 Welcome everybody again for this third day of the school. Today we're going to have three speakers and the first one is Matteo Marcilli who is a professor researcher here at ICTP. We'll give a tutorial on stochastic processes. So I leave the floor to Matteo. And before we start, just let me remind you a couple of rules. So if you have questions, you can either post them in the chat or you can raise the end with the zoom feature. And if you're following from YouTube, you can ask questions by posting the question on the YouTube chat and I'll keep track of that. Please, Matteo, thanks for. Okay, welcome everybody. So in this, this is a tutorial. And the idea of this tutorial is to give you an idea of how you introduce stochastic effects in theory and simulations of individual based models. Okay, so going from the ordinary differential equations that you have seen in many of the talks to models that also include stochastic effects. Okay, so. So the idea is say this is just an example of some data from epidemics. I don't remember exactly what the epidemics was, but as you can see, the dynamics of the number of infected in time is not at all a smooth function as you would get from the integration of differential equations for populations. So, which means that you have to include the other terms. So one of course is seasonality. So a term that depends on time explicitly, but the other one is also what I shall call generally noise. And the idea of this lecture is to discuss what type of noise and how this noise comes about. Okay. And, in particular, I'm going to discuss demographic noise. Okay, so if you take a simple model like MSI model, then these are the two differential equations that you have. And generally if you integrate these two equations you would get just a continuous curve. But if you are interested in a model that is going on in a real population, then the increase in the number of infected individual has to be an integer. And if you look closely at this curve, it should be like a step function. And these steps corresponds to microscopic events where a particular individual in the population get infected. So this is what we want to describe. So what we want to do is to modify these equations of this model in order to take into account this discreteness and these stochastic events. Okay, so let's start from the very simple, simplest case. Let's look at just one individual, and let's imagine that he may get infected in time with a rate which is constant in time. So this probability, RDT, that a new individual will get infected in time interval from T to T plus DT. And these events are independent. Okay, so you can represent this on a line and these events are represented by these star symbols. If you plot the number of individuals as a function of time, then each time there is an event, the number of individuals will increase by one. Okay, so how can one simulate this event? If you imagine you want to write a computer code that generates this function I of T, then one idea is to fix a small DT. For every interval with a probability, RDT, you will increase I of T by one. Okay, and you can do this for a particular interval T, T plus DT, and then you advance time to T plus DT, you consider the next time interval. Okay, and for each interval you should draw just a random number, and if this random number is less than R times DT, then you increase I by one, otherwise you don't. Okay, so now how should you choose this DT? So, the idea is that if you choose this DT too large, you may have that more than one individual gets infected in this time DT, so DT should be very, very small. But if DT is very, very small, then your simulation is going to be very small, very slow. Okay. So the question is, can we find a more say smarter way to simulate these problems, to simulate this process. And the idea is to ask, well, how much time given that I have an event at time T, how much time should I wait for the next event? Okay, so this is a little bit of math. So essentially, if this P0 of T, T plus S is the probability that there is no event that occurs in the interval between T and T plus S. And then because of time translation invariance, so this probability will only depend on S, it will be constant in T, it will be the same for every interval. And this probability can be written as the probability that the time that you have to wait has to be larger than S. Okay, so it will be the integral from S to infinity of the PDF of the probability density of this waiting time. The next observation is that if you ask this question for two interval consecutive intervals of length S1 and S2, then it must be that the probability that you don't get infected in an interval of length S1 and S2 is the probability that you don't get infected in the first interval times the probability that you don't get infected in the second interval. Okay, and this should be true for whatever is the size of the intervals. So this requirement implies that this probability, which is a cumulative probability of this waiting time distribution, must be an exponential, okay, must be e to the minus r times S. And then the PDF is just given by the derivative of this and it's the PDF of the waiting time is just r e to the minus r times S. Okay, so now we have this probability distribution. It is very easy to draw a random variable, which has this probability distribution is just given by this formula here where RAND is a uniform random variable. And so what you can do to simulate your process is just to draw a random variable TW with this PDF, then increase your I of t by one and then advance time and then you can repeat it in this way many times. Okay, this is a more efficient way of simulating this process because it requires you to draw fewer random numbers, and it's also exact is not approximate. Okay, so you can also do the same simulation. More efficient with a larger fix the team. If you observe that the probability to get K events in an interval of length. The probability is given by the Poisson distribution. Okay, so it's all in this case, essentially you can keep a DT fixed draw the number of new infected the distribution and then advanced time by by you or by town essentially. Okay. So, these, these all these mathematics that I've been describing is essentially what is called the Poisson process. And these are is a is a process that describes events that occur and that have no memory because essentially in every interval, you can have an event independently of whatever has happened before. Okay, so now let's go back to our problem. We have a set of simulating a population model like the SI model. Now, what happens is these are a little bit more complicated because now we have two population, the susceptible individuals and the infective infected individuals. So there are two compartments in our model. We can say that for every individual, sorry, for every individual, there is a variable Xi, which is either zero, if he's susceptible or one, if he's infected. And the model is say, it's a model where at each time interval t t plus dt, each susceptible can become infected with a certain rate and each infected can become susceptible. So, now what is the the important point is the rate at which a susceptible individual becomes infected depends on how many infected individuals are there, how many susceptible individuals are there, and also on the state of the individual. Of course, in order to be infected, an individual has to be susceptible in order to be become susceptible and individual has to be infected. So this rate generally depends, depends on the variables Xi. So this makes the simulation a little bit more complicated. But again, you have the, you can see at this simulation is a situation where you have n Poisson process, which are, which interact with one another. Okay. And so let us see how we can deal with this situation. Okay, so imagine that we are a time P. And then, and the, the each individual which is represented by a different line here is in a particular state Xi. And at each time, every individual, every each of these states can change. Okay, so now we can use the idea that we developed before and we can draw waiting time from the exponential distribution and ask ourselves, when is it that the state of Mr. I will change. And then you can ask yourself, what is the, what is the individual that will change next, this will be the one for which the waiting time for the next event is the smallest. And then you can simulate this process by just finding what is the, who is the guy with the minimal waiting time you draw waiting time for each of these persons. Then you'll find who is the guy with the minimal waiting time and then you advance to you, you change the state of Mr. I start, and you advance time. Okay. So this is exact. You need to draw n random variables, which is, which are the waiting times for each of the individuals. But, well, you can argue that when you, when you perform the, when you change the state of Mr. I, you don't need to redraw the waiting times for the other people, unless their state has changed. Because of the memory less property of the process. Okay, so can we do this simulation, even better. So, the idea is that essentially, we can figure out what is the time that we have to wait for the next event to occur, regardless of which is the variable that will change this state. Essentially that we are a time t and where this violet guy change the state. And then at this point, essentially, we are interested in understanding, which is the next event with that will take place. We don't need to keep the distinction between the different individuals. We can just deal with this as a single Poisson process. And, and we can just find what is the time that we will have to wait for the next event to occur in this combined Poisson process. And then once we find this time we can also ask, who is the guy, what is the probability that the guy that changed the state is a particular guy, I start. Okay, the way to do this is is simple. The probability that the minimal time that you have to wait is larger than T is the probability that all the times that you have to wait for the next events, the events on all individuals are all larger than T. And of course of this, because these are independent events, then you can factorize this probability, then you know that all these are exponentials. And then you can compute directly, the, what is the probability that the next event will occur later than T. And this is e to the minus this capital RT, but capital R is just the sum of all the rates of all the individual rates. So, you can just draw a exponential distribution or exponential random variable from using this simple formula. Okay, then the second step can be done easily, because if you ask yourself what is the probability that the next guy that will change the state is Mr. I, then this is the probability that is he's or her, he's waiting time is smaller than all the other waiting times. And you can compute this because this is just the expected value on his weight in time that of the probability that all other waiting times are larger. If you do this integral, you get a very simple answer, which is just the ratio of the rate of the process I divided by the capital R, which is the cumulative rate or the rate for the whole population. So you see that if you, you can do a simulation very efficiently in this way by just drawing the time for the when the next event will occur, and then drawing which is the individual that will change his state from this probability distribution. And this way of simulating processes like this is called the Gillespie algorithm. So if you go back to the SI model, then here we are interested in two variables, the number of infected individuals, which is just the sum over I of all these random variable Xi. This is just and the number of susceptible is just the total population minus minus I. So and the rates are equal to beta times the probability that if I pick, if I meet a person at random it will be infected. This is just I divided by N. This is the rate for all the people who are susceptible for which Xi is equal to zero. And otherwise the rate to recover is just me. Okay, and this is the rate that applies to all individuals whose state is infected. So for which Xi is equal to one. And the prescription that I told you before is that if you want to simulate the SI model at an individual level, you just throw a waiting time for the next event from a from an expansion distribution with formula by drawing a random number, taking the log and divided by air. And where are here is given by this simple formula and depends on I, and then you draw I star from the formula that I told you before and essentially corresponds to advancing time to TW by TW and to setting the number of infected individual at time t plus TW as the number of infected individual at time t minus one with probability mu R divided by R. So this is the probability that yes that sorry this I think there should be a plus here and the probability that I get an infected infection more should correspond to it plus one and otherwise it decreases by one. Okay. So if you write a program, then this is essentially is very easy to write the program and then you can compare what is the difference between the ordinary differential equation the user ordinary differential equation which is the black line. The curve that you get from this stochastic simulation which is the green one and as you see you get some stochastic fluctuations. Okay, so a couple of questions, a couple of comments, if you don't have questions. So one comment is why can't I use a finite d t and use the fact that this is a personal process so I know how many events occur in a time between a personal process because it's given by the Poisson distributions I told you where the problem is that when an event occur this changes the rate of other events that can occur. Okay, so one cannot use a finite d t in this case. So this way of simulating can also be written down, can be described mathematically by what is called the master equation. Master equation is the is an equation for the probability of having a high individuals infected at time t and the idea of this master equation is that essentially you want to understand if you want to understand how this probability changes in time, then you have to consider what are the events that are to increasing this probability and these are the events where either you have one less individual who is infected and then someone gets infected and then you go from I minus one to I, or you get I plus one infected individuals and then you get one of them who recovers, or otherwise you have infected individuals and some of them either recovers or some other susceptible individual gets infected so you go to I plus one. Notice that here these two terms that describe the transition from I to I plus one I to I minus one are proportional to the probability of having I individuals at time t and they have a minus sign because the probability then decreases. Okay. So, now, of course, in a real situation like the one I told you before there are other stochastic events. In particular, the rates themselves depends on time. So, you should also take this into account. However, typically, the, this does not change much the way in which we have been doing our simulation. Probably because the, the rates so the time the typical time over which one event occur is a border one of the total population over the size of the total population, and you don't expect rates to change that much over such a short. Time interval. Okay. So essentially, this less the algorithm can be used also in the case where there is a seasonality. This is an interesting observation that has been made by a McCain and co workers. Sometimes ago, and he says to do with the fact that you can get seasonality, even without seasons. And if you look at this master equations solution of this master equation for system system systems. Stationary state with a fixed point and essentially was deterministic evolution. That would be very smooth. Then instead, if you look at the finite population size at the evolution of a finite population you find these large stochastic oscillations. Okay. And this is just an effect which is entirely due to the stochastic nature of the system and this has also been applied to epidemics. As you can see in this plot which is taken by from this paper. Okay, so I would post here if there are questions because then I'm going to change a little bit subject. Yes, exactly. So it's perfect. So I think it's a good moment to stop and see if there are questions from the audience. So if you have questions, please use the raise hand or type it in the chat. Yes, please model and mute yourself and ask. I just try to understand your slides, the previous slide, when you show about stochastic. Yeah, the previous one. Yeah, this one. So meaning that here we can see the slide in picture. Yeah, that one. This one, meaning that meaning that he can see the discrepancy between prediction of the deterministic model and stochastic model is it. Because you can see that the stochastic can predict like the solution with like some sort of verdict solution or cycle or something. But it seems like the deterministic part giving you stable state. Right. That is what you are saying. Exactly. Yes. And the reason is because of the demographic stochasticity. Exactly. Exactly. Exactly. Because you have this demographic stochasticity, you can have this oscillation. So this is a property that depends on the hessian of the when you compute the hessian on the stationary state of the model. Okay, so if you want to know more, I think it would be ideal if you go to these two references where the problem is discussed in detail. Okay. Hello. Yes. Yes. Yeah. Is it okay? There is a question in the chat from Jordi who asks is the period of seasonality in the stochastic process the same as the period of the deterministic damping. In this case here, when in this case here, I think I cannot, I don't recall exactly, so I would not like to make statements that are not exact. And So, so this is essentially related. So indeed, if you see in these plots, the the damping or if you look at the period of the deterministic dynamics is pretty much synchronized with the period of the stochastic oscillations. So, so in this sense, these two things are clearly related. And, and as I told you, well, if you if you look at the fixed point condition, then, and you look at the hessian, then the this periodicity here of the deterministic dynamics will be related to that. And this is the periodicity which is explored by this stochastic fluctuations in a, in a finite system. There is another question by some mass who is asking in the method of adding noise to the master equation. Can we add noises with different distribution or different frequencies like pink or white noise. How can we change this kind of parameters in the algorithm. So the. So if you want to ask the population noise is is given by by is a discrete noise and is is given by this description that I gave you here. So if you want to add the other type of nice, which is exogenous noise. Then this cannot all this can so like a stochastic forcing with a particular distribution. That of course can be done but probably this is more relevant. So we can discuss this better in the next part of the lecture. Oh, sorry. Yes, if you want to ask a question please go ahead. Yeah. Can I ask a question. I understand mathematically that there is an equation, as you said before that we can find the process which accused next. But what does it mean ecologically like how a how we can translate that in what is happening like you can if the model you can predict how the distribution is going to go in the future for the next time. I'm sorry if I don't explain myself well. So here, this is a model, say the stochastic the master equation this is a model where you make a very simple assumption on the dynamics of what is going on. You say that this is a model where is a well mixed population of infected and susceptible they bump on each other and you have this. And for this model under this condition you can compute in this way, the probability distribution if you are a time T you can compute what will be the probability distribution at a later time. Okay. Now, the question is how, how much this model describe a real ecological, real technology. And this is something on which you will learn across all the, I mean, I think this whole winter school is about that. Thank you. I mean, the tutorial is just to give you a sense of, if you look into this model, what is the mathematics and what is the, what are the arguments that are that are on the basis of these models. Okay. Yeah, thank you. So if there is time. Yes, there is another, I will say there is another question by model so we can go ahead with that and then we Okay, thank you. I got another silly question for you about that picture right when you got this divergent between deterministic and stochastic prediction it seems like the deterministic curve should be like of stable spiral part solution. Can you repeat always the case, can you get that whenever you got like the stable case or something. No, no, you don't get it. So for example, if you, for example, for a simple as I model as the one I describe you don't get this, this dynamics. So there are specific conditions on the on the rates on this rate that make this phenomenon possible. Okay, and again, if you are interested in this I really recommend you go through this very nice papers that discuss this property in detail. Okay, thank you. Okay. Let's go ahead. Okay, so now what I would like to discuss is probably a more direct way of introducing stochastic effect into the dynamics of population dynamics by adding a term directly to the deterministic equations. Okay, so if you have an equation again for the SI model for the number of infected. So this would be the deterministic dynamics. So, so what you can ask is, what is the term that I should add to this model in order to account for population noise. Okay, so, and so this is essentially a subject that goes under the name of stochastic differential equations. The, the summary is, if you want the two line summary is that the, the effect of noise you think about the effect of noise on a finite and small time window is the accumulation of many, many infinitesimally small events. So essentially the noise that you should consider can be considered as the sum of many, many random variables. So it is essentially described by the central limit theorem. And because of this, the noise, the type of noise that you should add to this stochastic differential equation cannot be anything, but it has a very specific mathematical form which is essentially related to what is called the vener process. Okay, and the main thing that you should remember about this vener process that I call W here is that is different, it's different. So the increment in this stochastic function over a small time interval dt is of the order square root dt. Okay, and this is a very important consequences that really are important to take into account when you want to simulate this process. Okay, so let me go through this in some detail. Okay. So, as I told you what we want to do is to take a differential equation and add the noise to it. Okay, in this case I'm using this variable why which is the fraction of infected individual in a population. So in order to give a meaning to this noise, I say we should tell, well, how do I integrate this stochastic differential equation so how do I get the value of the random variable why from a time t from the variable at time t zero when I integrate this stochastic differential equation from t zero to t. Okay, so the first part is rather easy because this is just a normal integral, but what is this integral of the noise. So, now in order to describe this integral of the noise, you should really think about it as, as what you would get if you discretize the time interval between t zero and t into small intervals of size dt. Okay, so the number of intervals is of order t minus dt, t minus t zero divided by dt, and this sum is is the sum on the contribution of this noise over this small interval between of size dt. Okay, so now let's think about what what should be the properties of this random variable Xi, which is the integral of the noise over a small time interval of size dt. Okay, so, first of all, if this, if we think this, these are stochastic effects, then it's natural or reasonable to think that these are also independent and identically distributed. Because noise is expected to represent something which we don't know anything. And, and so if, if this is so then the sum of these small effects is going to be to obey the central limit theorem. And so we know that this sum is going to be n times where n is the number of intervals times the expected value of this Xi plus square root n times the variance times Gaussian variable. Okay, where the PDF of this variable Z is a Gaussian. Okay, so in our case I remind you that n, the number of intervals or the number of some of variables that we are summing here. It's inversely proportional to dt. And also the expected value of these increments because this is noise is going to be zero. Okay, so now, if you look at this term here, and if you want a finite limit when dt goes to zero and n goes to infinity, then you should have that the variance of these random variable x time divided by the t should be finite. Okay. Because otherwise you don't get a finite limit. Okay. Sorry. So this is what essentially defines the winner process. And the idea is that each of these Xi which is the integral of the noise. So if I want an integral dt that I call DW should be proportional to the square root of the team times a Gaussian variable Z. Okay, and so I want that if I integrate this noise over a time from T zero to T. Then I get a sum which will will be of the order of square root T minus T zero times a Gaussian variable Z. Okay. Okay, so this is the first lesson so that this DW should be proportional to square root dt. And this tells you that when you do a program and simulate this winner process, then when you advance time, you should rescale things by a square root dt. So your your increments should always be a square root dt. So notice here that this defines a curve, which is essentially this winner process. And which, which is a fine, which is a well defined limit when the T goes to zero. And this you can see graphically by the fact that I have been running this program with two values of dt, which is 10 to the minus three and 10 to the minus four from time equals zero to one. And you can see the function w of t that I have observed that I obtained, and you can see that, well, you cannot tell which is which actually I don't even remember which is which. Okay, which one was integrated with the DT which is 10 to the minus three and which one was integrated with the DT which is 10 to the minus four. Okay. So did the winner process, which is this stochastic random curve as is a very interesting object mathematical object and it has this process properties that you can show that it is continuous is a continuous path. And almost surely. And also it has independent increments if you look at what is the increment of the inner process in a time interval, and you know, another time interval that are not overlapping then these are independent. And also, as I told you, the most important thing that you have to remember is that the differential of these finite processes proportional to square root dt. And because of this, this is nowhere different. Okay, so we get some creative addition to the slides. Okay, so now let's go back to stochastic differential equation so what we have understood is that if we have a deterministic equation which is the first part. If y is equal to a times the team, then the noise that we should add is proportional to this vener process or to the differential of this vener process will be can be any function that also may depend on why. Okay. And if I want to define how what is the solution of this equation, then I should tell you how you integrate how you integrate it. Okay. So in order to do this, well, you have two parts one is just a usual integral is all the basic integral that you compute with a usual rules like. Whatever you like. But the other one is is stochastic integral is an integral that also involves DW. Okay. And so you can think, well, okay, I will compute these integral in the same way so define a time tau i discretizing in many many small intervals. I'm tau i between inside each of these intervals and just evaluate this function at these points and then some these function times DW and then that's it. Okay. However, if you think a little bit more closely, then what you find out is that the value of this interval integral. Depends on how you choose the midpoint. Okay, how you choose this tau i. In particular, what you can see is that if you look at this particular integral, which is w w d w between t zero and team, then you can define it as the limit as it goes to infinity of this discretize some. And if you take this tau i to be alpha times the endpoint plus one minus alpha times the beginning of the interval, then you get that the result actually depends on how you choose the midpoint. Okay. So it depends on this alpha. Okay, so then in order to give a precise meaning to these integrals, you have to specify how you choose these alpha how you do this. And how you compute the integrals. Okay, how you integrate differential equations. Okay, and this is called a prescription. Okay, and the most natural prescription is just to take alpha equal to zero, which corresponds to what you do when you do forward integration of stochastic of differential equations. Okay, so, so the consequence of this is that the, if you, if you look at the differential, so the rules that this mathematics for integrals and differential obey a little bit strange. Because, for example, if you compute this integral here, you get one part, which is what you would get if you just integrate if w were a normal function, which is just this part here. You get a new part, a new term here, which, which, which just comes from this, this prescription and the fact that w is a stochastic function. Actually, it comes from the fact that the DW is proportion to square root dt. Okay, so when you compute the differential of a function of w, then this does not only contain so this differential should be f of w plus the w minus f of w. So this does not you have to expand the f of w plus the w to second order there, not only first order term, because DW square is equal to dt, and then you'll get also this this term here. Okay, so this is consequences for how you deal with the stochastic differential equations and how you change variables. And because when you change variable you have to always take into account that DW square is equal to dt. And so there is another deterministic term that comes out in your differential equation as a consequence of the stochastic term. Okay, so you can do many exercises, but let me go back to the SIS model. And so what distance you so for the SIS model, you can derive a stochastic differential equation, which is, which is this deterministic part. And then the stochastic part, the fact that you have this stochastic part can be derived from looking at what should be the variance of the noise that you add here. Okay, and, and the variance of the noise for each interval. Okay, so this form and depends on dt. So here you should put the square root of the variance of this term here which is given by this, this object here. Okay, then essentially you can run a simulation just integrating forward this stochastic differential equation. And what you get is a, is a path that approximate the deterministic solution, but it has this population noise added to it. Okay, and now how you can relate to these two things that I've been telling you about population noise, the Poisson process and master equation and the stochastic differential equation. There is one way of doing this, which is rather natural and it's called the fun camp and system size expansion. I don't think we have time to go through it. But essentially, what I want to tell you is just these two different, what I've been telling you are not two different ways of two different stochastic processes is the same stochastic processes discussed in two different ways. Okay. So here are a couple of references to which are rather accessible for with a minimal and mathematical say education so and where you can find all the things that I've been telling you. Okay, thank you very much. Thank you very much Matteo for the very nice tutorial. So I think we have time for a few questions. So there was one in the chat that I can start to ask by pleasure I think you sort of answer to that already. How does the stochastic prescription affect simulation. Does it even affect simulations. It does. So if you because say, just to make you give you some intuition. Okay, so when. So, okay, so sorry. Let me go back to this point here. Let's take this example here. Okay. So you see, when you take the expected value of this integral here. d w. If if time is computed at the beginning of the interval, then d w and w are independent. If you take the expected value of this, this is going to be zero. If instead, you take a midpoint, which is not the starting, not not the beginning of the interval, you don't take t i as, as, as the t i equal to t i minus one, sorry, which is alpha equal to zero. Then, if you don't take the, the, the starting point of the interval, then these two variables here w and d w are not independent. And so when you take the expected value, this is not going to be zero. So, when you integrate, and so this means that when you integrate these stochastic differential equations. So you know, when you have an integration, when you have a, when you have a normal differential equation, you can choose any midpoint you want. Okay, you can have say implicit methods where you essentially solve this when we're essentially here you estimate your function x of t at the end of the interval so this becomes an implicit equation, etc, etc. So, and for normal differential equation, this makes no difference. Okay, it only changes the accuracy. You can only affect the accuracy with which you, you can integrate this equation. So, but if you have stochastic differential equation instead, you should be very careful how you choose the midpoint, how you integrate this, this equation. The idea of the ETO prescription is that the noise is in a time t is independent of x at time t. Okay, is a, is a, is an effect which is, which is, yeah, is independent of what was developed. Is this clear? Yes, that part is clear. But my question more was along how does it show in simulation? How does this prescription show up in simulation? It shows in this way. So, say for example, if you, yeah, so, so when you, when you have a stochastic differential equation like this one, if you take alpha equal to c, then you integrate it, you would get, you, I mean, you would get a contribution which is like this one. Say for example, if you take alpha equal to one half, alpha equal to one half correspond to a difference prescription, which is called Stratonovich. Then these are the prescriptions such that you don't have these other terms here. Okay, you don't have the second term here. Okay. And so the rule that you apply to stochastic differential equation are the same as the rule that applied to differential equations. Okay. And so the results are completely different. Also, if you do numerical simulations, the results are, well, not completely, but they are different. Okay. Okay, so there is something you should care about. You should be always specific about what prescription you are using when you're dealing with stochastic differential equation. And I tell you this because also in, in papers that you, that you can read, this is not always specified. Okay. But if you want your result to be reproducible, you should say what is the prescription that you are using in a stochastic differential equation. Okay, so there are a couple of more questions in the chat. So I think we have five minutes more to leave like two minutes for a break. So, next question from Zoret. Can we study bifurcation in a stochastic model. Yes, of course. So now bifurcation, so you always think of, so if you think at a discrete map, then this is like, I don't know, if you look at the map or the logistic map, then you can add noise in different ways. But then if you look at it, like something like Lawrence system, where you have stochastic differential equations, couple differential equations. You can add stochastic noise to that. And, and see, and I think that many people that have started these problems. Yes. Another question about typically noise by John Luca was, are there models for which the stochastic differential equation involved a noisestorm that depends on the state of the system at that time. Say DW depends on X. Yes. So typically, you, you, you do this by saying that, well DW is always an independent at least in the prescriptions so DW is always the differential of the final process. So DW depends on X at time T is the, is the variance of the noise is B of X and of XT. Okay, so the, so this is the way in which you can build this dependence of X and T of, of the, of the noisestorm on on the state of the system. Okay. Great. So, is there any other question. I have, I have a question. Yes, please. I have a question that if the paper does not say what alpha they, they used. It's not reproducible. But then I was wondering if I can reverse engineer and given a solution or some, some characteristics of the solution. I can find what alpha or, or make, or maybe a distribution of probability of alphas they use. Yeah, so let's see. Yes, you can. In many cases in fair, what is the prescription that has been used also because different communities, typically use different prescriptions so theoretical physics. Many people use strato which prescription. And whereas for example in, I think in epidemiology, the natural prescription is more the ego prescription. But yes, I mean, the principle to, to, to figure out what is the prescription use but I mean when you write a paper you should ensure that your results are reproducible so you should state what is the prescription essentially. It's a, yeah, it's on the side of the authors to specify this. Great. So, I think it's time to take a short break before the next tutorial so thanks a lot there again for this very nice introduction. And this is what this introduction is available on YouTube so again you can go back and rewatch these lecture this tutorial as many times as you want. So, now we're going to take this small break and