Epidemics workshoprise

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  1. 1. Dr. Mayte Cruz-Aponte RISE WORKSHOP March 20,2015 BIOMATHEMATICS: EPIDEMIOLOGY OF THE SPREAD OF DISEASES Source: http://consensus.nih.gov/IMAGES/ Art/EndOfLifeCareSOS024HIRESsmall.jpg
  2. 2. OVERVIEW Epidemiology Epidemiological compartmental models Basic Reproductive number Real life applications from epidemics to life sciences Simulations Group work
  3. 3. Interdisciplinary work needed! WHAT IS EPIDEMIOLOGY?
  4. 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. 5. How disease spread Experiment SPREAD OF DISEASES http://weddingsabeautiful.com/files/2012/12/1326591.jpg
  6. 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. 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. 8. BASIC REPRODUCTIVE NUMBER FOR KNOWN DISEASES Source: http://www.bbc.com/news/health-29953765
  9. 9. MORTALITY RATES FOR KNOWN DISEASES Source: http://www.bbc.com/news/health-29953765
  10. 10. MORTALITY FOR KNOWN DISEASES Source: http://www.bbc.com/news/health-29953765
  11. 11. Epidemic Models Social Science Models and more EPIDEMIOLOGICAL COMPARTMENTAL MODELS
  12. 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. 13. SIS SIMPLE EPIDEMIC MODEL BASIC REPRODUCTIVE NUMBER: R0 =
  14. 14. SIS MODEL SIMULATION
  15. 15. SIR EPIDEMIC MODEL Can you write down the differential equations of the model? Can you write down the Basic Reproductive number relation?
  16. 16. SIR MODEL R0 = + + Basic Reproductive Number
  17. 17. SIR MODEL SIMULATION
  18. 18. SEIR EPIDEMIC MODEL R0 = + Mathematical Model Basic Reproductive Number
  19. 19. SIS NETWORK MODEL BETWEEN TWO CITIES R1 0 = q12 , R2 0 = q21 BASIC REPRODUCTIVE NUMBER:
  20. 20. Zombies Bibber Fever Gangs Rumor Spreads Viral Videos NON TRADITIONAL MODELS
  21. 21. http://www.fanpop.com/clubs/justin-bieber/images/22035621/ title/bieber-well-soon-photo
  22. 22. PARAMETERS AND MATHEMATICAL MODEL FOR THE BIEBER FEVER
  23. 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. 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. 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. 26. NEGATIVE MEDIA COVERING July 13, 2013! Protagonist of Glee Cory Monteith found death at a hotel in Canada - CNN! Dont Stop Believing Journey, Interpretado por los actores de Glee" Death of actor Cory Monteith!
  27. 27. ZOMBIES!!! DR. ROBERT SMITH? http://mysite.science.uottawa.ca/rsmith43/
  28. 28. The truth shall set us free METAPOPULATION MODEL OF INFLUENZA AH1N1 THE CASE OF MEXICO
  29. 29. INFLUENZA AH1N1-2009
  30. 30. " 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
  31. 31. Data of conrmed cases (black curve). ! Simulation of our normalized model by total number of cases (green curve). ! Infection, incubation and recuperation rate where adjusted to the rst outbreak before the epidemic rst 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
  32. 32. STATES CLASIFICATIONS Initial influenza outbreak and the historical influenza corridor A/H1N1 epidemic outbreak in Mxico 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 (Acua-Soto, MD).!
  33. 33. Mathematical Model Flow Chart! ! S susceptible per city, ! E infected individuals incubating the virus, ! C Conrmed infected cases, ! U Unconrmed infected individuals, ! R recovered individuals. ! The parameters qijs 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)!
  34. 34. PARAMETERS OF THE MODEL
  35. 35. SYSTEM OF ORDINARY DIFERENTIAL EQUATIONS
  36. 36. BASIC REPRODUCTIVE NUMBER OF OUR MODEL
  37. 37. 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 Mxico City. B and C - shows cases in which the contribution of strongly connected populations is large relative to the weakly connected contribution.! Transportation Dynamics!
  38. 38. Multiple waves of the same pandemic!
  39. 39. CONFIRMED CASES PER STATE
  40. 40. 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.
  41. 41. Scilab SIMULATIONS
  42. 42. SIS SIMPLE EPIDEMIC MODEL
  43. 43. SIS MODEL SIMULATION
  44. 44. 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');
  45. 45. 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.
  46. 46. http://www.phdcomics.com/comics.php?f=1789