Mathematical Modeling of Infectious Disease

Transcription

1 Mathematical Modeling of Infectious Disease Preview Day Mock Class April 19, 2015 David J. Gerberry Assistant Professor of Mathematics Xavier University

2 Who is this idiot? PhD in Mathematics from Purdue Postdoc in Medical School at UCLA modeled HIV and Tuberculosis My job today Teach you some mathematical epidemiology Demonstrate how mathematics plays an important role in other fields

3 Ebola 2014 largest ebola epidemic in history ~ 26,000 total cases ~ 10,600 deaths

4 Ebola 2014 Epidemic of Panic: Interest in ebola as measured by google searches Why the interest in Aug and Oct?

5 Ebola 2014 Epidemic of Panic: Interest in ebola as measured by google searches Why the interest in Aug and Oct? Aug: Two American health workers infected in Africa were brought to US for treatment Oct: Thomas Eric Duncan first ebola case in US (Dallas, TX)

6 Ebola 2014 on Twitter

7 Xavier Take Home #1: We suck! Americans didn t care until ebola came to US Ignatian Heritage à people for others, social justice, reflection on our place in the world Appropriate conversion in a Xavier classroom Take Home #2: Technology is amassing quantitative data like never before Math/Statistics/CS increasingly important Not just Physics and Engineering anymore Politics, Social Sciences, Biology, Medicine, Business, Economics Liberal Arts Education à experience allows us to see connections between different different areas of study

8 What is a mathematical model?

9 What is a mathematical model? What is a model? Answer: a SIMPLIFIED version of something using mathematical ideas (equations, formulas, functions, rules, etc.)

10 Mathematical models of infectious disease Simplified description of how a disease spreads through a populations using mathematical ideas Mathema'cal Epidemiology (sounds impressive, tell your parents this) Ex; equations for numbers of susceptible, infected, recovered Ex; Rules for how many people each person infects Susceptible Infected Recovered Transmission Recovery

11 Basic Reproductive Number, R 0 Definition: Average number of people an infectious person infects at the start of an epidemic R 0 is a measure of how quickly an epidemic will take off or if it will at all R 0 > 1 Cases increase each step disease dies out disease persists

12 An Incredibly Simple Disease Model One person starts out infected. (Patient 0) They infect 2 people. Those two each infect 2 new people. Each newly infected person infects 2 more and so on.

13 1

14 1 2

15 1 2 4

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18 n new infections on step n

19 Discussion How many steps do you think it would take to infect everyone in the world?

20 1 Step 0 2 Step 1 4 Step 2 8 Step people are infected on the 10th step

21 1 2, , , , , , , , , ,048, ,097,152 How long until everyone in Cincinnati is infected? 300,000 people 19 steps

22 1 2, , , , , , , , , ,048, ,097, billion people in the world How many steps until we infect the entire world?

23 1 2, , , , , , , , , ,048, ,097, billion people in the world

24 Plot the number of new infections as a graph R 0 =2 =) total human annihilation What isn t realistic about our model?

25 Plot the number of new infections as a graph Deaths Early part of epidemic This model would really only work well at the very beginning of an epidemic Weeks

26 Discussion So we can understand the start of an outbreak, but what happens next? Why does the number of cases peak and then decrease?

27 Discussion So we can understand the start of an outbreak, but what happens next? Why does the number of cases peak and then decrease? We need a new mathematical model

28 Including Stochasticity (i.e. Randomness) For each infected person, roll a die to determine how many people she/he infects. Example Rules for Infection:

29 Doesn t stay fun for long Kind of repetitive We re just following simple rules/commands over and over and over again If there was only some kind of machine that could do repetitive commands over and over

30 Doesn t stay fun for long Kind of repetitive We re just following simple rules/commands over and over and over again If there was only some kind of machine that could do repetitive commands over and over Computers love this kind of stuff!!! e-counterplague-1plus

31 Questions / Discussion Why does the number of cases peak and then decrease? Can we find R 0 for our new stochastic models? Do they behave as we expect them to?

32 How to find R 0 for real diseases? R 0 determines how quickly the epidemic takes off e (R 0 1)t infectious period infections See which value of R 0 matches the observed data most closely R 0 estimates for ebola: Liberia: 1.59 Sierra Leone: 2.43 Guinea: time

33 Estimates of R 0 for several diseases R 0 estimates for ebola: Liberia: 1.59 Sierra Leone: 2.43 Guinea: 1.51 number of new infections in 6-7 days R 0 estimates for HIV: 2.5 in some regions number of new infections over years ² Can have the same R 0 but very different infection rates per day

34 Is R 0 still useful after an epidemic has already started? Would a vaccine have to be 100% effective to eliminate ebola? Would we have to vaccinate 100% of the population to eliminate ebola?

35 Is R 0 still useful after an epidemic has already started? Would a vaccine have to be 100% effective to eliminate ebola? Would we have to vaccinate 100% of the population to eliminate ebola? ² NO!! just need reproductive number < 1 R eff = R 0 (1 (proportion vaccinated) (protection rate))

36 Is R 0 still useful after an epidemic has already started? Would a vaccine have to be 100% effective to eliminate ebola? Would we have to vaccinate 100% of the population to eliminate ebola? ² NO!! just need reproductive number < 1 R eff = R 0 (1 (proportion vaccinated) (protection rate)) Will a 75% vaccine work if we can get 50% of the population vaccinated?

37 Vaccination proportion vaccinated Won t Eliminate Disease R0 = 1.5 R0 = 2 R0 = 2.5 Eliminates Disease Will a 75% vaccine work if we can get 50% of the population vaccinated? MAYBE protection rate of vaccine

38 Anti-Vaccination Movement Eliminates Disease proportion vaccinated R0 = 18 R0 = 12 R0 = 7 Won t Eliminate Disease protection rate of vaccine

39 Anti-Vaccination Movement proportion vaccinated R0 = 18 R0 = 12 R0 = 7 Eliminates Disease Won t Eliminate Disease Disease Lower Respiratory Infections HIV/AIDS Diarrheal Diseases Tuberculosis (TB) Malaria Measles Pertussis Deaths 3.9 million 2.8 million 1.8 million 1.6 million 1.3 million 600 thousand 290 thousand Tetanus Meningitis 210 thousand 170 thousand protection rate of vaccine Syphilis 160 thousand

40 If we have time we won t Some more models to play with Spatial model where people move around and infect others Excel Spreadsheet Model Does everyone who is susceptible catch the disease? Does everyone recover? How quickly does it last before dying out completely? What difference does it make if the disease can be fatal? Does the whole population ever die out? What difference does natural immunity and/or immunization make? What figures on the spreadsheet give the reproductive ratio, R 0? What value of R 0 do we require for an epidemic not to occur? For a given value of R 0 what level of vaccination do we need to prevent an epidemic occurring? What was R 0 for SARS? What is R 0 for measles? So what level of vaccination do we need for measles to prevent epidemics?