User talk:Hardiked

1 introduction Mathematical modelling is technique used to solve many practical problems in terms of mathematics by making some assumptions. In this project we are going to understand how to formulate any problem and solve it by one simple example and then we see one simple paradox given much time before by famous mathematician. Simple steps involved in mathematical modelling are following: (i) select the modelling approach (ii) formulate the model (iii) solve the model 2 calculus and its application calculus is most powerful weapon of thought yet devised by wit of man. -Wallace B. Smith calculus is the word originated from Latin word 'calculus' .It's meaning in latin is small pebble used for counting.It is the study of change of variables with respect to some other variables.And we know that everything in this world is changing nothing is stationary and this is the reason why this branch of mathematics got this wide range of application everywhere in this world.Now, let's look at some application of calculus in di�erent �eld shortly. 1. biology (a) to determine the exact rate of growth in a bacterial culture when di�erent variables such as temperature and food source are changed. (b) help to decrease the rate of growth of harmful bacteria. 2. electrical engineer (a) to determine the exact length of cable needed to connect two power substation. 3. architect (a) to determine the amount of material needed to build the curved dome. 4. space ight engineers (a) to plan lengthy mission. 5. physics (a) to determine the center of mass of any object. Physics is impossible without calculus, it is backbone of entire subject. now let's start our main problem. 3 SIR model This model basically developed to understand how fast particular diseases spread in certain region. It is widely accepted and famous model used worldwide to make prediction about any diseases. Spread of many diseases like ebola,polio etc. are predicted using this model only. Here, I have presented the simple approach to this model. Let,s understand it in further few secton. 3.1 assumptions This problem states that particular diseases attack the certain region, people start to get a�ected by it. Let's say its e�ect last for some day that we are going to assume further.then how fast this diseases spread in that region.It needs to include some basic assumptions as listed below: 1. Population remained constant during the time interval we have considered. 2. the person who is a�ected once is not going to su�er from that diseases once again. 3. rate of contact of particular person with those who are infected remained constant throughout all the days. This assumptions are made to simplify our calculations. allthough in actual model all this cases are considered in special case. 3.2 variables Many variables are included in this model. So, �rst we are going to de�ne all the variables, then classify them as changing with time or not. list of variables are: 1. N=Total population of that region 2. S=Number of susceptible1 peoples 3. I=Number of infected peoples 4. R=Number of recovered peoples 5. �=rate of contact of infected with susceptible people. 6. =average time period for which diseases last. So out of this 6 variables de�ned above it is clear that N,S,I are changing with respect to time, whereas �, are going to remain constant with respect to time2. So, now we have to writ equations of only those variables which are changing with time.and before doing that lets draw few graphs to understand how this variables are changing with time. 3.3 Graphs First, we have to understand that initially there are no infected peoples in region i.e. all are susceptible but slowly number of infected start to grow and susceptible start to fall. Same thing is depicted in the graph shown below: �gure 1:number of susceptible person vs days �gure 2:number of removed person vs days 1those who are still not a�ected by diseases 2refer to assumptions 4 Equations Now ,we are going further to see challangig part of given model.We have to make equation of three variables that we have decided as depend on time one by one. Just observe the process and you understand each and every thing that I have written so far. 1. equation of S:- Some of the few thing which we know about this quantity is that this number is going to decrease with time.so negative sign must appear in its expression.now rate of decrease of this quantity depend on the rate � at which susceptible people come in contact with infected person. dS dt = 􀀀� � S(t) � I(t) .......(1) 2. equation of R:- We know that is the average period for which diseases last. So, 1/ is the rate at which people are recovering. so it is understood that rate at which people are recovering depends only on number of infected people at that time and rate at which people are recovering. dR dt = I(t)

.......(2) 3. equation of I:- rate of change of infected person is equal to rate at which susceptible people got infected but on the other hand we have to consider that infected people are also recovering from the diseases. Hence, rate at which susceptible people get infected is given by � � S(t) � I(t)3. And rate at which infected people are recovering is given by I(t)

4.so overall expression for I(t) is given by � � S(t) � I(t) 􀀀 I(t)

.......(3) 3see one 1 4see 2 5 Understanding The Equations Now let's think about how to apply this equations. All three equations just give rough sketch of what is going on in real ground. This provide us the number of susceptible people reducing, number of increase in infected people and recovered people.minus sign in front of equation no.(1) is for our reference that whatever is the answer of this quantity is to be subtracted from total population. 6 example I am going to end this assignment here only for example you can refer to the link that I am going to provide you soon 7 dicothomy paradox This famous paradox is given by famous mathematician zeno. it states that If you want to reach the goal then you �rst have to reach half of the distance then quarter then one eigth and so on. as this is in�nite sequence one can never reach to the end. With the help of this example Zeno had argued that motion is not posiible on Earth. Analytical way of understanding this problem is D = 1 2 + 1 4 + 1 8 + � � � + 1 2n 7.1 solution Now lets try to understand how the approsch of sequence and series can solve it X1 n=1 1 2n This is geometric series containing in�nite terms, and we know how to sum the in�nite series.so,it's sum is X1 n=1 1 2n = 1 2 1 􀀀 1 2 = 1 Which shows that this sum has �nite value. So, this sequence must be convergent and hence argument of jone proved wrong abd one who start motion according to his paradox must reach at his destination as proved in our solution.