Mathematical Modeling of Infectious Diseases

Mathematical Modeling of Infectious Diseases Breakthrough Cincinnati s Super Saturday November 22, 2014 David J. Gerberry Assistant Professor of Mathematics Xavier University www.cs.xavier.edu/~david.gerberry! david.gerberry@xavier.edu!

Who is this idiot? from Mineral Ridge, Ohio Youngstown State! Purdue! UCLA! Xavier as a 6 th / 7 th grader Wanted to be a Major League Baseball player as it turned out PhD in Mathematics, Professor at Xavier

Why am I here? Mathematical Models of Infectious Disease Worked in Medical School at UCLA modeled HIV and Tuberculosis My job today: show how math can be used to study diseases

Infectious Diseases Can be spread from one person to another Examples? Diseases that aren t infectious? What causes infectious diseases? How are infectious diseases spread? Which infectious diseases are most deadly?

Most deadly infectious diseases? Disease Creutzfeldt-Jakob disease Case Fatality Rate (left untreated) 100% Rabies ~100% Smallpox 95% HIV/AIDS 80-90% Ebola 65-85% Disease Lower Respiratory Infections HIV/AIDS Diarrheal Diseases Tuberculosis (TB) Malaria Measles Pertussis Tetanus Meningitis Syphilis Deaths 3.9 million 2.8 million 1.8 million 1.6 million 1.3 million 600 thousand 290 thousand 210 thousand 170 thousand 160 thousand Ebola in 2014: 5,741 deaths as of November 16

Some fancy words Epidemiology: The study of disease in a popula1on Types of disease outbreaks: Sporadic: occasional occurrence Endemic: cases in a region are regular and o9en Epidemic: unusually high number of cases in a region Pandemic: global epidemic Today: Mathema1cal Epidemiology (sounds impressive, tell your parents this)

Ebola Everyone starts here: not yet infected Infected but no symptoms, feel healthy, can infect others Have symptoms, are sick and can infect others Healthy again, immune to infection Susceptible Incubation Period Infected Recovered Transmission Symptom onset Recovery

Ebola Relatively difficult to transmit: only thru direct contact with bodily fluids Susceptible Transmission Incubation period: can be up to 21 days but is usually 7-14 days Symptoms: fever, flu-like, nausea, vomiting, diarrhea, bleeding 65 85% die immune to infection Incubation Period Infected Death 25% Recovery 75% Recovered Symptom onset

Ebola Ebola is no joke but I will make jokes (bad ones) Real joke is the panic that went around about ebola in the media and in the American public. There never was and will not be an ebola outbreak in the US. The real joke (joke, as in, makes you ashamed to be a human) is that thousands of people have died in West Africa since last spring and the average American didn t know/care until someone in the US got it.

What is a mathematical model?

What is a mathematical model? What is a model?

What is a mathematical model? What is a model in general?

What is a mathematical model? What is a model?

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

Mathematical models of infectious disease Simplified description of how a disease spreads through a populations using mathematical ideas Ex; equations for numbers of susceptible, infected, recovered Ex; Rules for how many people each person infects Susceptible Infected Recovered Transmission Recovery

Activity: The Standing Disease Everyone starts sitting down. One person stands and is the first case. They pick two still sitting to infect. Those two stand up and each pick two others from those still sitting. The next generation stands up and each pick two more and so on. How many steps did the disease take to infect whole class? What would happen if 3 instead of 2? What if the class was bigger?

1 2

1 2 4

1 2 4 8

1 2 4 8 16

1 2 4 32 8 16

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

1 Step 0 2 Step 1 4 Step 2 8 Step 3 16 32 64 128 256 512 1024 1024 people are infected on the 10th step

1 2,048 2 4,096 4 8,192 8 16,384 16 32,768 32 65,536 64 131,072 128 262,144 256 524,288 512 1,048,576 1024 2,097,152 How long until everyone in Cincinnati is infected? 300,000 people 19 days

1 2,048 2 4,096 4 8,192 8 16,384 16 32,768 32 65,536 64 131,072 128 262,144 256 524,288 512 1,048,576 1024 2,097,152 7.125 billion people in the world How many steps until we infect the entire world?

1 2,048 2 4,096 4 8,192 8 16,384 16 32,768 32 65,536 64 131,072 128 262,144 256 524,288 512 1,048,576 1024 2,097,152 7.125 billion people in the world

Plot the number of new infections as a graph What isn t realistic about our model?

Plot the number of new infections as a graph Early part of epidemic Deaths This model would really only work well at the very beginning of an epidemic 800 600 400 200 5 10 15 20 25 30 Weeks

So we can understand the start of an outbreak, but what happens next? Why does the number of cases peak and then decrease? Discussion Make a mathematical model to explore what is happening

26-Card Epidemic 1. Put the 26 black cards down in a pile face up. 2. Put the 26 red cards in a pile face down this is your population. 3. Pick one card from this population and put it facing up on the table this is the first infection (step 0). 4. It will be red, so replace it with a black card this represents that person, now recovered and back in the general population. 5. Shuffle the population cards, and put two cards face up on the table. These are the new infections (step 1). 6. Put any black cards back into the population pack these people are now immune, so won t get the infection again. Replenish the population pack with black cards to replace the red ones you ve put down. 7. Repeat items 5 and 6 (picking 2 cards for each infected person) until you pick only black cards. This is the end of the epidemic, there are no new infections. 8. Plot a graph to show how many new infections there are at each step. Infections Step 0 1 2 3 4 Time

26-Card Epidemic: Questions Does the infection start to drop off? Why? Does your epidemic play out the same way every time? Why or why not? What is the probability that no one new is infected at step 2? How might things change if: more than one person is infected at the start each infected person infects 3 or 4 people instead of 2 Is this a realistic model? How could we improve it?

Possible Improvement By shuffling the deck and picking random cards, we re assuming that everyone is equally likely to get infected by someone else. Not completely realistic but not necessarily bad either Let s make a model that includes the fact that some people are at higher risk than others.

The Network-Standing Disease Like the Standing Disease but: - before starting, everyone writes down the names of the two other people in the room that they know the best. The first case picks the two they ve written down to infect. The next generation stands up and each pick their two and so on. How is this different from the standing disease? How many steps to infect everyone?

Now let s incorporate some a lot more randomness (fancy word STOCHASTICITY) (another STOCHASTIC MODEL) - get out dice - get out counters

Counter Ebola 1. Does your epidemic take off or die out? 2. Roll a die the result tells you how many people this person infects. 3. Put the appropriate number of counters in the column of step 2 for the disease. 4. For each infected person at step 2, roll the die to see how many people they infect, putting that number of counters in the column for step 3. 5. Repeat 4 until no one is infected, recording how many infected people you have at each step.

Let me do a few examples

Counter Ebola: Questions Does your infection play out the same way every time? Does the epidemic take off or die out? How often did it take off or die out? How many total infected people did you have? Did we lose some element of realism that we had in our 26-Card epidemic?

Counter Ebola: BIG QUESTION Did we see differences among the different groups? ANY IDEAS ON HOW WE COULD EXPLAIN THE DIFFERENCES WE SAW? (i.e. What caused the differences?)

R 0 Reproductive Ratio R 0 R 0 R 0 Definition: Average number of people an infected person infects at the start of an epidemic.

R 0 Reproductive Ratio R 0 R 0 R 0 Definition: Average number of people an infected person infects at the start of an epidemic. What is R 0 for the Standing Disease?

R 0 Reproductive Ratio R 0 R 0 R 0 Definition: Average number of people an infected person infects at the start of an epidemic. R 0 = 2

R 0 Reproductive Ratio R 0 Definition: Average number of people an infected person infects at the start of an epidemic. R 0 = 2 Deaths 800 600 400 200 R 0 = 2 5 10 15 20 25 30 Weeks

R 0 Reproductive Ratio R 0 R 0 is a measure of how quickly an epidemic will take off

R 0 Reproductive Ratio R 0 R 0 is a measure of how quickly an epidemic will take off R 0 > 1 Cases increase each step

Let s verify this conclusion Running this model by hand is OK but kind of repetitive we re just following simple rules/commands over and over and over again if there was only some time of machine that we could do repetitive commands over and over computers love this kind of stuff!!! e-counterplague-1plus

Importance of R 0 In general, we can eliminate a disease or prevent it taking hold in the first place if we can reduce the basic reproductive number below 1. R 0 < 1 " disease dies out

Immunization / Vaccination What is the point of vaccines? Could we incorporate vaccination into our previous models? Standing disease, 26-card epidemic, network standing disease, counter ebola What would be the effect if we did? play with e-counterplague to see What is the effect of vaccination on R 0? Does everyone in the population have to be vaccinated to eliminate a disease?

Vaccine Controversy What do we know about it? Where did it start? Not vaccinating is dangerous! Measles Pertussis not only for your kids but for others 600 thousand 290 thousand some people can t get vaccines because of certain medical conditions they rely on R 0 < 1 to prevent disease from taking hold

Back to Ebola How can we stop the ebola epidemic?

Back to Ebola How can we stop the ebola epidemic? What is R 0 for ebola? How could estimate this?

Back to Ebola How can we stop the ebola epidemic? What is R 0 for ebola? How could estimate this? Plot the number of new infections as a graph See which value of R 0 matches the observed data most closely R 0 determines how quickly the epidemic takes off Deaths 800 600 400 200 5 10 15 20 25 30 Weeks

Back to Ebola R 0 for ebola is approximately 1.5-2.5 reducing transmission by how much would eliminate ebola?

Back to Ebola R 0 for ebola is approximately 1.5-2.5 reducing transmission by how much would eliminate ebola? ANS: 33-40%

Back to Ebola R 0 for ebola is approximately 1.5-2.5 reducing transmission by how much would eliminate ebola? ANS: 33-40% If a vaccine were developed, would need to vaccinate 33-40% of the population.

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?