Lecture 38: Analysis of infectious disease dynamics continued...

Welcome back. Let us continue our discussion  on SIR model for infectious disease dynamics.   We will have a quick look into the model.

The model is in front of you, a third order system   d S, given by d S by d t is equal to minus r S  I d I by d t is equal to r S I minus a I and d   R by d t is equal to a I where r is the  infection rate, always positive, a is   the removal rate always positive, S0 the initial  number of susceptible members of the population,   I0 the initial members of the populations which  have been introduced as infected ones and the   initial condition for removed ones was S0. Now, in the last lecture, we had a look into   the behavior of I-S projection of our phase  portrait and we saw how we can use the a by   r ratio and the initial susceptible population  to make a prediction of whether or not pandemic   would prevail. Now, let us do a further analysis  to see whether we can answer some more questions.  

So, in the last lecture, we saw that  under certain conditions, you can have the   maxima in the projection which means that the  number of infections would rise with time,   reach a peak, that is what we are referring  as peak, all of these days and then the wave   starts declining. Can we make an estimation about  the number of members which would get infected   when the peak reaches. Let us see this.

So, I have   dI by dt is equal to r S I minus aI this was the  model equation for I and when you have a maxima   for the function I is equal to I of t, then  the derivative should become 0. So therefore,   at the peak the   dI by dt is equal to 0, from where I can write r  S I minus a I is equal to 0 which means r S minus   a multiplied by I should be equal to 0. Now, I  cannot be 0 because, if you do not introduce any,   if you do not have any infected members,  further infection cannot spread.

So, therefore,   I can write r S minus a should be equal to 0. Now, this gives me a condition for the value   of S the susceptible population at which you  will get a maxima, which means if you remember   the I-S projection then we had a system like  this we had a phase, line like this. So, this   is S max.

S max does not mean that S is max.   S max means the max corresponding to I max.  So therefore, S max would be equal to a upon   r.

So, when the population of the susceptibles  becomes equal to a by r you get the peak.   So, now it is not very difficult to see how  to make a prediction on when you will reach   the maxima. So, you will reach maxima when the  population of susceptibles in your system becomes   equal to a upon r.

That is the condition. So,  whenever this condition is satisfied, you will   reach the maxima in number of infected persons.  So, now, if this is the case can we determine the   number of infected persons in the peak, I max.

So, in the previous lecture, we wrote an   expression for I in terms of S. So, I so, let  me see the expression for I, I was nothing but   minus S plus a upon r ln S  plus N minus a upon r ln S 0.   This is what we derived in  the last lecture.

So, for   maxima, I have S should be equal to a upon r from  where I can write I max is equal to minus a upon r   plus a upon r ln a upon r  plus N minus a upon r ln S 0.   So, can I do some rearrangements. so, let me write  this as a upon r minus 1 plus ln a upon r S 0   plus N is equal to I max.

So, now, the question  is that what does, so, let me write the question,   what does I max   depend upon? The answer would be, it  depends upon N from equation number 2.   Does this make sense?

Obviously, it makes sense  because, if you have smaller population the number   of infections which it might reach at maxima  would be smaller. Countries which have very   large population, the corresponding number,  corresponding N is large and the number of   infected persons at maxima will also  be large. Intuitively this is correct.  

It depends upon S 0, susceptibles. Now, the  question is that if you have a population in   which everyone is susceptible versus a population  in which the number of susceptibles is smaller,   which population is more likely to have higher  number of infectives? Obviously, the ones in which   there are more number of  susceptible, quite simply.  

It also depends upon a by r. a  is the removal rate, r is the   infection rate, and in fact what you  have done is you have clubbed this   in a r S 0 which is what, which is 1 over R 0,   reproduction number. So, this is an interesting  observation.

So, if I have I max which is given as   a upon r multiplied by 1 minus 1 plus ln  1 upon the reproduction ratio plus N.   So, now, the question is what good can come out  of this equation. This is a very important and   interesting equation because this simply  says that if I know my total population,   if I know my population of susceptibles in the  beginning and very safely, it can be approximated   during the early stages of the pandemic as to  be equal to the number, the population itself.  

And if I know the reproduction ratio in  the beginning, then I already know how much   of the population is going to be infected at  the peak of the pandemic, I already know that.   The only thing which is unknown this whole  business, the population would be known.   So, S 0 can be approximated, what  is to be determined with accuracy is   a and r.

So, if you can know from the initial  data, the infection rate and the removal rate,   then during the beginning of the pandemic itself,  you can know or you can have an estimate of the   number of infected persons which would the  number which it would reach when you reach the   maxima of your peak, interesting. Now, finally, we are also had an expression   which is given as d R by d t is equal to a  I. Now, what happens when the pandemic where   S, time t tends to infinity, when you are at  the far end of your pandemic, you would like   to have some more information.

So, now, I have  my flow scheme as S going to I going to R.   And this is very similar to the reaction scheme  which we saw in one of our previous lectures as   a going to b going to c. So, what is the ultimate  fate of S and I and R, this is what we want to   know.

From a going to b going to c analysis, we  already have a fairly good idea about this. So,   I saw that if I write the evolution of I with time   with some I 0, then I have 2 options. I may go here positive gradient depending   upon S 0 R upon a greater than 0, I  can have this option negative gradient   for the same quantity, reproduction number  to be less than 1.

So, in either case, this   is only the initial population and we saw the S-I  projection and we know that there are cases where   your I goes asymptotically to 0, and  even the cases where infections rise,   you have a peak and then again the  infections go to 0. So, therefore,   limit t tends to infinity I is equal to 0. You will not be left with any infected persons   as time t tends to infinity.

Why would  this happen? Well this may happen because   there are no susceptibles, everyone has got  infected and those who have got infected,   they have either been recovered or in worst case,   the population has gone to extinction, everyone  is dead. So therefore, eventually I will go to 0.  

What about S? Is it guaranteed that every single  person in the population will become infected.   So therefore when you have a compartment  S and from this compartment you go to I,   the fate of I is that it must go to 0.

But  what about S? Will it always go to 0? So, limit   t tends to infinity S is equal  to 0, this is the question.  

And we know at least by today’s experience that  this is, this may not be the case. It is not,   there is no guarantee that every single person  in the compartment S will be infected. It is   a bulimics population S alright, but you still  have the parameter R and therefore, R gives you,   it gives you a qualitative idea of whether  the fraction of, in discrete time domain,   the fraction of persons would which would get  infected.

We will come to the discrete time   model in a few moments, but that is  basically what the significance of R is.   So, therefore, we would like  to know the total population   which would get infected or in other words,  the population which would not get infected,   which would remain susceptible, but which  have not got the infection. And that can be   established by the fact that S infinity  S at infinity plus, R infinity,   total recovered population is equal to N.

S at infinity plus R infinity is equal to   the total population N, or in other words S  infinity is equal to N minus R at infinity.   So, therefore, if you know R at infinity, then  you can simply subtract R infinity from N and that   will give you the population S infinity, which  means the susceptible population at infinity.   In other words, this is the population  which has not contracted the disease.  

And this can be obtained by this equation, N is  equal to S minus N is equal to S plus I plus R   which means R is equal to N minus S  minus I from where I can write dR by dt   is equal to, sorry let me do one small  change. I can write this as I and this as R.   So, this is I.

This is R. So, dR by dt  is equal to a times N minus S minus R.   So, now you have an equation  which is in front of you,   and now you would need to solve this equation. 

So, if you know the time evolution of S,   you can know, you can substitute that here. You  can solve for R, and from here you can solve for R   infinity from where you can get S infinity is  equal to N minus R infinity. This is one of the   approaches to determine this.  

Now, we have been following all of  these methods in which you try to solve   the equations analytically, turned out to be  a little difficult. It was not impossible,   but turned out to be a little difficult. You may  want to solve this these equations numerically   and that proves to be much easier  task than solving them analytically.  

So, let us see what can be the strategy.  I have d S by d t is equal to minus r S I   d I by d t is equal to r S I minus a I and d   R by d t is equal to a I. So, how would I solve  this numerically?

Let me adopt Euler method.   So, when I am solving this   numerically, it is basically the algorithm  is evolving the system in time. So, therefore   I will have the vector S, I, R and  this would be the steps in which   my discretized algorithm would evolve  the system, and this signifies time,   n would signify the time.

So, following the Euler method   what is the algorithm? I will have S at the step  n plus 1 is equal to S at the step n plus delta t.   And what is delta t, this is delta t,   multiplied by the definition of the derivative  which comes from the equation minus r S n   I n.

Similarly, I will have I n plus 1 is equal  to I n plus delta t r S n I n minus a I n.   And finally, capital R n plus 1 is  equal to capital R n plus delta t a I n.   This method will give you the evolution of  the system in time, but how would you start,   evolution means if you know the value at a given  instant of time and discretize time like this,   the method to go to the next step is in front of  you.

But how will you determine the first step.   First step is very simple, you know  S 0, you know, I 0, you know R0.   So, therefore, if you know R  and a and you have set delta t,   if you know R and a and you have set delta t,  then the value of S at first instant of time   would be simply what S 0, which is known  to you plus delta t, when you have set it   minus r is known, S 0 I 0.

So, this is known  to you. So, therefore, what will happen to S2,   you will determine S1 and you will plug it here. So, S1 plus delta t minus rS1 I1 and so on.

So,   in this way, in this manner, you can evolve your  system. There is a larger theory which goes on   analyzing the complexity of solutions,  the error analysis convergence and so on   when you adopt a numerical method like Euler  technique versus an analytical method. We   will not go into those details in this particular  course.

I encourage you to look into those aspects   before you solve this problem numerically.  But this is basically the algorithm.   Now, we saw previously that one may model  a problem in continuous time domain or   it is possible to model it in discrete time domain  and especially, for population dynamics.

Now, we   saw that how one particular approach may be  similar or different from the other approach.   The SIR model which we discussed till  now was analyzed in continuous domain.   Can we do this in discrete domain as well?

So,  what we have here in front of us are the model   equations in which you have SIR model in discrete  time domain. So, what, how do you do this, you   do this like this that you have, you define the  population S, I and R at given instant of time.   And as I mentioned during the population  dynamics in the previous lectures,   it is not very difficult to see that the  next population is going to be dependent   upon the current population as we also find Euler  technique.

So, the next population is going to be   the dependent on the current population. And therefore, one of the methods can be to   model the system like this that S at time t plus  1 will be equal to S at time t plus some quantity   plus. Plus or minus depending upon  whether the population in that particular   section, that particular compartment  is increasing or decreasing,   but that can be taken care by the sign  itself.

Plus or minus some quantity.   Now, that quantity should be in  correspondence with the time rate of change.   And we saw in the previous case that the time  rate of change is given for S as multiplication of   S and I.

So multiplication of S at that instant  of time and multiplication of I at that instant   of time. But, as we saw previously, you cannot  simply take this, because there can be a fraction   of the population only, which would satisfy  this condition of binary fission. For example,   if you remember our previous case.

So, therefore, you must multiply it with   a constant. And the constant here in this case is  r. So, therefore, here in this case you have minus   r S t I t.

this is the analysis, this is the  model equation discrete domain. Similarly,   following the same logic which we did in  the previous continuous domain analysis,   I can write I t plus 1 is equal to I t,   the population of the current incident plus  the influx contribution from the influx.   So, S t, I t, and the contribution from the out  flux, minus a I t and finally, for R t plus 1,   in an analogous manner, I can write this as R  t.

So, this should be capital R, R t plus a I t.   This is the discrete time model. So, if you  have again this method is very similar to   the Euler’s solution technique by the way,  if you see there is an extra delta t.  

And that delta t is now absorbed in minus  r here or other constants here. So, now,   if I have this model and if I try to determine  the time evolution, I should in principle get   the same time evolution as what you get in  continuous time domain model. So, if you look   at the text reference which has been given here.

I have borrowed two results from them the figure   on the left hand side gives you the evolution  of S, evolution of I, evolution of R in time.   And it is not very difficult to see that this  continuously decreasing curve would be your S,   this is your S.   We saw in the previous analysis that is a  peak in the number of infective, so, this   is I and this is the recovered persons.  

So, you will in fact have a match between what you  saw in the continuous domain versus what you saw   here. Similarly, we saw the SI projection, and  you have the same projection here, you have the   peaks. All these three cases and so, you  basically see that the pandemic prevails   when the infections go up, reach a  maxima, have a peak and they come down.  

So, now, whether you do this analysis in  discrete time domain or continuous time domain,   the characteristic remain unchanged, what  you will basically see is that there is a   infection rate, there is a removal rate, there  is an initial population and depending upon   these values, you can expect that if you  introduce certain members which are infected,   whether the pandemic would  increase with time or decrease   with, whether the pandemic would occur or not, or  the infections will simply die down with time.   And what also we saw very interestingly is that it  is possible to make predictions about the epidemic   or pandemic, I am using epidemic and pandemic  interchangeably because in today’s scenario, the   way we have interactions in different geographical  locations in the world, every epidemic is very   likely to be converted to a pandemic. So, now, we can say that, we can simply   use just the term pandemic.

So, we also  saw that it is possible to make predictions   about the evolution of the pandemic during the  beginning of the pandemic itself, by knowing   the principles of dynamical systems. And we are  also seeing we are also witnessing that there   are a lot of models which have come up which make  predictions about the evolution of the pandemic.   We see many of them to be really very successful. 

The SIR model, which we saw is the first   and the very preliminary model, the models  now have become much more sophisticated,   but even a simple model is more or less,  simple as SIR model, actually gives us a   very good idea about the future projection. Now, the last point which is worth discussing   here is that what is the importance of modeling,  why should we bother about modeling? Why so many   groups in the world doing this modeling?

And  when do they fail? Why do they fail? Is there a   specific physics behind them, which makes them  bound to work or makes them bound to fail?  

We saw that there are very simple parameters,  which in the beginning itself would tell you   whether the pandemic would, I mean, whether the  infections would grow with time or not, but all of   those dependent upon the parameters a, r and S0.  a is the removal rate, S was the infection rate.   So, during the beginning of the pandemic itself,  if you have an access to good quality data,   accurate data such that you will be able  to make correct measurements of the removal   and infection rate, then you may expect the  model predictions also to be correct.

Now,   this is important, and that is why it is important  to have the correct accurate collection of data   because if your data itself is not accurate, the  information that you are coming to you is not   what is going on in reality in the population. Then you would probably make a wrong prediction   because the data available to you was wrong  and therefore, correspondingly the parameters   that you drew out of the data were wrong and  obviously, the predictions would be wrong.   One particular situation happens which is worth  paying attention is that many times we make a   projection that pandemic would happen there  would be a peak, but the peak does not come.  

What should we infer from this?  Should it be considered as a failure   or it is something else? Well, one should pay  attention to the motivation behind developing   these models.

The motivation behind developing  these models is to prepare ourselves to combat   such situations. So, therefore, if one  is projecting a wave, then the government   as well as people have the responsibility to  take actions such that the wave can be avoided.   So, therefore, if a projection is made and the  wave does not come, that in fact is the success   of the entire modeling business because you were  doing this modeling precisely to avoid occurrence   of this wave.

Yes, but if you predict the wave not  to occur and a wave comes, then that particular   scenario is in fact the failure of modeling. And the failure of modeling may not necessarily   because of the failure of the modeling  procedure but there can be several processes,   several reasons report responsible behind that.  The reasons may be physical that the model   assumptions are not correct, the modeling  process is not correct.

The mathematical   procedure is not correct. One of the reasons can  as well be the inaccurate availability of data.   But in any case, a knowledge of dynamical system  can actually help us bring about these wonderful   results, which can help us and the society  cope up with future evolution of the pandemic.  

We will take another example of physical systems  in the next lecture. Till then, goodbye.