P Kandaswamypgkswamy@yahoo.co.in Mathematics Prospects and Future

Many mathematical principles are based on ideals, and apply to an abstract, perfect world. This perfect world of mathematics is reflected in the imperfect physical world, such as in the approximate symmetry of a face divided by an axis along the nose. More symmetrical faces are generally regarded as more aesthetically pleasing. Mathematics might seem an ugly and irrelevant subject at high school, but it's ultimately the study of truth - and truth is beauty! You might be surprised to find that mathematics is in everything in nature from rabbits to seashells

Fibonacci's Rabbits

Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series.

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Can you see how the series is formed and how it continues?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:1/1 = 1, 2/1 = 2, 3/2 = 15, 5/3 = 1666..., 8/5 = 16, 13/8 = 1625, 21/13 = 161538... It is easier to see what is happening if we plot the ratios on a graph:The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1618034 , although we can find an even more accurate value.

We now proceed to construct the equations governing the interaction between the above four factors during a disease. A small growth V in the population of viruses due to multiplication in an interval of time t is proportional to V and (the coefficient of virus multiplication). The number of virus neutralized by the antibodies will be proportional to both the number of virus V and antibodies F and hence equal to FV where is the coefficient connected with the probability of neutralization of viruses upon encounter with antibodies. Thus we get V = V t - FV t

That is dV/dt = ( - F)V(1)Secondly we proceed to construct the equation to describe the growth of plasma cells. The immunocompetent B lymphocyte is stimulated by an antigen coupled with T cell receptor (TV complex) and initiates the cascade process of forming cells which synthesize the antibodies neutralizing antigen of this kind. Since in our model by antibodies we mean substrates capable of binding with viruses (including the T cells receptors), the number of lymphocytes stimulated in this way will be proportional to VF.

Then the relation describing the increment of plasma cells over a normal level C* which is constant in a normal organismThe first term in the RHS describe the generation cells: denotes the time during which a cascade of plasma cells is formed, denotes the coefficient allowing for: the probability of an encounter of antigen antibody, the stimulation of the cascade reaction and the number of newly generated cells. The second term represents the fall in population of the cells due to ageing and C is the coefficient equal to the inverse of the plasma cells of the life time.

We now proceed to calculate the balance of the number of antibodies reacting with antigen. Let dF denote the change in the number of antibodies during on interval of time t.

Then we have dF = Cdt - FVdt - f Fdt

The first term on the RHS represents the generation of antibodies by plasma cells and is the rate of production of antibodies by one plasma cell. While deriving equation (1) it was noted that the number of viruses eliminated during the time interval t due to their neutralization by antibodies was given by FVt. If the neutralization of one antigen requires antibodies then FVt is the number of antibodies neutralized.

Thus dF/dt = C - FV - f F(3)

The last term stand for the fall in population of the antibodies due to aging where f is the coefficient inversely proportional to the time of decay of an antibody.

Finally we shall construct an equation to describe the characteristic damage to vital organs. Let M be the characteristic of a normal organ and M1 be the corresponding characteristic of a normal part of the damaged organ. Let

m = 1- M'/M

designate the relative characteristic of damage to the target organ. For the intact organ m is zero and for the completely damaged organ it is one. During the disease the increased relative value of the damaged part of the organ is proportional to the number of antigens and decrease in this characteristic is due to the recuperative capacity of the organism and hence we have

that is, dm/dt = V - m m(4)

Where is a special constant for each particular disease and m is the organ-damage time constant.

Usually for equations with delay the initial conditions are given on an interval [t0-, t0]. But for the problem in hand until the moment of infection t = t0 there were no virus in the organism: V(t) 0 for t < t0; and therefore the initial conditions can be given at the point t = t0.

Thus we have V(to) = Vo , C(to) = Co (5) F(to) = Fo , m(to) = mo

as the initial conditions. Thus equations (1) to (4) along with initial data (5) constitute the simplest model of a disease.

Thus we havewith

2.2 Qualitative Study of the Simple Model:

For all t > 0 equations (1) to (4) has a unique solution satisfying the initial conditions (5).2. Non-negativity of the initial conditions imply the non-negativity of solutions.3. If the initial conditions are nonnegative and furthermore C0 C*, then C(t) C* for all t 0.4. The system possesses two stationary states specified by I.V1 = 0, C1 = C*, F1 = (C*)/(f) , m1=0 II. V2 = [c(f - C*)]/[(-c)], C2 = [f-c2C*]/[(- c)] F2 = / , m2 = /m V2

While the first always describes the state of a healthy organism (V1 = 0, m1= 0) the second describes the chronic form of the disease only of V2 > 0. That is only when > C and f > C*or < C and f < C*

1. The first stationary state is asymptotically stable when < F*

2. If Vo < [f ( F*-)] / [] ,

for < F*, V(t)0 as t

Two Limiting Cases. The organism produces no antibodies of a particular specificity to neutralize a particular kind of virus. Then we have F(t) = Fo = 0 for all t > 0 and = 0. In this case (1) assumes the form dV/dt = Vwhich implies V(t) = V0 et where V0 is the initial concentration of viruses. The dynamics of the damage of the organ is specified by dm/dt + m m = Vo etwhich for the initial condition m(0)=0 possesses the solution m = (Vo / +m )(et-e-mt) is (V,F,m) (V0 e-t,O,((V0)/())(et-1)

(m set to zero) (no restorative capacity to the organ)

Clearly such a solution corresponds to the course of the disease with leathel outcome since the are no factors compensating the growth antigens. This is certainly a limiting case and rare to encounter in reality but serve as a good approximation for situations developed in some old people whose immune system fail to have an expressed reactivity to antigens or in people with acquired or congenital immune defects. The other limiting case involves a dominant immune response when the level of antibodies present in the organism specific for the given antigen is sufficient for all the antigens that penetrate the organism without activating the antibody producing mechanism. In this case we have F always at a constant level F* and F* yielding dV/dt = (-F*)Vpossessing V(t) = V0 e( - F*)t as solution.

This implies that the antigen population in the organism will decrease exponentially. When is set zero we have V = V0 e- F* t as the second limiting solution. Thus, we have found two limit solutions corresponding to the lethal outcome and a high immunologic barrier. For given coefficients of the model and initial conditions, the entire family of diverse dynamics of the disease lies in the dotted area in the figure below

V0 e(t) V0 e(-F*t)ln V0ln VtDynamics of a disease, High immunologic barries and lethal outcome as limit solutions.

VV maxV0t1 t1 t

t1 t2 tFFmax /

F#

The acute form of a disease

t1 t2 t3 t4 t VmaxVV0

F t1 t2 t3 t4 t The chronic form of a disease

The time rate of change of mass is where k is a positive constant of proportionality and the ve sign implying that the mass is decreasing at the time. Equating (2) and (3)Physical ModelsConsider the melting of a snow ball. Let r be the radius of the snow ball. Let half of the snow ball melt in one hour. How long will it take for the remainder to melt? Conditions remain unchanged. If is the density of the snow ball, M(t) its mass, V(t) its volume and time t in hours.Then

We get Thus according to this model the radius of the snow ball decreases uniformly with time. Integrating (4) and using the initial condition, we getwhere is the time required for the snow ball to melt, which occurs when the radius is zero. We need to know the value of , to get . For which we have to use the condition that after one hour half the snow ball melted.

So that Also from (5) easily get Thus Finally it takes close to 4 more hours to melt away completely.

Institutions where Mathematics can be perusedIISER Pune, Bhopal, Mohali, Tiruvanandapuram, Kolkatta - 5 year BS-MS dual degree with major in biology, chemistry, mathematics and physicsChennai Mathematics Institute. Integrated MSc( Mathematics ) ISI Bangalore, Kolkatta, Delhi.Indian IInstitute of Science Bangalore (Four year BS Course Common for all science subjects)IITs Central Universities: Hyderabad, Pondichery, Kochin University of Science & TechnologyTATA Institute of Fundamental Research. Mumbai, Bangalore, Hyderabad,Institute of Mathematical Sciences (MATSCIENCE), Chennai, State University Departments: Ramanujan Institute, Chennai, Bharathiar University Coimbatore, Bangalore University Bangalore etc

Applications of Mathematics:

DRDO, ISRO, NAL, C-MMACS, CDOT, IAF, IITM, National Institute of Immunology, CSIR Labs etc.

Awards

Fields Medal, Abel Prize, Millennium Prize, Shanti Swarup Bhatnagar Prize for Science and Technology Young Scientist Award, National Academy Sciences Scopus Award, Nobel Prize for Economics etc.

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