mathematical models

1. Dr. Mayteé Cruz-Aponte
RISE WORKSHOP
March 20,2015
BIOMATHEMATICS:
EPIDEMIOLOGY OF THE SPREAD
OF DISEASES
Source: http://consensus.nih.gov/IMAGES/
Art/EndOfLifeCareSOS024HIRESsmall.jpg

2. OVERVIEW
• Epidemiology
• Epidemiological compartmental models
• Basic Reproductive number
• Real life applications from epidemics to life sciences
• Simulations
• Group work

3. Interdisciplinary work needed!
WHAT IS EPIDEMIOLOGY?

4. EPIDEMIOLOGY: the branch of medicine that deals with the incidence,
distribution, and possible control of diseases and other
factors relating to health.
Source: http://www.epiwork.eu/

5. … How disease spread …
Experiment
SPREAD OF DISEASES
http://weddingsabeautiful.com/files/2012/12/1326591.jpg

6. BASIC REPRODUCTIVE NUMBER
The basic reproductive number R0 is the number of secondary cases
a single infectious individual generates during the period of infectivity
on a completely susceptible population.

7. BASIC REPRODUCTIVE NUMBER
FOR KNOWN DISEASES
Source: http://www.npr.org/blogs/health/2014/10/02/352983774/no-seriously-how-contagious-is-ebola

8. BASIC REPRODUCTIVE NUMBER
FOR KNOWN DISEASES
Source: http://www.bbc.com/news/health-29953765

9. MORTALITY RATES
FOR KNOWN DISEASES
Source: http://www.bbc.com/news/health-29953765

10. MORTALITY FOR KNOWN DISEASES
Source: http://www.bbc.com/news/health-29953765

11. Epidemic Models
Social Science Models and more …
EPIDEMIOLOGICAL
COMPARTMENTAL MODELS

12. WHAT ARE COMPARTMENTS?
• Compartments or urns are classifications of subjects or individuals in a model.
• Epidemiological classes
• Susceptible, Infected, Recovered, Exposed or Latent, Vaccinated …
https://plus.maths.org/issue14/features/diseases/Faces.gif

13. SIS SIMPLE EPIDEMIC MODEL
BASIC REPRODUCTIVE NUMBER:
R0 =
β
α

14. SIS MODEL SIMULATION

15. SIR EPIDEMIC MODEL
Can you write down the differential equations of the model?
Can you write down the Basic Reproductive number relation?

16. SIR MODEL
R0 =
β
δ +µ +σ
Basic Reproductive Number

17. SIR MODEL SIMULATION

18. SEIR EPIDEMIC MODEL
R0 =
β +µ
α
Mathematical Model Basic Reproductive Number

19. SIS NETWORK MODEL BETWEEN TWO CITIES
R1
0 =
β
α − q12
, R2
0 =
β
α − q21
BASIC REPRODUCTIVE NUMBER:

20. Zombies
Bibber Fever
Gangs
Rumor Spreads
Viral Videos
NON TRADITIONAL MODELS

21. http://www.fanpop.com/clubs/justin-bieber/images/22035621/
title/bieber-well-soon-photo

22. PARAMETERS AND MATHEMATICAL MODEL FOR
THE BIEBER FEVER

23. The Viral Disease in
this Age of
Technology?
Jacob Perez - Shadow Mountain High School!
Meilia Brooks - Agua Fria High School!
Keyaanna Pausch - Flagstaff High School !
Yulian Chulovskiy - San Miguel High School !

24. SOCIAL MEDIA SHARING
Mathematical Model!
Susceptible!
Has not seen the video!
!
Infected!
Has seen the video and
share it on social media
sites!
Recovered!
Has seen the video but
is not sharing it on
social media sites!

25. POSITIVE MEDIA COVERING
July 15, 2013!
“‘Gangnam Style’ is one
year old, and music is
forever different.”!
- Mat Honan!
“Gangnam Style” – PSY"
Anniversary of song release!
https://www.youtube.com/trendsmap!

26. NEGATIVE MEDIA COVERING
July 13, 2013!
“Protagonist of Glee Cory Monteith
found death at a hotel in Canada” - CNN!
“Don’t Stop Believing’” – Journey, Interpretado por los actores de Glee"
Death of actor Cory Monteith!

27. ZOMBIES!!!
DR. ROBERT SMITH?
http://mysite.science.uottawa.ca/rsmith43/

30. The truth shall set us free…
METAPOPULATION MODEL OF INFLUENZA
AH1N1 THE CASE OF MEXICO

31. INFLUENZA AH1N1-2009

32. " Primer brote en La Gloria, Veracruz, entre el 10 de marzo y 6 de abril 2009.
" Primer caso probable de SARS alrededor del 5 de marzo de 2009

33. • Data of confirmed cases (black curve). !
• Simulation of our normalized model by
total number of cases (green curve). !
• Infection, incubation and recuperation
rate where adjusted to the first outbreak
before the epidemic first peak. !
• With our adjustment we predict that the
epidemic started on day 73 of the year
i.e. March 14, 2009. !
• Day 117 corresponds to April 29,2009.!
PARAMETER ESTIMATION
THE START OF AN EPIDEMIC

34. STATES CLASIFICATIONS
Initial influenza outbreak and the
historical influenza corridor
• A/H1N1 epidemic outbreak in México by
region 2009.
• The Mexican States that contributed with
more than half of the total cases during the
initial spread of A/H1N1 up to June 4, 2009
are shown in dark gray.
• The remaining States (light gray) were the
main contributors to secondary outbreaks
later in the year.
• The red dots mark states in the historical
influenza corridor (Acuña-Soto, MD).!

35. Mathematical Model Flow Chart!
!
• S – susceptible per city, !
• E – infected individuals incubating the virus, !
• C – Confirmed infected cases, !
• U – Unconfirmed infected individuals, !
• R – recovered individuals. !
• The parameters qij’s are the rate of individuals
that travel from one city to another.!
Network connectivity !
!
• We consider terrestrial transportation between the cities in
Mexico and the metropolitan area (Federal District)!

36. PARAMETERS OF THE MODEL

37. SYSTEM OF ORDINARY DIFERENTIAL EQUATIONS

38. BASIC REPRODUCTIVE NUMBER OF OUR MODEL

39. Influence of transportation on the time
course of the epidemic outbreak
• The solid and dashed curves are,
respectively, the total of infectious people in
strongly and weakly connected populations.
• The dotted line is the epidemic curve in the
originating state, Veracruz.
• A - shows simulations in which
strongly and weakly connected
populations contribute, nearly the
same, to the traffic through México
City.
• B and C - shows cases in which the
contribution of strongly connected
populations is large relative to the
weakly connected contribution.!
Transportation Dynamics!

40. Multiple waves of the same
pandemic!

41. CONFIRMED CASES PER STATE

42. CONCLUSIONS
• Transport and the partition of the population into weakly and strongly
connected states induces a delay in the dynamics.
• The modulation of the infection rate by social-distancing, school
closures and the academic calendar is enough to explain the
emergence of the multiple waves of infection.
• Early arrival of the vaccine will have had a significant impact on the
time course of the epidemic if they where available.
• The intervention at the beginning of the April outbreak did mitigate
the spread of the disease, but as a consequence generate two more
waves and hence determined the shape of the data collected.

43. Scilab
SIMULATIONS

44. SIS SIMPLE EPIDEMIC MODEL

45. SIS MODEL SIMULATION

46. SCILAB CODE
//Code by Maytee Cruz
//Version 1 March 20, 2015
//RISE Workshop
function xdot = SISmodel(t,x,b,a)
S = x(1);
I = x(2);
N = S+I;
Sdot = -b*S*I/N + a*I;
Idot = b*S*I/N - a*I;
xdot = [Sdot; Idot];
endfunction
// Set the parameters
b = 0.9;
a = 1/7;
// Set the initial conditions and
// put into the vector x0
S0 = 1000;
I0 = 1;
x0 = [S0; I0];
//
// Create the time samples for the
// output of the ODE solver.
//
tfinal = 30.0;
t = linspace(0,tfinal,201);
//
// Call the ode solver
//
x = ode(x0,0.0,t,list(SImodel,b,a));
//
// Plot the solution (as functions of t)
clf;
figure(1);
lines(0) // disables vertical paging
a1=get("current_axes")//get the handle of
the newly created axes
a1.title.font_size = 4;
a1.x_label.font_size = 2;
a1.y_label.font_size = 2;
a1.labels_font_size=2;
a1.thickness = 3;
plot(t,x(1,:),t,x(2,:));
tstr = msprintf('Solution of the SIS Model
(b=%.2f, a=%.2f)',b,a);
title(tstr)
xlabel('Time in days');
ylabel('Individuals');
legend('Suceptible','Infected');

47. ACTIVITY
• With the basic SIR Model construct an SEIR model and add the
necessary lines of code to produce a graph with the 4
compartments.
• Think of a situation you want to model. Construct a model either
epidemiological or social and identify the compartmental
definitions and parameters needed for it.

48. http://www.phdcomics.com/comics.php?f=1789