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Understanding Epidemic SI (Susceptible-Infectious) Model from Scratch

March 3, 2020


testing… Let me slowly move into
the camera well okay this is not me
sorry it’s wrong this is not either… great! that’s me
good so I’m gonna start this video with several questions about this disease: What would happen when a disease can spread within a certain amount of
population? What if we know the original number of infected people and suspected
people at the very beginning, would that be possible for us to foresee the change of numbers of infected people in a certain time frame? And what are those
papers saying by using what they called SIR or SEIR model? Does that model mean that was an extremely accurate result? If it’s not what are the
constraints for the application of those models? Now we’re going to see
whether it’s possible for us to try to understand how the basic epidemic model was made. Assume that the total number of the
population is N and then the number of susceptible people was S(t) And the number of infected people was I (t) What is happening in between S(t) and I(t) is that the susceptible group, because of their contact with the rest of the people, they are becoming infective through time so let’s see how the model can be made…
little bit adjustment…. so the total number of N can be divided
into I(t) infective and S(t) susceptable and normally they have a recovered amount of
people R(t) but to simplify the model at this stage we assume the recovered is zero
meaning that it’s really hard and there’s no chance for them to get
recovered. I know it’s really cruel but for example this kind of model we call s
I model can be applied into the HIV. let’s see… then we’re going to also define at the same time two parameters: we defined a number of contacts that each infected people can approach per
unit time as γ, meaning that for each one of the people in the infective group they can approach γ this many people from the overall (including the suspected guys) and then we define the probability of disease transmission per contact
(remember per contact) as β what did that mean? meaning that for each contact between the infective people and suspected people the success rate for the suspected people who become infected is β. (well I know it’s really hard to
pronounce anyway) so what is the correlation of the number of the two groups S(t) and I(t) ? (since the R is zero now) let’s see by the definition of gamma and
beta for each of the one person in S let’s say it’s one person the
possibility of their meeting infected people is I (t)divided by N (Constant Number) for each person It ‘s like: if they meet another one guy in society N, the possibility of this guy becoming in the infected group is this so while we define the number of contact per unit time as γ and the probability of
disease transmission was β then the number of person of each one in the susceptive group may lead to: (γ ) times (β) times (I/N) amount of people to get infected per unit time. we remember this… so the total number of people in susceptible group is S(t) so that we know one person can lead to (γ )* (β) * (I/N)amount of people get infected per unit time meaning that if the total number (of people to be potentially transformed into infective) is S(t) so we have to multiply S(t)… meaning that the total number of people can lead to this amount of increase in the infective group I(t) per unit time, meaning that the changing rate (
the speed of raising/increasing) of I(t) equals (S)* (γ )* (β) * (I/N) in this case we know that N equals I (t) plus S(t) then we got a number equals well
let’s use only the variable I(t) for left and right of the equation so (the right side is): (N-I)* (γ )* (β) * (I/N) we know that (γ ),(β) , (N) are all constant numbers within a certain time frame so we get the changing speed of I equals N minus I times I times another constant so in this case we find a correlation of
the I. Using differential equations we know that dI/dt equals n minus I and then a constant we call it gamma beta divided by N and also
times I that’s how they get to this step! so remember they’re all of their
assumptions: a fixed number they divide into this two and then the recovered is
zero then we got the SI model so what do we do? we give these models to mathematicians and they will tell us how the function I in relation to T was
going to be like… which is kind of like this model… if we got the time or more importantly if I got the knowledge to understand the mathematical problems we can try to infer from this step to this but now that’s approximately the basic
understandings of S-I Model. that’s it so yes thank you for watching!
and yep! so see you next time! bye-bye

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