and here our next talk is going to be from Glenn letter from the University of nebraska-lincoln and the title is mathematical modeling for epidemiology and ecology so I will give before to Glenn thank you so much for being here so I'm going to talk about uh mathematical modeling for epidemiology and ecology which is the the title of a book that will be out soon um so it's a reworking of my first mathematics for the life sciences book it should be out in April um it was written for two purposes one was to be a textbook for a possible undergraduate course in either mathematical modeling or mathematical biology but also to be a source of problems and methods to supplement differential equations in modeling courses which is the the role that I'm talking about now so in the talk I'll Identify some important special features of the book that you probably won't find anywhere else I'll show briefly show an overview of the contents and then I'll spend the rest of the talk highlighting some of the topics again picking topics that you aren't likely to see anywhere else um okay so I'm putting a focus on mathematical modeling and Tracy's talk was a great setup for me because she Illustrated what mathematical modeling really is it's not word problems it's not applications of mathematics it's designing mathematical models analyzing the mathematical models by thinking about the effective parameters on the behavior of the system and then analyzing and then looking back at the real world to see whether your Model results made sense so um in the book many model derivations appear in the text and the problem I discuss alternative choices for biological assumptions a lot of of problems contain symbolic parameters as in Tracy's modeling many problems ask for a biological interpretation of results and there are numerous projects that also require modeling um I also have an emphasis on scientific computation the the book can be used without doing scientific computation but you're missing about a third of modeling if if you don't modeling requires simulation as well as analysis um and it in my view students need to learn to use the programming environment themselves and so I've tried to make that as as easy for students as possible the book comes with 22 Matlab programs no prior programming experience is needed I've used these programs with students before and it is in it is true that no prior programming experience is needed there's an appendix that has a minimal Matlab tutorial that does a line by line explanations of the two simplest programs the programs are structured to make them easy to modify for new experiments or to alter for different models because all the model specific and scenario specific code is in a small section at the beginning of each program and one of the topics I talk about later in this in this talk I'll show you a snippet of my simplest program um this is the general table of contents in the in my presentation of table of contents I've grayed out topics that are not relevant for differential equations and I've highlighted in blue topics that I will say at least a little bit more about in the talk so I won't be talking about discrete linear systems the the main point there for differential equations is uh just to introduce eigenvalues for students who haven't had a linear algebra course uh chapter 3 on mechanistic modeling is the longest chapter in the book and you can think of it as divided up into sub portions the first two sections are just about specific processes that are incorporated into models transition processes like disease recovery and interaction processes like disease transmission and I'll be talking about two of the transition processes later my vaccination model and multi-phase transition processes then there is a section on compartment analysis which uses the seir epidemic model to illustrate the the uh the method and a section that does model analysis but it doesn't do dynamical system analysis that's something you do for an endemic model instead it looks at things like how many people are still susceptible when the epidemic ends um then there's a case study that includes two covid-19 scenarios which I've used in teaching materials I consider section 3.6 to be one of the most important in the books in the book because this is where you learn scaling and then the chapter ends with three case studies the last one adding demographics to a disease model is again something I will talk about later in this talk uh chapter four is on single populations and I will just talk about one topic from here and uh chapter six uh there are a couple of topics here that will be in in my talk tomorrow so I'm not going to spend time on them now so let's start with vaccination and so here's an sir vax model that includes vaccination and I've grayed out everything that isn't about vaccination uh so the part that isn't grayed out we can write down is it isolated single cohort vaccination model and in the standard formulation of vaccination it's just the same as radioactive decay s of 0 is equal to one because the population is measured in cohorts so what's wrong with this implementation of vaccination so if I had time I'd ask for for suggestions from the audience but instead I'll tell you the three things that are wrong with it it's applied to the entire susceptible class when in reality there are people who are unable to take the vaccine and many people who are unwilling it assumes an unlimited Supply when in reality at the beginning of a vaccination campaign there's hardly any Supply at all and it assumes instantaneous distribution when in reality distribution has to be accomplished by a limited number of health care workers so all three of these things are flaws in the isolated in the using the the radioactive decay model so let's correct them one at a time limited acceptance is pretty easy let's assume a fraction a of the population is willing and able to be vaccinated and let's let WBA the population fraction that is waiting for vaccination this is not part of the epidemiology model though it interacts with it because when vaccination is scheduled it there is no effort made to limit vaccination to people who are still susceptible people who are recovered can get vaccinated too so w is not a subset of s it's a different category altogether it's a non-epidemiological category next let's look at limited Supply this is fairly easy to deal with as well instead of a constant rate const instead of a rate constant fee let's put in a rate function that has a maximum of fee so this rate function starts out at zero it increases over time until sometime Tau at which at which it achieves the maximum value of fee the simplest such model assumes that g is piecewise linear in in my when I wrote this down in the the first time I thought this is too simple but you'll see it works great and the last thing which is a little more challenging is limited distribution capacity and the idea here was that and here's where modeling really comes in Tracy will appreciate this particularly the idea is that vaccinators are like enzyme molecules and enzyme kinetics and the people who get vaccinated are like the substrate and so the Michaelis Menton kind of model should work and so if you take P equal to 1 in this formula that's the Michaelis metin model with the extra factored G of T so the idea is that there's a a bound to how large the vaccination rate can be if there's a huge number of people you can still only vaccinate people at the speed that is possible based on Health Care capacity P equal 2 corresponds to the ecological model of hauling type 3 which I actually uh which is actually the one I'm going to use so these are these limits are only needed for scenarios with a new vaccine okay so how does it fit to data well the plot in B is the the black is the model and the red which you really can't see is the CDC data so it fits the data really really well I don't have time to talk about the plot on the left so I'll just leave that here is a program that students can use to reproduce this graph so my program doesn't actually produce that graph what this program does is it takes the simplest vaccination function and uses two different values of a so in order for students to change this they need to get rid of one of the values of a they need to add lines for K and Tau the additional parameters and they need to replace this right hand side here with the appropriate function that's not a lot of work for students to have to do the rest of the program where the extra coding is doesn't change and this essentially is a cut down version of a general program that's in the group of 22 that does just owed any ode simulations any systems of ODS okay next let's talk about uh transitions so a single phase transition is like radioactive decay if the mean time is 1 over mu then the rate constant is Mu a two-phase transition would have the same mean time but now it has two different phases so you might be in the first phase waiting to join the second or you might be in the second phase waiting to be finished and so the rate constant for each phase is 2 times mu because you're only spending half the time in each of the two phases uh phase one is just like the single phase transition phase two starts out with an initial population of zero because everybody starts in Phase One and Y2 is populated by people who are leaving phase one and then people leave phase two in the usual way and so you can solve that system of differential equations and get this answer and you could have any value for K you could write down the general formula for a k phase transition and here's what the results look like this is the remaining fraction infected from an initial cohort the dark blue one is the single phase transition and it's not very good for disease recovery this would say that more people in this cohort recover on day one than on day eight even though the mean well say day five the mean recovery time is five days but more people recover on day one than day five that's clearly wrong the purple curve for k equals 6 is much more realistic not that many people recover right at the beginning and not that many people recover late most people recover in the middle it's okay to use this for teaching but you should consider a multi-phase transition for some research problems and whether you actually need to do that is an issue explored in one of the projects in the book okay next let's talk about adding demographics to a disease model so this is a simple sir model in which I have natural death in each category and I have the possibility of disease-induced death if Alpha is greater than zero and I want to add a birth rate to make a sensible model so what do I do well the usual choice is a constant rate usually Capital Lambda is the symbol used for that I like to choose mu for the constant rate as I showed earlier so that the Baseline population is one then you don't need to scale the populations it's silly to use a different value and and then have to scale the populations you might think that if you want a density dependent birth rate you should just make it proportional to the population but this doesn't actually do anything because it exactly compensates for natural deaths so if Alpha is greater than zero the birth rate is too small to replace the people who die from the disease and the population crashes eventually if Alpha is zero then you didn't need the factor of n which is always equal to one so under no circumstances is Mu n a good choice but if you want density dependence you could try a birth rate based on logistic growth so here's the logistic growth equation applied to the total population which now is is not going to be constant the total death rate is still mu n so growth is birth minus deaths so if I know the growth that I want and I know the death that I want well then green is just blue plus red so I get a birth rate function that has two components it has the component that replaces the people who die naturally plus a component that tries to adjust for people dying of the Disease by pushing the population towards a carrying capacity this is a better mechanistic model but as in other model modifications it turns out it only makes a noticeable difference if the death rate is high so under most circumstances we're justified in using the constant birth rate model that most people use here though we've actually studied the question in using mathematical modeling and come up with the conclusion not on the basis of mathematical convenience but on the basis of good modeling okay lastly I want to spend a fair amount of time talking about using structures to do phase line analysis so many differential equations courses now include phase line analysis for single differential equations here's a single differential equation it has logistic growth and it has holling Type 3 consumption so it's similar to the vaccination model C represents the number of consumers so this is essentially the prey equation from a predator prey model but I'm thinking of C as being a constant rather than a variable that turns the two-dimensional Predator prey model into a one-dimensional model so the usual way to study this with a phase line is to plot the whole function f of x and see where F changes sign uh that's not so good here because this F is a complicated function with two parameters so we'll need to select specific pairs p and c and worse yet it's hard to do any modeling because we'll have to start over if we change either of the two parameters we'd like to see what effect one of the parameters has on the model but if I have to start over when I change either of the parameters then that's a lot of extra work it's not the only option we could think of this function f as being an increase part minus a decrease part and we could draw a blue graph for the increase and a red graph for the decrease and then the equilibria are where they cross and whichever one is bigger determines whether we're increasing or decreasing so this is every bit as good as following F it's just as easy to tell whether blue is bigger than red than it is to tell whether f is positive and in fact it's more intuitive but we can do even better than that by removing a common non-negative factor to make the functions easier so for example here if I remove the factor x that's common to both terms then I've changed the first term from quadratic to linear that makes it simpler so in principle it seems like removing a common non-negative factor might help us then what's left is not actually the increase function but it's a factor of the increased function and it represents the increase function which is why I have it in quotation marks so structurally uh symbolically f of x is now being factored into a w of x times uh G of x minus h of X and I want to choose W so that it's 0 when x is 0 and positive else otherwise and the reason I'm doing that is that now I don't need to think much about w equilibria will have G equal H H or they'll have W equals 0 but W equals 0 simply means x equals zero so I just have to make sure I remember that x equals 0 is an equilibrium even though G is not equal to H there and to determine whether X is increasing or not I don't need to worry about W because it's never negative so let's look at two possible implementations of this idea because factoring here is not unique so I mentioned that you could Factor this by removing a factor of X that's the obvious way to do it but it still has the problem that if I want to graph the red curve I still need to to do a different graph every time I change either C or p instead though I could grab I could factor out the extra Factor P plus x squared that's in the denominator this is by no means obvious but once you see it then it's clear why it's going to be beneficial because now the blue curve is non-linear and the red curve is linear but the red curve has a parameter in it so if I want to fix p and choose a family of C values I only need one graph for the whole family I don't need to change the graph just because I've changed C and so here's how it plays out in practice this is my last actual slide so in these graphs the blue curve is the fixed increase function when p is 0.01 and the red Curves in the first graph are for different values of c so you can see there's some bifurcation going on here there are C values that have only one intersection for a large x there are C values that have only one for a small X and then there are C values that have three let's look in more detail at one of those so for that one the equilibria are where the blue and red curves cross and also at the origin for each of the regions between equilibria you just look to see which is bigger blue or red if blue is bigger then the arrow goes to the right if red is bigger then the arrow goes to the left and just like any other phase line analysis the arrows tell you which of the equilibria are stable and which are unstable so in this system we have two stable equilibrium equilibria we have a very important bifurcation Point here in which if you start above it you'll go to the higher equilibrium and if you start below it you'll go to the lower and so finally I want to end with a little bit of Shameless self-promotion which I apologize for uh so this this material was from my book which is going to be out by April and another feature of the book that I didn't mention in this talk is linked problem sets and so I'm going to talk about linked problem sets and present two of these in detail in my talk tomorrow find me during the Expo or shoot at me an email if you want to chat uh uh or or as I said shoot me an email um thank you so I I'll take questions hey Glenn I found your uh uh pandemic modeling your epidemiology modeling interesting um I I actually I actually used to work for IBM recently and I used an seir model to predict uh covet case counts um for different countries as well as the state of Hawaii for uh for the joint Artificial Intelligence Center and Indo paycom and things like that and I think the biggest the biggest challenge for us was um you know predictive having an accurate forecast Beyond two to three weeks because uh because we didn't know how the the reproduction number was going to change you know because it changed a lot based on lockdowns and and things like that yeah so that's something that I didn't have time to talk about but in my section 3.5 I have two scenarios from the covet pandemic uh and the first of those scenarios I created in March 2020 when the covid pandemic was really fresh uh and then just modified it a little later when we knew more uh it's it's a modification of the seir model in which the Infectious class is broken into three groups asymptomatic regular infectious and pre-hospitalized and the reason for that is that the interventions that appear in the model are handled differently for each group um so testing only in that time only applied to people who had symptoms so if you had symptoms if you were in the Infectious group then you might have gotten tested in which case you might have gone into isolation uh whereas if you weren't tested but you did have covid then you might have uh had worn a mask or or limited your contact with people and so each of those features is built into the model and so instead of having to change the basic reproduction number over time what you have to do is just pick the values of three specific parameters and then you observe how the basic reproduction number changes over time yeah we we did use a moving window to try to identify the the you know the reproduction number and how it changes over time still was it was quite difficult to predict accurately beyond beyond three weeks which were they you were trying to do it empirically from data whereas right whereas if you do it mechanistically then you then you can still use data but then you don't need as as many num as many parameters in your model the more parameters you have in your model the less predictive value it has yeah so that's another reason why you couldn't go beyond three weeks because your model had a huge number of parameters and uh I mean you may not have written them in the model but no we didn't have we actually only I think only the reproduction number was the parameter the other parameters that have to do with the incubation period and Recovery times those are Well published and right yeah I misspoke what what I meant was that if you had done a mechanistic model for the basic reproduction number that would have had a lot of parameters in it yeah yeah and another another challenge we had was the we you know we used the reported case counts to to kind of um fit our a model um but you know the other challenge there was that not you know not all covet cases are reported so we had absolutely parameter in there that was a scaling factor and right you know there was a lot of confidence interval work right so again if you look at my covid model that's uh that for March 2020 that's that's in my book um I put some effort into how to pick the parameters for the model including parameters that are really hard to know like what fraction of in of infectious people are pre-hospitalized it's really hard to know that because some infectious people have symptoms but they don't get tested so they're in class i1 but uh you don't know which of the whether they're in a i1 or still susceptible yeah so it's very difficult to identify the class for an individual and so you have to use what little data you have that's Fairly reliable and then back out what the model parameters must have been uh anyone else uh Edom you have your hand up thank you for the presentation um I wanted to know if there is um a program to train modelers or mathematicians to be able to communicate their finding their results to non-technical people because oh that's a really good uh we had a story that all about these people are going to die and everyone was Panic like we're like oh is human race coming to an end are we all coming to die yeah but because they couldn't communicate the results well um I don't think journalists have um knowledge in mathematical modeling or whatever so whatever it is I once did a research program with very talented pre-freshman so they just graduated from high school uh they had really high SAT scores really good at math good grades uh and the first year that we did it we had them do a research project and they were completely unable to write a good report of their research project I'm sure those that all of those students could write a good history paper so the next summer we hired a graduate student from our English Department whose specialty was teaching technical writing and uh that year and every year after we got really good reports from the students my colleague who or who organized that program with me and I wrote a paper on that along with the English graduate student so shoot me an email and I'll point you to where to find that paper it won't tell you everything you want to know but it'll give you at least a few pointers uh let's see somebody asked uh quantify the determine uh the low predictive value of accounting for certain factors um I don't know about that in in specific cases but another thing that I will point you to is that um most of us have have heard of Occam's razor which says that things shouldn't that don't make things more complicated than they absolutely need to be there is a statistical implementation that's equivalent to its equivalent to an input to a implementation of Occam's razor called the akaika information criteria if you haven't heard of that try to find out about it it doesn't appear in statistics books yet and I don't know why it was discovered in 1974 so we've had a lot of time to get it into statistics books but it isn't yet and essentially what it does is it assigns each model a score for how act for how how inaccurate it is and another score for how complex it is and it adds those two scores together lowest score wins so if you have a model with a huge number of parameters it might predict the data that the data set that you have better than a model with only three parameters but it won't win the AIC battle because its complexity is too high that's another topic that's in my book that I didn't talk about today or even mention because it's empirical modeling but I do have a chapter on empirical modeling and AIC is in there yes um on that note thank you so much for presenting and also answering all those questions thank you next