Computational Epidemiology of Malaria by Cellular Automata and Stochastic Integer Linear Programming

1. 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

2. an investigation for 2014-2017…
• Computational Epidemiology of Malaria by
Cellular Automata and Stochastic Integer
Linear programming

3. Why malaria again?
OMS
Milenium goals
P. falciparum
P. vivax
Predominance
-----

4. the know-how...
www.pmt.es www.upc.edu
epischisto.org
THE staff www.ufrpe.br
www.cesar.org.br
www.fiocruz.br
www.ines.org.br

5. Fundamentals
• 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.

6. Fundamentals
An inflection point is a point on a curve at which the sign of
the curvature (i.e., the concavity) changes. Inflection points may
be stationary points, but are not local maxima or local minima.
The first derivative test can sometimes distinguish inflection points
from extrema for differentiable functions
The second derivative test is also useful. A necessary condition
for to be an inflection point is
A sufficient condition requires and to have opposite signs in
the neighborhood of (Bronshtein and Semendyayev 2004, p. 231).

7. Fundamentals – Mathematical
Epidemiology

8. Fundamentals – how to solve these Differential Equation systems? Some are
unsolved in analytic form, but in numerical one…
Adams-Bashforth-Moulton Method
Adams' Method
Collocation Method
Courant-Friedrichs-Lewy Condition
Euler Backward Method
Euler Forward Method
Galerkin Method
Gauss-Jackson Method
Gill's Method
Isocline
Kaps-Rentrop Methods
Milne's Method
Predictor-Corrector Methods
Relaxation Methods
RK2
RK4
Rosenbrock Methods
Runge-Kutta Method

9. and for neglected diseases?
with sparse
data

10. Aitken Interpolation Chebyshev Approximatio... Moving Median
B-Spline Cubic Spline Muller's Method
Berlekamp-Massey Algor... Gauss's Interpolation... Neville's Algorithm
Bernstein-Bézier Curve Hermite's Interpolatin... Newton's Divided Diffe...
Bézier Curve Internal Knot NURBS Curve
Bézier Spline Interpolation NURBS Surface
Bezigon Lagrange Interpolating... Richardson Extrapolation
Bicubic Spline Lagrange Interpolation Spline
Bulirsch-Stoer Algorithm Lagrangian Coefficient Thiele's Interpolation...
C-Determinant Lebesgue Constants Thin Plate Spline
Cardinal Function Moving Average
Solving sparse systems
Interpolation?
and some hidden scenarios…

11. 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|+1Z (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

12. Disease cycle:
http://www.wellcome.ac.uk/en/labnotes5/animation_popups/schisto.html (animation 5´24´´)
http://wwwnc.cdc.gov/travel/images/map3-14-distribution-schistosomiasis.jpg
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 more deaths...
Mar, 27th. 2009 

13. a REALLY neglected disease in
Brazil...
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...
http://200.17.137.109:8081/xiscanoe/infra-estrutura

14. 2006 starts a new monitoring
Praia Carne de Vaca
Praia Enseada dos
Golfinhos
Praia do Forte
Praia Pau Amarelo
Praia do Janga
Lagoa do Náutico
Praia Porto de Galinhas
BRAZIL

15. 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.

16. 2006 – 2007, data collect in-loco

17. 2006 – 2007, data collect in-loco
http://200.17.137.109:8081/xiscanoe/infra-estrutura/expedicoes

18. Figure 1.
Adjusted Prelavence
0to 10 (3)
10to 20 (32)
20to 30 (11)
30to 50 (3)

Stream
Prevalence per 100 hab
0 to 1 (15)
1 20 (17)
20 60 (14)
60 80 (2)
80 100 (1)
Breeding sites
to
to
to
to
water-collecting tank
Riacho Doce
1a. Prevalence 1b. Adjusted Prevalence
Male Female Total
Age group Pop1
Posit2
Prev3
Pop Posit Prev Pop Posit Prev
up to 9 99 7 7.1 100 3 3.0 199 10 5.0
10 to 19 109 26 23.9 99 24 24.2 208 50 24.0
20 to 29 76 31 40.8 90 21 23.3 166 52 31.3
30 to 39 88 18 20.5 103 23 22.3 191 41 21.5
>= 40* 141 14 9.9 168 18 10.7 310 32 10.3
unreported 16 3 18.8 10 2 20.0 26 5 19.2
Total 529 99 18.71 570 91 15.96 1100 190 17.3
* 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

19. 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.
http://dx.doi.org/10.1590/S0073-47212010000100003
Collecting
Sites
Alive Dead Positive to
S. mansoni
% de
infection
I 0 0
II 1707 129 4 0,23
III 297 198 0 0
IV 0 0
V 0 0
VI 0 0
VII 2355 322 37 1,57
VIII 76 125 3 3,95
IX 0 0
Total 4435 774 44 0,99

20. 2009-2010, modelling with 15 real
parameters (?)
Paremeter Ranges (avg) How were obtained?
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...

21. Mechanistic epidemic models
Two alternative approaches
Top-down Population-based Models (PbMs)
Bottom-up Agent-based Models (AbMs)
PbM AbM

22. 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

23. 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
N
rN
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).
rt
e
N
NC
C
tN



0
0
1
)(
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



ISM
I
SSM
S
rM·I·Mp=
dt
dM
rM·I·Mp=
dt
dM


(3a)
(3b)

24. 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)

25. 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

26. 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)

27. Computational vs Statistical models
Day 26 Day 43 Day 88
Day 106 Day 132 Day 365
Color 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 365
Color 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”

28. Predictive scenarios
2012 2017 2022 2027
Color 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

29. STATE OF ART – some of our
production on it
• http://www.systems-
journal.eu/article/view/172
• http://dl.acm.org/citation.cfm?id=2488022&d
l=ACM&coll=DL#
• http://www.ij-
healthgeographics.com/content/11/1/51
• http://dx.doi.org/10.1590/S0102-
311X2013000200022

30. Consequences…

31. INNOVATION on collecting DATA: automatic proposal for
diagnosis of schistosomiasis, malaria... (patent in progress)
SEE PROJECTS
http://200.17.137.109:8081/xiscanoe/projeto/graduate-projects

32. some world initiatives on automatic
diagnosis…
• http://www.fastcoexist.com/3026100/fund-this/a-handheld-device-
that-can-diagnose-diseases-and-drug-resistance-in-15-minutes
• http://www.jove.com/blog/2012/05/04/crowd-sourcing-for-
malaria-game-on
• http://g1.globo.com/jornal-nacional/noticia/2013/01/empresa-de-
israel-cria-celular-que-faz-exames-medicos.html
• http://lifelensproject.com/blog/
but no one on mobile simulation of cellular automata for disease
spreading…

33. INNOVATION on collecting DATA:
an integrated plataform www.ankos.com.br
http://ankos.sourceforge.net/

34. INNOVATION on collecting DATA:
an integrated plataform SchistoTrack (patent in progress)
We are Health Map in PE-Brazil!
http://healthmap.org/

35. INNOVATION on mathematical and
computational simulation using Cellular
Automata

36. What are we doing
now? Running
simulations on Mobile
platforms and these
simulations will guide
the data collect

37. 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

38. So, what is the problem for now?
an investigation for 2014-2017…
• Computational Epidemiology of Malaria by
Cellular Automata and Stochastic Integer
Linear programming

39. 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?

40. Stochastic Integer Linear Programming
Sparse points captured
by stochastic scenarios
and inflexion points
by statistical noises

41. @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…
• What else?
– Contact us www.epischisto.org

42. Thanks!
We have several other projects by other researchers...
www.epischisto.org