Aditya Ghosh, Author at Adamas University

Mathematical Modelling – a fresh opening for research due to COVID-19 ?

However, COVID-19 had spread out a long ago, but until now, the whole world is not getting rid of the disease. Every day the number is increasing throughout the world and of course, India is not an exception. At present India is at no 3 position based on the covid-19 cases. If we unable to control or predict the proper lockdown criteria, we may end up by top of the list soon. However, biological and medical research are going on, but Mathematical research is also not very lagging behind in this case. There is a powerful tool in Mathematics, namely, Mathematical Modelling that is working as a bridge between two extreme opposite subjects, Biology and Mathematics and give the opportunity for the researchers of both field to work together.

What is the need of Mathematical Modelling in this scenario?

Biology and medical research is going on to invent medicine and vaccine for COVID-19. Then the common question may arise, “what is the need of Mathematics then?” Sometimes, it is not sufficient to get the idea of the biological system without understanding thoroughly between various parts of the system and Mathematical modelling exactly does that. Therefore, this works as a block diagram to the overall scenario.

Until now, we have no medicine or vaccine other that Lockdown. Now by analyzing the data of patients from various countries, a more suitable suggestion can be given for imposing the lockdown. Modelling comprises of Differential Equations, Statistics that help us to know about the stability of the situation. Therefore, for non-mathematics person, a brief idea is given about Mathematical Modelling.

What is Mathematical Modelling

Using mathematical symbol and logic a real world phenomena can be represented mathematically, which is called the Mathematical Model for that. The process of constructing a mathematical model is called Mathematical Modelling.

After creating a mathematical Model, the model can be analyzed and the derived results can be justified with real world data. If the result match with real world biological data, then we may end up with a successful model. A schematic diagram can be more appropriate in this regard,

There are some limitations also of Mathematical Modelling. In earlier time, there were manual mode of data taking, which leads to data loss or unavailability of data. Due to that, justification of Models with real world data was sometimes failed. Now days each patient data is computerized in most cases and so availability of data is more and we can consider the data we want. Mathematical Models are more realistic for that reason.

Again, biological interactions within a body is very much nonlinear in nature. For representing these terms we got huge nonlinear terms which is not suitable for Mathematics. Some of the terms may be simplified using various mathematical software but most of the cases are very tough for calculations. In this regard, sometimes we use a simplification of nonlinearity by ignoring higher order terms or by linearizing those equations. By doing that, the nature of the original equations were changing which is not desirable.

Modelling is then also very much important to predict to the nature of a problem apart from its difficulties. For any branch of science formulation of situations are very important mathematically and it will give rise to a good research article. So basically, modelling is important thought branches of sciences and even in some other disciplines.

Opportunities of research in Modelling on COVID 19

Many people often asked, “What is the direct applications of Mathematics in the real world?” The answer is COVID-19. Before COVID-19, many people were not aware of mathematical modelling and prediction. However, after this disease happens many researches are going on about the lifting of lockdown, namely unlock-1. Again, in many states including West Bengal, fresh lockdown is given to control the spreading of this disease. For this kind of decision, proper mathematical calculations will be there, which is coming from Mathematical Model. Now for various country, the nature of the same disease is different, so different models need to be implemented. Those models can be validated with data of that country only. For researchers in bio mathematics, this kind of disease is a golden opportunity for doing a fresh research. This unknown disease gives rise to many different aspects of symptoms every day which will enrich the research aspects of a researcher in bio mathematics. Till now no drug or vaccine is present. Once it will be discovered, the drug aspects can be incorporated in the mathematical model, which will make the model more realistic and help to predict the future upto some level.

If anybody want to do research in this field, he/she need to know the idea of differential equations and mathematical software like MATLAB, MATHEMATICA etc. This kind of research is very realistic now a days and interesting.

Readers are requested to understand that Mathematical modelling is not capable of invent any drug or vaccine and it will not be able to remove any disease from society. It is basically a block representation of a real life phenomena which will help researcher from other discipline to do their job easily.

TEACHING MATHEMATICS – CONCEPT MOTIVATION VS ACTION LEARNING -THE CONTROVERSY!

The key of success is the ability to solve problems- this may be a perfect phrase for the most difficult and necessary subject Mathematics. In the present era of technology, a student need both conceptual and practical exposure of Mathematics. For this purpose the teachers (instructors) has a great role. The subject is compulsory up to class X, and after that, it is optional. However, for today’s world every technological, scientific, venture needs the idea of Mathematics a lot.

For this subject many concepts are needed and a student can easily handle this concept if we, the instructors are able to provide some real world phenomena to them based on that concept. This concept has to be given in the beginner level so that students will used to the situation and they can upgrade their thinking capability and cognitive mind. In college / university level, some open problems and classic results can be introduced to the students, which will motivate not only the student but also the teacher. Future teachers of Mathematics are students now. Hence, action learning or real life problems can be introduced to all levels of students from their higher secondary level. Probably this will grow some craze of Mathematics among them.

Some facts:
For the purely theory oriented subject Mathematics, Some of the following facts are very useful. There is a controversy about the teaching of Mathematics. Some teachers follow the conceptual approach with purely theory oriented teaching and after that the application. Some may try to teach theory based on applications. In any case, the basic idea is to trigger
the cognitive mind of the young students to generate a passion for the subject.

Curiosity– This is the main criteria. Regardless the age of the learner, if
curiosity is there, then learning a subject is very pleasant experience. Normally we have seen the child in primary level is more curious rather that an adult now a days.

Motivation– The curiosity will turn into motivation and this help to grow
people as a high quality professional. Teacher has a great role to motivate students. Especially for an engineering or non-math major student, only teacher can motivatehim/her. Initially the motivation will be technology or some real life facts. How Mathematics play a role for the real life problems, if we can able to make a student understand, that may be enough for motivation. Motivation is a bit more in adult learners rather than primary
level due to the lack of visualization of the future of the topic. For example, teacher can motivate a student for differential equation by introducing a real life problem “Vibration of a string” or “Population scenario”.

Deep Learning– For a motivated student, reward does not matter. Sometimes they work for the sake of their passion towards the subject. Students through deep learning try to get the complete knowledge for Mathematics. This will reduce the anxiety among them related to the topic. This can be done through asking questions, drawing diagrams, preparing charts etc.

Real life problems– The common question from student is that “What is the
application in real world?” This question is valid also. For some of the topics in Mathematics (Pure Mathematics), giving a practical example is a challenge. However, for remaining part of Mathematics, we can give some real life problems easily and justify the concept. For example, we can give a pry-predator situation from biological aspect and easily generate some simplified model using differential equation. In optimization we can give the example of work of a delivery boy in a day and how to optimize that properly which leads us to travelling salesman problems in LPP.

Software/ Practical– Now a days, many software are available in the market
to do our job easier. Using MATLAB, MATHEMATICA finding eigen value, characteristic equations, graph generations are very easy and students will be attached to the subject through this. Even in mobiles, we can have various apps of Mathematics, which will make the subject easier for student. Situation of Mathematics in India: Manjul Bhargava, mathematics professor at Princeton University and winner of the Fields Medal(2014), one of the highest honors in math, says India can’t hope to lead the fourth
industrial revolution, “If we don’t have strong mathematical talent coming up very soon”
(Source-https://economictimes.indiatimes.com/jobs/mathematics-is-on-decline-in-india-reasonexcessive-focus-on-engineering/articleshow/69938324.cms?from=mdr) .

“Without the math, the procedures may be just experimental, may work sometimes, sometimes not. Only mathematics will tell you under what circumstances it would work,”- Srinivasa Varadhan, mathematics professor at New York University. (Sourcehttps://economictimes.indiatimes.com/jobs/mathematics-is-on-decline-in-india-reason-excessivefocus-on-engineering/articleshow/69938324.cms?from=mdr)
Both these statements are regarding the two very popular topic of modern era, AI (Artificial Intelligence) and ML (Machine Learning) which is the future all over the world as well as in India. There are various problems I education of Mathematics in India. Specially the ideology of Pure and Applied Math. This divides the whole scenario of Mathematics in India though actually these two topics are not separated. They are very much interlinked and stand together. The study of higher mathematics will be incomplete without the help of research, which is missing in this part of the world. Here we consider teaching and research are two separate quantity.

Action Learning: For these problems, Action Learning may give a ray of hope. This is a problem solving approach, which emphasize on a particular problem and that will reflect on the result. The probable best method is to initially focus on particular topics of the attraction of the students.
Action learning is used traditionally in business management, social science, research based problems. However, this can be introduced to teach Mathematics also. Through this action learning, teacher may be able to find the creativity of the student, which was hidden. It will be the job of the teacher to find the hidden talent through this action learning.

The problems in Mathematics is that it’s language is very abstract and in the higher education it is much more complicated, abstract and it has been taught by create a distance with real world problems. Hence, people are unable to visualize mathematics in real life.

For example, 2? = 4 implies ? = 2, because we divide both side by 2 . Actually, this is a concept of field theory. According to field theory as 2 ≠ 0, so it’s inverse exist and we multiply both side by the inverse, which is 1/2.

When some non math major subjects are taught, the applications of Mathematics in their particular fields need to be discussed, which will attract the students towards it. For non major students, Mathematics need to be taught through action learning and then concepts will be developed based on the applications. So we have to follow “middle of the road” policy.

Hence, finally we can say that Action Learning and Conceptual Mathematics is not different and they will be always together. However, probably we need to fix the order for most of the cases. Without concept, there is no progress. Concepts can be gained through action learning and that will remain with us for a long time.

COVID-19: Can Mathematics help???

In recent months, two words are very much familiar with every citizen of the world, “COVID-19” and “Pandemic”. Across the globe, we are fighting with this invisible powerful virus without any medicine but LOCKDOWN.

Background: As per the report of WORLD HEALTH ORGANIZATION, on 7 January 2020 in the case of Wuhan in China, the first Novel Coronavirus was identified. It was named severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) by the International Committee on Taxonomy of Viruses on 11 February 2020.

World Scenario: At present in the world, huge number of cases are there with several deaths recovered cases. Until now there are large numbers of active cases including mild and serious cases, there are active cases and closed cases (including deaths and recovered) also.

Scenario in India: INDIA is also very much affected in COVID-19 which is increasing day by day. The following 6 diagrams may help us to understand the nature of growth of COVID-19 in India and its various angles.

Role of Mathematics in COVID-19: Now the question is how mathematics can help in this regard. Firstly, let us look into the following diagram,

CYCLE OF A MATHEMATICAL MODEL

At present, every disease can be converted to mathematics using Mathematical Modelling. A modelling is a process to convert a real life problem into a mathematical phenomenon using its’ logic. The cycle can be understood from the above diagram.

We need few parameters to form a mathematical model based on a real life situation. Regarding COVID-19, also people are working on various model but until now as the situation is not very much clear, so no such well-established model has come up.

Model parameters and its importance in COVID-19:

The following probable parameters can be considered for forming a model regarding this disease.

Susceptible (S) – who are affected by others
Exposed (E)- who are not showing any symptoms but able to infect others.
Infected(I)- who have fully developed symptoms and can easily infect others.
Recovered (R)- who recovered normally or by some other medicines.

As soon as some drugs will be available in the market, that also can be taken into consideration to make the model more realistic. This kind of model may be applicable in India where various transmission dynamics will be considered. We can visualize the interrelation among these parameters as follows:

What is the Goal of this type of model?

Here after successfully forming the model, we can study its stability analysis of the model both locally and globally. In addition, we try to find the Basic Reproductive Number – which describes the expected number of cases directly generated by one case of the population where all individuals are susceptible to infection. Based on this number we can generate conditions by which the model can be stable or asymptotically stable or unstable.

Also based on we can also predict the risk factor to the society due to this invisible virus effect. At present, there are no drugs for this virus. As soon as some drugs will be there, then drug parameter may be incorporated in the model to make the model more realistic. Based on the drug, Reproduction number will change and then the scenario will be changing.

Limitations of the model:

As there are various significant biological factors, which may not be included in the model as solving a non-linear mathematical model may be very much time consuming and complex. Therefore, for simplifying we may linearize some of the factors, which will deviate the model from its original shape. Hence, a model will always help us to understand this overall scenario but not the particular one.