space
3
Spatial moment closure
4
Going to the (spatial) limit
SFM-16 2 / 60
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SFM-16 3 / 60
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large: at least 100 distinct locations if not more regular: convenient to work with grid connectivity: full, von Neumann neighbourhood, Moore
neighbourhood
spatial homogeneity to heterogeneity full connectivity, homogeneous parameters and initial values smaller neighbourhood, homogeneous parameters and initial
values
smaller neighbourhood, homogeneous parameters, varying initial
values
smaller neighbourhood, heterogeneous parameters, varying initial
values
PDE analysis: step size of grid tends to zero SFM-16 4 / 60
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Assuming n subpopulations and p locations, then X (j)
i
is the size of the subpopulation i at location j. Xi = (X (1)
i
, . . . , X (p)
i
) Xi = p
j=1 X (j) i
X(j) = (X (j)
1 , . . . , X (j) n )
X (j) = n
i=1 X (j) i
X = (X1, . . . , Xn) X = n
i=1
p
j=1 X (j) i
= p
j=1
n
i=1 X (j) i
Xi = 1/p p
j=1 X (j) i
In the case of the n × m grid, p = n × m and Xi = 1/p n
j=1
m
k=1 X (j,k) i
SFM-16 5 / 60
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non-spatial model
S + I → I + I rate : kSI I → R rate : rI
spatial model with movement, assuming full connectivity
S(i) + I (i) → I (i) + I (i) rate : k(i)S(i)I (i) I (i) → R(i) rate : r(i)I (i) S(i) → S(j) i = j rate : ms(ij)S(i) I (i) → I (j) i = j rate : mi (ij)I (i)
defines a population CTMC with locations smaller neighbourhoods: X (i) → X (j) for j ∈ N(i) SFM-16 6 / 60
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deterministic fluid approximation with assumption of parameter
homogeneity across locations and full connectivity between locations dS(i) dt = −kS(i)I (i) −
p
msS(i) +
p
msS(j) dI (i) dt = kS(i)I (i) − rI (i) −
p
miI (i) +
p
miI (j) dR(i) dt = rI (i)
equivalent to adding location attribute to each subpopulation increases number of ODEs from 3 to 3p SFM-16 7 / 60
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deterministic fluid approximation with assumption of parameter
homogeneity across locations and specified neighbourhood dS(i) dt = −kS(i)I (i) −
msS(i) +
msS(j) dI (i) dt = kS(i)I (i) − rI (i) −
miI (i) +
miI (j) dR(i) dt = rI (i)
same number of ODEs but fewer terms in each SFM-16 8 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: no connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 9 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 10 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 11 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Deterministic: full connectivity
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
SFM-16 12 / 60
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Stochastic simulations
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001 No connectivity Von Neumann full connectivity neighbourhood
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Stochastic simulations
Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001
20 40 60 80 100 20 40 60 80 100 spatial average time <S>: no connectivity <I>: no connectivity <R>: no connectivity <S>: von Neumann <I>: von Neumann <R>: von Neumann <S>: full connectivity <I>: full connectivity <R>: full connectivity SFM-16 14 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 15 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except border, I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 16 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third k = 0.05
SFM-16 17 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 10, R(i)(0) = 0 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05 except bottom third mi = 0.001
SFM-16 18 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 19 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
www.quanticol.eu
Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
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Stochastic: von Neumann neighbourhood
Initial values: S(i)(0) = 30, I (i)(0) = 0, R(i)(0) = 0 except top left corner I (i)(0) = 10 Parameters: k = 0.01, r = 0.2, ms = mi = 0.05
SFM-16 20 / 60
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technique that abstracts from space provides spatial averages for each subpopulation closure of ODEs for averages give approximations useful for large discrete space models no general theory yet; model specific potential use large model with a few interesting locations hybrid treatment of locations approximate most locations with spatial moment ODEs use other techniques for interesting locations SFM-16 21 / 60
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Moments are the expected values of products of variables
considered at a single time point
E[X] is the first moment of X and the average value of X E[XY ] is a second joint moment and the average value of XY Var[X] = E[X 2] − E[X]2 is a second central moment and the
variance of X
Cov(X, Y ) = E[XY ] − E[X]]E[Y ] is a second central joint
moment and the covariance of X and Y
In general, E[X n1
1 X n2 2 . . . X nk k ] is the nth joint moment when
n = k
j=1 nj
SFM-16 22 / 60
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For a population CTMC, the behavior of each moment is
defined by an ODE (Engblom, 2006) dE[M(Y)] dt =
E[r(Y) · (M(Y + v) − M(Y))]
Typically, each moment ODE involves higher moments Infinite system of ODEs which can’t be simulated Use moment closure to truncate system SFM-16 23 / 60
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Stochastic linearisation approximate E[XY ] by E[X]E[Y ] (or similar products) equivalent to the assumption that Cov(X, Y ) = 0 Distribution assumption approximate higher moments with moments from chosen
distribution
log-normal is good for populations since it has non-negative
support
Assume higher-order moments are negligible approximate them with zero SFM-16 24 / 60
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Spatial moments are the expected value of products of variables
average over all locations, at a single time point
Typically, leads to infinite system of ODEs Apply moment closure techniques [Marion et al, 2002, 2005] No general expression for ODEs (yet) Consider for SIR model with homogeneous parameters and full
connectivity
SFM-16 25 / 60
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dE[S]/dt = −kE[SI ] dE[I]/dt = kE[SI ] − rE[I] dE[R]/dt = rE[I] dE[SI ]/dt = k(E[S2I ] − E[SI 2]) − rE[SI ] −msp(E[SI ] − E[SI]) −mip(E[SI ] − E[SI]) dE[S2]/dt = −2k(E[S2I ] +msp(E[S2] − E[SS])) dE[I 2]/dt = 2(kE[SI 2] − rE[I 2] −mip(E[I 2] − E[II])
SFM-16 26 / 60
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dE[S2I ]/dt = k(E[S3I ] − 2E[S2I 2]) − rE[S2I ] −msp(E[S2I ] − E[S2I]) −mip(E[S2I ] − E[S2I]) dE[SI 2]/dt = −kE[SI 3] + 2kE[S2I 2] − rE[S2I ] −msp(E[SI 2] − E[SI I]) −mip(E[SI 2] − E[SI I])
SFM-16 27 / 60
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Rate S(i) S(j) I (i) E (j) R(i) kS(i)I (i)
+1 rI (i)
+1 msS(i)
+1 miI (i)
+1 S(i)(t + δt) − S(i)(t) = δS(i) = [−kS(i)(t)I (i)(t) −ms
p
S(i)(t) + ms
p
S(j)(t)]δt
SFM-16 28 / 60
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Rate S(i) S(j) I (i) E (j) R(i) kS(i)I (i)
+1 rI (i)
+1 msS(i)
+1 miI (i)
+1 I (i)(t + δt) − I (i)(t) = δI (i) = [kS(i)(t)I (i)(t) − rI (i)(t) −mi
p
I (i)(t) + mi
p
I (j)(t)]δt
SFM-16 29 / 60
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Rate S(i) S(j) I (i) E (j) R(i) kS(i)I (i)
+1 rI (i)
+1 msS(i)
+1 miI (i)
+1 R(i)(t + δt) − R(i)(t) = δR(i) = [rI (i)(t)]δt
SFM-16 30 / 60
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S(i)(t + δt) − S(i)(t) = [−kS(i)(t)I (i)(t) −ms
p
S(i)(t) + ms
p
S(j)(t)]δt Take spatial averages 1/p
p
(S(i)(t + δt) + S(i)(t)) = [−1/p
p
(kS(i)(t)I (i)(t) −ms
p
S(i)(t) + ms
p
S(j)(t))]δt
SFM-16 31 / 60
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Rewrite with spatial average notation S(t + δt) + S(t) = [−kSI (t) −ms
p
S(t) + ms
p
S(t)]δt = −kSI (t)δt Take expectations E[S](t + δt) + E[S](t) = −kSI (t)δt Divide by δt and take the limit as δt → 0 dE[S](t) = −kSI
SFM-16 32 / 60
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Expand S(i)I (i)(t + δt) = S(i)(t + δt)I (i)(t + δt) = (S(i)(t) + δS(i))(I (i)(t) + δI (i)) = S(i)I (i)(t) + S(i)δI (i) + I (i)δS(i) + δS(i)δI (i) Hence S(i)I (i)(t + δt) − S(i)I (i)(t) = S(i)δI (i) + I (i)δS(i) + K = S(i)[kS(i)I (i) − rI (i) − ms
p
S(i) + ms
p
S(j)]δ + + I (i)[−kS(i)I (i) − mi
p
I (i) + mi
p
I (j)]δ + K
SFM-16 33 / 60
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Taking spatial averages SI (t + δt) − SI (t) = [k(S2I − S2I ) − rSI − ms
p
SI + msS
p
S(j) − mi
p
SI + miS
p
I (j)]δ + K = [k(S2I − S2I ) − rSI − msp(SI − SI) − mip(SI − SI)]δ + K Following the same procedure as before dE[SI ](t + δt)/dt) = k(E[S2I ] − E[S2I ]) − rE[SI ] −msp(E[SI ] − E[SI]) −mip(E[SI ] − E[SI])
SFM-16 34 / 60
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Taking spatial averages SI (t + δt) − SI (t) = [k(S2I − S2I ) − rSI − ms
p
SI + msS
p
S(j) − mi
p
SI + miS
p
I (j)]δ + K = [k(S2I − S2I ) − rSI − msp(SI − SI) − mip(SI − SI)]δ + K Following the same procedure as before dE[SI ](t + δt)/dt) = k(E[S2I ] − E[S2I ]) − rE[SI ] −msp(E[SI ] − E[S]E[I]) −mip(E[SI ] − E[S]E[I])
SFM-16 35 / 60
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dE[S]/dt = −kE[SI ] dE[I]/dt = kE[SI ] − rE[I] dE[R]/dt = rE[I] dE[SI ]/dt = k(E[S2I ] − E[SI 2]) − rE[SI ] −msp(E[SI ] − E[SI]) −mip(E[SI ] − E[SI]) dE[S2]/dt = −2k(E[S2I ] +msp(E[S2] − E[SS])) dE[I 2]/dt = 2(kE[SI 2] − rE[I 2] −mip(E[I 2] − E[II])
SFM-16 36 / 60
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dE[S]/dt = −kE[SI ] dE[I]/dt = kE[SI ] − rE[I] dE[R]/dt = rE[I] dE[SI ]/dt = k(E[S2I ] − E[SI 2]) − rE[SI ] −msp(E[SI ] − E[SI]) −mip(E[SI ] − E[SI]) dE[S2]/dt = −2k(E[S2I ] +msp(E[S2] − E[SS])) dE[I 2]/dt = 2(kE[SI 2] − rE[I 2] −mip(E[I 2] − E[II])
SFM-16 36 / 60
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dE[S2I ]/dt = k(E[S3I ] − 2E[S2I 2]) − rE[S2I ] −msp(E[S2I ] − E[S2I]) −mip(E[S2I ] − E[S2I]) dE[SI 2]/dt = −kE[SI 3] + 2kE[S2I 2] − rE[S2I ] −msp(E[SI 2] − E[SI I]) −mip(E[SI 2] − E[SI I])
SFM-16 37 / 60
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dE[S2I ]/dt = k(E[S3I ] − 2E[S2I 2]) − rE[S2I ] −msp(E[S2I ] − E[S2I]) −mip(E[S2I ] − E[S2I]) dE[SI 2]/dt = −kE[SI 3] + 2kE[S2I 2] − rE[S2I ] −msp(E[SI 2] − E[SI I]) −mip(E[SI 2] − E[SI I])
SFM-16 37 / 60
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stochastic linearisation for third order moments approximate E[S2I ] with E[S]E[SI ] or E[S2]E[I] approximate E[SI 2] with E[S]E[I 2] or E[SI ]E[I] stochastic linearisation for fourth order moments approximate E[S3I ], E[SI 3] and E[S2I 2] log-normal distribution to approximate third order moments
E[S2I ] = E[S2]E[SI ]2 E[S]2E[I] E[SI 2] = E[I 2]E[SI ]2 E[I]2E[S]
SFM-16 38 / 60
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Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001 Stochastic versus deterministic
20 40 60 80 100 20 40 60 80 100 spatial average time S: stochastic I: stochastic R: stochastic S: deterministic I: deterministic R: deterministic SFM-16 39 / 60
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Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001 Stochastic versus stochastic linearisation of third moments
20 40 60 80 100 20 40 60 80 100 spatial average time S: stochastic I: stochastic R: stochastic S: SL of third moments I: SL of third moments R: SL of third moments SFM-16 40 / 60
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Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001 Stochastic versus stochastic linearisation of fourth moments
20 40 60 80 100 20 40 60 80 100 spatial average time S: stochastic I: stochastic R: stochastic S: SL of fourth moments I: SL of fourth moments R: SL of fourth moments SFM-16 41 / 60
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Initial values: S(i)(0) = 49, I (i)(0) = 1, R(i)(0) = 0 Parameters: k = 0.011, r = 0.1, ms = mi = 0.00001 Stochastic versus log normal third moments
20 40 60 80 100 20 40 60 80 100 spatial average time S: stochastic I: stochastic R: stochastic S: LN for third moments I: LN for third moments R: LN for third moments SFM-16 42 / 60
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techniques for approximating average behaviour over space generality: when does it work? different parameter ranges for SIR
general definition of ODEs applications for global behaviour to abstract “uninteresting” parts of a model
SFM-16 43 / 60
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SFM-16 44 / 60
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Current approach for scalable modelling
SFM-16 45 / 60
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Current approach for scalable modelling State-space
fluid approximation
− − − − − − − − − − − − − − → ODEs explosion
SFM-16 45 / 60
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Current approach for scalable modelling State-space
fluid approximation
− − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space?
SFM-16 45 / 60
www.quanticol.eu
Current approach for scalable modelling State-space
fluid approximation
− − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? Add space
SFM-16 45 / 60
www.quanticol.eu
Current approach for scalable modelling State-space
fluid approximation
− − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? Bigger
fluid approximation
− − − − − − − − − − − − − − → PDEs state-space explosion
SFM-16 45 / 60
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Current approach for scalable modelling State-space
fluid approximation
− − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? Bigger
fluid approximation
− − − − − − − − − − − − − − → PDEs state-space explosion Yes, here is how it can be done
SFM-16 45 / 60
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population discrete-space model on grid partial differential equation model results for population discrete-space model fluidisation of space PDE solver uses discretisation of space [Tschaikowski and Tribastone, 2014]
SFM-16 46 / 60
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Consider agents which are moving on a lattice in the unit square. SFM-16 47 / 60
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Consider agents which are moving on a lattice in the unit square. The lattice consists of (K + 1)2 regions. SFM-16 47 / 60
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Consider agents which are moving on a lattice in the unit square. The lattice consists of (K + 1)2 regions.
0, 1 1, 1 0, 0 1, 0
SFM-16 47 / 60
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Consider agents which are moving on a lattice in the unit square. The lattice consists of (K + 1)2 regions.
0, 1
1 2, 1
1, 1 0, 1
2 1 2, 1 2
1, 1
2
0, 0
1 2, 0
1, 0
SFM-16 48 / 60
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Consider agents which are moving on a lattice in the unit square. The lattice consists of (K + 1)2 regions.
0, 1
1 4, 1 1 2, 1 3 4, 1
1, 1 0, 3
4 1 4, 3 4 1 2, 3 4 3 4, 3 4
1, 3
4
0, 1
2 1 4, 1 2 1 2, 1 2 3 4, 1 2
1, 1
2
0, 1
4 1 4, 1 4 1 2, 1 4 3 4, 1 4
1, 1
4
0, 0
1 4, 0 1 2, 0 3 4, 0
1, 0
SFM-16 49 / 60
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agents moving on a grid using von Neumann neighbourhood and
with random, unbiased walk
with many agents, we know how to approximate behaviour with
ODEs
the number of ODEs is proportional to number of grid points the finer the granularity of the space, the larger the ODE system we can approximate the ODE system by a system of partial
differential equations that does not depend on the number of grid points
technique supported by Spatial Fluid Extended Process Algebra
(SFEPA) [Tschaikowski and Tribastone, 2014]
SFM-16 50 / 60
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The fluid approximation applied to agents give the ODEs for the
nonspatial model dS/dt = qrSR − sS dR/dt = zR − rRS
SFM-16 51 / 60
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The fluid approximation applied to agents give the ODEs for the
nonspatial model dS/dt = qrSR − sS dR/dt = zR − rRS
Considering this fluid approximation over a grid gives the ODEs
dS(x,y)/dt = qrS(x,y)R(x,y) − sS(x,y) + µS△dS(x,y) dR(x,y)/dt = zR(x,y) − rS(x,y)R(x,y) + µR△dR(x,y)
SFM-16 51 / 60
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The fluid approximation applied to agents give the ODEs for the
nonspatial model dS/dt = qrSR − sS dR/dt = zR − rRS
Considering this fluid approximation over a grid gives the ODEs
dS(x,y)/dt = qrS(x,y)R(x,y) − sS(x,y) + µS△dS(x,y) dR(x,y)/dt = zR(x,y) − rS(x,y)R(x,y) + µR△dR(x,y)
These can be approximated by the following PDE system
[Okuba and Levin, 2001] ∂tS = qrSR − sS + µS△S ∂tR = zR − rSR + µR△R
SFM-16 51 / 60
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generalise model to d predators results
d = 1 d = 3 d = 5 K Error ODE Time Error ODE Time Error ODE Time 4 0.082 0 s 0.086 0 s 0.090 0 s 8 0.026 0 s 0.030 3 s 0.034 11 s 16 0.014 5 s 0.018 68 s 0.021 241 s 20 0.013 13 s 0.017 195 s 0.020 764 s PDE Time 0 s 1 s 2 s
The ODEs and PDEs were solved using ode15s and parabolic,
respectively
SFM-16 52 / 60
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generalise model to d predators results
d = 1 d = 3 d = 5 K Error ODE Time Error ODE Time Error ODE Time 4 0.082 0 s 0.086 0 s 0.090 0 s 8 0.026 0 s 0.030 3 s 0.034 11 s 16 0.014 5 s 0.018 68 s 0.021 241 s 20 0.013 13 s 0.017 195 s 0.020 764 s PDE Time 0 s 1 s 2 s
The ODEs and PDEs were solved using ode15s and parabolic,
respectively
PDEs solver is faster because the underlying spatial
discretization is governed by the PDEs and not by K
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−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 (a) ODE Solution −1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 (b) PDE Solution
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ODE$with$8x8$points$
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ODE$with$16x16$points$
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ODE$with$32x32$points$
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PDE$
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spatial representations discrete versus continuous
spatial analysis techniques dependent on representation general overview two in some detail choice for modelling CAS discrete space with population aggregation questions to be answered by modelling SFM-16 58 / 60
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Energy-efficient computing for wildlife tracking: Design tradeoffs and early experiences with ZebraNet, SIGPLAN Notices, 37:96107, 2002
spatially heterogeneous environments, Journal of Theoretical Biology, 232:127142, 2005
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316:12981301, 2007
Spatial Stochastic Process Algebra, Proceedings of VALUETOOLS 2014
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