Good day youtube welcome to mathgotserved.com in this clip were going to be going over how to use the calculus of the differential equations to understand the spread of an epidemic so one that's on the mind of many right now is ebola do not forget to visit our email@example.com for access to a wide variety of math tutorials ranging from algebra all the way to calculus so the hypothetical problem were going to use to understand the spread of an epidemic using calculus is as follows is a family of five are infected with Ebola how long will it take take it let's take the spread to 50% of LA's population so let's assume that the family of five players on the Los Angeles that have the population of three point eight million how long will it take to get 50% of the population infected with this disease okay right so what were going to use to do were going to be using the SIR model to basically understand the complexity of the situation right so using the SIR model okay the other one that can be used to model the spread of an epidemic, not the stochastic model what this is the model that were going to be using today so the SIR model basically involves the the the of dividing up the population into different groups of cool compartment okay so you might wonder what does SIR Mean? Well SIR basically represents the three functions that, basically used to define the rate at which the different variables or function change when you're looking at the spread of disease okay so let's go ahead and explain what this word SIR means that the first letter represents a function okay this function tells you the number of people that are susceptible oryet to be infected by the epidemic okay so let's call this number of people susceptible or yet to be infected okay I we have the S down the next letter in the SIR acronym is I I of T this function represents the number of people that are infected with with the X with the disease okay so I represents number of people infected we can and then lastly R R of T is very interesting on function r of t represents the number of people that have recovered from the outbreak with all immunity okay the number of people that recovered with immunity okay so immunity is important here because if yourecover from the disease interest you'll susceptible then you you classify as an S in the equation but if you, with immunity then you you are classified as being R of t okay now not on people that fall into the RP group is the people that basically passed away as a result of infection and are not exposed to infect others okay so are to represents those who recovered with immunity and also includes those who passively as a result of infection and are not in a position where they can infect others okay so on this SIR model can be used to generate an equation that basically groups everybody in a population sample into the respective categories okay so the equation is N equals S S plus I plus R okay so there you have the SIR now you might wonder what is N will and represents the total number of people in the population areas that you looking at okay so in this case in this problem we looking at Los Angeles for example so N is going to be three point eight million or the total amount of people living in Los Angeles okay so let's do that here N of t is total population now in order for us to answer this question how long does it take for 50% of people in let's in Los Angeles to get infected with this disease we need to know how fast the disease spreads okay the rate of infection so that's what we need to know so if the rate of infection where three point eight million people in a week then we know that in half a week 50% of the population will get infected with Ebola okay so how fast does it spread if we can determine the rate of spread and we can tell exactly how long is take for half of the population to get infected okay so take a look at this equation right here we're going to differentiate this equation implicitly okay and we looking for the rate at which the infection function changes okay so that's what were going to be solving for we differentiated implicitly because we looking for the instantaneous change of the infected function with respect to time in this function is not solved explicitly forI so we are just going to differentiate implicitly okay so we are gonna find the derivative with respect to time off the entire equation N= S+I+R remember these three of functions of time so we differentiate implicitly will going to have this is a constant the population a constant so that will be zero we have zero equals the X DT plus PI T plus the our DT okay only looking for the rate of infection so let's isolate the IDP because this function represents the rate of infection so simply subtract the SDT of the RDT from both sides from the remote side of the equation in in isolating the IDT for that have the IDT equals negative BS e minus the our DT
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