Point: 10 Scientific Computing by Joa on epischisto.org • Scientific computing (or computational science) is the field of study concerned to the construction of mathematical models and techniques of numerical solutions using computers to analyze and solve scientific and engineering problems. • Typically, such models require a large amount of calculation, and usually run on computers with great power scalability (parallel and distributed machines) • Scientific computing is currently regarded as a third way for science complementing experimentation (observation) and theory. http://www.springer.com/mathematics/computational+science+%26+engineering/journal/10915
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Computational Epidemiology as a scientific computing area: cellular automata and SILP for disease monitoring? Why not?
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Point: 10 Scientific Computing
by Joa on epischisto.org
• Scientific computing (or computational science) is the field of study concerned to the construction of mathematical models and techniques of numerical solutions using computers to analyze and solve scientific and engineering problems.
• Typically, such models require a large amount of calculation, and usually run on computers with great power scalability (parallel and distributed machines)
• Scientific computing is currently regarded as a third way for science complementing experimentation (observation) and theory.
• Epidemiology is the study of the distribution and determinants of health-related states or events (including disease), and the application of this study to the control of diseases and other health problems.
http://jech.bmj.com/
• The Tipping Point, Epidemics are a function of
the people who transmit infectious agents, the infectious agent itself, and the environment in which the infectious agent is operating. And when an epidemic tips, when it is jolted out of equilibrium, it tips because something has happened, some change has occurred in one (or two or three) of those areas.
a cellular automaton Cellular automaton A is a 4-upla A = <G, Z, N, f>,
where • G – set of cells • Z – set of possible cells states • N – set, which describes cells neighborhood • f – transition function, rules of the automaton:
– Z|N|+1Z (for automaton, which has cells “with memory”)
– Z|N|Z (for automaton, which has “memoryless” cells)
Statistical mechanics of cellular automata Rev. Mod. Phys. 55, 601 – Published 1 July 1983
Simple initial conditions: Homogeneous states or Self-similar patterns Random initial conditions:
Self-organization phenomena
Moore Neighbourhood (in grey) of the cell marked with a dot in a 2D square grid
Research? for example, with schistosomiasis... we would like to provide an almost real-time and future risk map for it... by monitoring the self-organization endemics states...
No data No case reports No statistical series No reliable data Only poor comunities Fiocruz (Schistosomiasis Laboratory) works to discover, to control and to report Fiocruz starts a new study in 2006...
why Carne de Vaca? Tourism interest Isolated population Identified cases Not analysed yet FIOCRUZ starts a new study Near from UFRPE Local support: politicians, population
The village comprises around 1600 people in 1041 households distributed in 70 blocks and covering approximately 4 km2.
* No information on sex for one individual. 1 population. 2 Number of positives. 3 Prevalence
per 100 inhabitants.
Spatial pattern, water use and risk levels associated with the transmission of schistosomiasis on the north coast of Pernambuco, Brazil. Cad. Saúde Pública vol.26 no.5 Rio de Janeiro May 2010.
http://dx.doi.org/10.1590/S0102-311X2010000500023
2008 – 2009, data analysis and reports... Parasitological exams on 1100 residents
2008 and 2009 data analysis and reports... Summary data for molluscs collected...
Ecological aspects and malacological survey to identification of transmission risk' sites for schistosomiasis in Pernambuco North Coast, Brazil. Iheringia, Sér. Zool. 2010, vol.100, n.1, pp. 19-24.
Susceptible human population 0-23 social inquires (Paredes et al, 2010)
Infected human population 0-23 croposcological inquires (Paredes et al, 2010)
Recovered population of humans 0-23 social inquires (Paredes et al, 2010)
Rate of mobility of humans 0-26% social inquires (Paredes et al, 2010)
Rate of mobility of molluscs 0-2% malacological research (Souza et al, 2010)
Population of healthy molluscs 0-1302 malacological research (Souza et al, 2010)
Population of infected molluscs 0-11 malacological research (Souza et al, 2010)
Area susceptible to flooding 0-45%
LAMEPE - Meteorological Laboratory of Pernambuco (lamepe, 2008)
and environmental inquires (Souza et al, 2010)
Connection to other cells 0-100%
LAMEPE - Meteorological Laboratory of Pernambuco (lamepe, 2008)
and environmental inquires (Souza et al, 2010)
Rate of human infection 0-100% croposcological inquires and social inquires (Paredes et al, 2010)
Rate of human re-infection 0-100% croposcological inquires and social inquires (Paredes et al, 2010)
Recovery rate 0-100% croposcological inquires and social inquires (Paredes et al, 2010)
Mollusc infection rate 0-100% malacological research (Souza et al, 2010)
Rate of sanitation 0-93% social and environmental inquires (Souza et al, 2010)
Rainfall of the area 39-389mm LAMEPE - Meteorological Laboratory of Pernambuco (Lamepe, 2008)
From one year (population 1 snapshot, molluscs 12 snapshots) without previous historical...
Mechanistic epidemic models
Two alternative approaches
Top-down Population-based Models (PbMs)
Bottom-up Agent-based Models (AbMs)
PbM AbM
one proposal: a top-down approach using a cellular automaton
a b
1 km
a ba b
1 km
simulation space, a 10x10 square grid
the dynamics
Mollusk population dynamics a growth model for the number of individuals (N) that
considers the intrinsic growth rate (r) and the maximum
sustainable yield or carrying capacity (C) defined at each
site (Verhulst, 1838):
)1(C
NrN
dt
dN
Human infection dynamics (SIR - SI)
This model splits the human population into three compartments: S (for susceptible), I (for infectious) and R (for recovered and not susceptible to infection) and the snail population into
two compartments: MS (for susceptible mollusk) and MI (for infectious mollusk).
Socioeconomic and environmental factors
environmental quality of the nine collection sites in Carne de Vaca, according to the criteria of Callisto et al (Souza et al, 2010).
rteN
NC
CtN
0
01
)(
the model calculates the local increase of population using equation 1 and calculating N(t+1) out from N(t). The values for r and C are set at each site and each time step, using monthly meteorological inputs and considering the ecological quality of the habitat
(1)
αRχI=dt
dR
χI·S·Mp=dt
dI
αR+p·S·M=dt
dS
IH
I
ISMI
SSMS
rM·I·Mp=dt
dM
rM·I·Mp=dt
dM
(3a)
(3b)
Cells and infection forces
states black: rate of human infection = 100%; red: 80% ≤ rate of human infection < 100%; light red: 60% ≤ rate of human infection < 80%; yellow: 40% ≤ rate of human infection < 60%; light yellow: 20% ≤ rate of human infection < 40%; cyan: 0% ≤ rate of human infection < 20%.
Infection forces Human
S -> I (infected molluscs contact, pH)
I -> R (if treated (1-α), χ) Molluscs
S -> I (infected human contact, pM)
the algorithm
1. Choose a cell in the world;
2. For each human in the cell perform a random walk weighted by the “probability of movement" defined
at each site.
Repeat these steps for every cell in the world. Then update data.
3. Choose a cell in the world;
4. Call the “Events” process;
5. Return the individual to his original cell after the infection phase;
6. Choose a cell in the world;
7. For the mollusk population in that cell, perform a diffusion process weighted by the “rate of movement"
defined at each site;
Repeat these steps for every cell in the world. Then update data.
1. Increase the population of mollusks using the growth model described in Section 3.1;
2. Compute the transition between population compartments of humans using the set of equations (3b)
defined in Section 3.2;
3. Compute the transition between population compartments of humans using the set of equations (3a)
defined in Section 3.2;
Update local data of the spatial cell.
Events process
Main
sumulations
Mathematica 7.0 (Mathematica, 2011) with a processor Intel i5 3GHz, 4MB Cache, 8GB RAM.
Computational costs of a complete simulation when assuming a fixed world size (10x10 cells) and extent (365 time steps) and an increasing number of parameters being swept for rejection sampling (from 1 to 15)
Computational vs Statistical models Day 26 Day 43 Day 88
Day 106 Day 132 Day 365Color Legend
I = 100%80% ≤ I < 100%
60% ≤ I < 80%
40% ≤ I < 60%
20% ≤ I < 40%
0% ≤ I < 20%
(I = percentage of
infected humans)
Temporal
evolution
Day 26Day 26 Day 43Day 43 Day 88Day 88
Day 106Day 106 Day 132Day 132 Day 365Day 365Color Legend
I = 100%80% ≤ I < 100%
60% ≤ I < 80%
40% ≤ I < 60%
20% ≤ I < 40%
0% ≤ I < 20%
(I = percentage of
infected humans)
Temporal
evolution “according to the risk indicator, in the scattering diagram of Moran represented in the Box Map (Figure 2), indicated 18 areas of highest risk for the schistosomiasis, all located in the central sector of the village. Areas with lower risk and areas of intermediate risk for occurrence of the disease were located in the north and central portions with some irregularity in the distribution”
Predictive scenarios
2012 2017 2022 2027Color legend
I = 100%
80% ≤ I < 100%60% ≤ I < 80%
40% ≤ I < 60%
20% ≤ I < 40%
0% ≤ I < 20%
Predictive scenarios generated with the parameter calibration of the year 2007 that show endemic schistosomiasis. I stands for the average percentage of infected humans per spatial cell predicted by the model
INNOVATION on mathematical and computational simulation using Cellular Automata
What are we doing now? Running
simulations on Mobile platforms and these
simulations will guide the data collect
the codes of the humans, 2013
by Conway, Cellular Automata are “not just a game”, 1970
by epischisto.org , Schistosomiasis by mobile phones and social machines and simulators based on Cellular Automata, 2011
So, what is the problem for now? an investigation for 2014-2017…
• Computational Epidemiology of Malaria by Cellular Automata and Stochastic Integer Linear programming
Fundamentals
• To find the possible scenarios that match inflection points as optimal conditions for epidemic trends…
• a NP-Complete Problem and polinomial reductions to SILP is possible and… how to solve it?
Stochastic Integer Linear Programming
Sparse points captured by stochastic scenarios and inflexion points by statistical noises
@work…
• a PhD Thesis on this direction (feb, 2014): Statistical confidence of Cellular Automata rules for Schistosomiasis by Genetic Algorithms (PPGBIO-UFRPE)
• IFORS 2014 for solving the SILP by an old approximation…
• Interior-point nethods for solve it? A giant deterministic one by relaxation… maybe…