 We will begin today's class where we look at and play with the bus diffusion model. So, last couple of lectures we have been looking at models to do to understand the diffusion process which is available which applicable in many scenarios like spread of epidemics, spread of new, sales of new products and such. So, this particular model is attributed to Frank Baas, he developed a model for diffusion of innovations to overcome the startup problem. That is in the diffusion model that we have seen till now, there has to be initial value for the adopters. The adopters if it is 0 means that no innovation will occur, there has to be at least one person who has purchased the product and then a word of mouth effect happens because of which the products are all sold and all the potential adopters end up buying the products. But Baas improved on that model by introducing what is called as advertising effect where in the initial adopters can be 0, but through advertisement the message is sent to the potential adopters which makes them buy the product. So, that is the key contribution done by Baas. Along with you know various solutions and derivations of that which then became his thesis work. Now, the stock flow diagram model that you see on this slide is the Baas diffusion model represented as a stock and flow model. So, we have two things that affects our adoption rate, again the population divided into two potential adopters and adopters. So, that means eventually all the potential adopter will become adopters. Now, adoption rate is governed by now two factors, one is adoption for advertising and then adoption from word of mouth. And here you can see the adoption from advertising has some potential adopters coming in and advertising effectiveness also affecting it. And this adoption from word of mouth is affected by adopters total population and fraction adoption fraction contact rate as well as the potential adopters. So, this model is very similar to the diffusion model that you saw where the adoption rate was equal to the word adoption from word of mouth, the underlying equation continues to remain the same. So, when P is equal to 0 we will have a basic logistic innovation or a diffusion model sorry logistic diffusion model, but if P is greater than 0 then we call it as a Baas diffusion model where adoption rate is nothing but a sum of adoptions from advertising and adoptions from word of mouth. So, let me just write it out. The Baas diffusion model we have adoption rate is nothing but adoption from advertisements plus adoption from word of mouth for WIM and short AR, adoption from advertising is given as a product P times A, sorry we are using potential adopters as P, small P times capital P plus sorry the notation we are using word of mouth is C into I into P into A divided by N where P is your potential adopters, A is your adopters, AR is your adoption rate, N is the total population, C is the contact rate, I is the adoption fraction and P is small P is the advertising effectiveness. In popular models we you know consider we usually set say Q is equal to C into I as a product. So, this is the coefficient for the word of mouth, so that will make our AR is equal to P small P into capital P plus Q into P into A by N. This is the equation that underlies a Baas diffusion model where small P is the advertising effectiveness and small Q is the word of mouth effectiveness or coefficient of advertisement and coefficient of word of mouth. So, given these two parameters as you can see if even if adopters are 0 still adoption rate can occur because of some effectiveness in advertisement happens to the potential adopters whereby we can simply overcome the startup problem. So, given this we can definitely model it as demodel. So, now in this Baas model there are only three parameters that we need to define one is P the coefficient of innovation or the advertising effectiveness and Q is also called as coefficient of imitation or the word of mouth effectiveness right. So, words are self-explanatory when you say imitation it is through word of mouth that means, you just want to copy from others what they have innovation you are just looking at advertisement and want to try out the product. So, that is P and Q. So, the various phrases used to define P as it is shown here it represents coefficient of innovation or as it called as external effect or advertising effect effectiveness advertising various phrases are there. So, this is just these three parameters we can actually model the diffusion of innovation. Now, the simple thing that we want to try out is what happens when P is less than Q and what happens when P is greater than Q to understand the dynamics that is going to come into play. So, that we can see when P is less than Q means the effectiveness of advertising is lower than the effectiveness of word of mouth. P is greater than Q means of course, effectiveness advertising should be greater than the effectiveness of word of mouth. When P is less than Q the adoption rate will show an increase and then it will fall with a peak value while if P is greater than Q then you get a simple goal seeking system right. So, what will be the why intercept can you guess see initially the potential adopters are all the total population are all potential adopters we are going to assume. So, why intercept is nothing but P into N here also it is P into N. So, let us just observe the first graph there is graph on the left side when P is less than Q it is going to peak at some point and then it is going to fall down. So, let us denote that as T star where T star is the time at which AR peaks or this is the whatever time of peak adoption. So, in this case actually T star is a time 0 or rather if you go for analytical solution then T star is actually going on the negative side. So, the peak is in the past so which does not make sense so we will assume peak at happens at time 0. So, this simple graph shows that only word of mouth can actually have my sales increasing my sales is nothing but an adoption rate continues to increase while only if it depends on advertisement then my sales are going to keep falling down. Again by word of mouth I mean a positive word of mouth and so that means we want the word of mouth happening much earlier and much quicker so that we can actually have a growth in the products as come out advertisement where initially might have large, but then you will get a continuously diminishing sales and no marketing department where you might want to show this saying that every month their sales are actually reducing rather than increasing right. So, we need to get this going as soon as possible. So, in this case continuing the discussion this case we can call it as a successful product let us say yeah we call it a successful product and this case we can call it as unsuccessful product unsuccessful product the sales is pretty much is dominated by the advertising effects rather than word of mouth. Assume the same p values are there right the starting point is the same, but if q in this case lesser than this one in that case then here I am going to reach the population much faster as compared to this case it is going to take me longer time to reach the population.