Mathematical Interpretation of Covid-19 using Population Model in the Indian Scenario

Abstract

The Navel corona virus has created an atmosphere of fear all over the world; fear is also inevitable because it is rapidly spreading from human to human. The virus, born in China's Wuhan city, has engulfed almost the entire world. India has also felt its virility, scientists from the country and abroad is engaged in the discovery of the vaccine against the virus, but no breakthrough has yet been achieved. India ranks second after China in terms of population. States, Countries and abroad are working together to save their country from community transition, medical personnel as well as police, army and voluntary institutions are being trained.

Various states in the country have introduced a variety of measures to prevent community proliferation, so the mathematical avalanches presented in this paper can be helpful to prevent corona infection as well as to raise awareness. This paper describes the impact of viruses on population dynamics. Also, this paper explains the interpretation of the population model, the infection of covid-19 based on the prey-predator model.

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Keywords: Ecology, population model, prey-predator, least square method. S’I’R’ model.

Introduction

The population model talks about two species of organisms, the predator eats his prey and prey depends on other types of foods. By studying population ecology, we can find answers to a few questions, for example, what are the factors that affect the population of organisms? What is going to change the population over time and how much is going to change? Etc. The history of population ecology is considered to be quite profound, and its best development can be seen clearly in some subjects such as population growth, exchange and mobility.

Studying “human population” has been one of the most important topics for population ecologists, a very important issue in this century. Thus, the population ecology explains the distribution of different species, as well as the way in which their distribution will change in the future.

Some of the questions that have been popular in the species are why they tend to live in particular areas, as well as how to restrict their growing boundaries. The answers are similar to the answers to questions about climate change.

Under population ecology, changes in the population of plants, animals and other types of organisms over time and changes in their habitat in the environment, etc., are considered paramount for study. Population means the organization of organisms of the same species at a particular place, at a particular time .Population of any organism is characterized as below:

• Population Size (N(t)): This sign shows the number of persons contained in the population.
• Denseness of Population (a(t)): Number of individual residing at the particular area
• Population growth (P): It indicates the change in the holding of the population along with time t.

Under ecology, there is a statement that the behavior between predators and prey is based on population density and the relationship between them is discussed in two respects. The first type of approach suggests that the predator has a strong desire to invade and kill his prey [15], stressing that the victim should be treated as under the predator. Considering on behalf of the other side, it is believed that the prey senses an attack on him and brings a slight change in his behavior to protect himself from the predator by dodging the predator to protect his life from it. So, it can be said that the change in the behavior of the prey to save his life falls within his rights [8].

If it comes to the hunting-hunter model, the predatory population kills its prey, the effect of which lasts for a long time, a new change is being observed at the present time, 'the indirect impact on the predatory population, also known as fear effects.' This indirect effect is quite impressive with direct influence ([12], [14], [16], [19]). There is always a fear of poachers being hunted in their pastures in the prey, so they abandon their pasture and go to another place to stay safe with their families. Such a tendency of prey reduces their highest rate of being hunted ([12], [17]). It is a battle of survival for prey, their lives soon come to an end when they are hunted, so it can be said that their fight does not last for long or, in other words, it is not possible to survive for the long term. Fear of being hunted affects their reproductive tendency, this leads to inconsistency in their reproductive tendency. ([2-3], [6], [14], [19]) When studying [1], an example is that three spine stickback men (gaserosius aculatus) (a small telost fish found in the northern hemisphere) realise the risk of poaching and plan to breed when conditions adapt to them [10].Theoretical research and microscopic inspection of that area have been very helpful in understanding the ecological population comprehensively.

To prevent the infection of epidemics in the ecosystem and to understand the changes in various items due to their wideness, both mathematicians and ecologists are jointly engaged in continuous research, at present, Covid - 19 is causing havoc throughout the world. It started in December 2019 from Wuhan, China, which has now engulfed almost the world; Scientists in the world are engaged in vaccine research to prevent this. Mathematicians and epidemiologists are working together to prevent the epidemic. The tendency in species is to live in their habitats and to eat or to say that they depend on other types of species for their livelihood, this trend is a major reason for the increase of infection, which is why, in this era of epidemics like Covid-19, human species are avoiding taking meat as food. This type of mobility in species is analyzed by using proper mathematical models with the help of differential equations. Mathematical Epidemic Model [4] based on the principles given by Kermack and McKendrick, several number of researchers are working on Covid-19 to rein in the infection of the epidemic at the earliest. It is very important that there is always a synergy between the organisms and their environment, as well as the biological activities that are in their development, their population, their living conditions, etc., are related to the areas of ecology and biology, so the mathematical model has been applied as a powerful tool to study the biological processes and changes in the dynamic activities of species ([5], [9]). From a mathematical point of view, it is evident that they can be classified as determinant models in different classes, including models dependent on simple differential equations [11].

Mathematical interpretation

Let's talk about the simple homogenous population model:

Consider N as the number of persons contained in the population at the time t i.e N is a time-dependent function, and the rate of change with respect to time t in the population size N is given by _dN/dt.

Also, the per capita rate of change is given by _{1/N (dN/dt)}

Therefore we have

_{1/N (dN/dt) = η-φ (1.1)}

Here η represents the rate of nativity and the mortality rate in the population is considered as φ. The difference η-φ is known as the intrinsic rate of growth which is equal to ρ i.e.

_{ρ = η-φ (1.2)}

Thus equation (1.1) can be written as

_{1/N (dN/dt) = ρ (1.3)}

With the initial condition i.e. at t=0

_{N(0) = N0 (1.4)}

Solution of the ordinary linear differential equation (1.1) can be obtained as:

_{N(t) = N0 e^ρt (1.5)}

It is obvious that in the population of any species, the number of individuals contained in the population N(t) at the time t has the exponentially increment for positive intrinsic growth.

The following data shows the number of infections per day from corona virus in our country , the data has been taken from 25th April 2020 to 25th May 2020, it is demonstrated on the graph, the graph clearly shows that the number of infected is continuously increasing exponentially.

Clearly the table helps to study the nature of infection. By analyzing the table and the tabulated data, the infection rate can be obtained, which helps to make strategies to overcome the infections.

Using the least square method or group averages method, the number of infected people at t=0 and intrinsic growth rate ρ can be calculated. In the least square method, equation (1.1) will be converted into linear form. Then, by taking the logarithm on both sides, we can get:

logN = ρt + logN₀

Or

Y = ρt + A (1.6)

Where Y = logN and A = logN₀

Then the normal equations for constants A and ρ will be taken, and two simultaneous equations for A and ρ are solved. By taking the antilog of A, N₀ is obtained. Putting ρ and A in equation (1.1), the best-fit line can be obtained.

In this paper, the next mathematical model is based on the analysis of the effect caused by a virus in any species. The effect caused by a virus on a single species is analyzed by a mathematical model, where we assume the following:

a(t) represents the density of the population at any time t,

b(t) indicates the density of viruses contained in the population at time t,

c(t) is the density of viruses contained in the environment at time t.

The model specified here in the paper has the characteristics given as below:

• Due to the virus infection in human bodies, population dynamics are affected.
• The population growth rate is calculated by assuming the birth rate of the virus p-fa(t) and the death rate of virus q+αb(t).

Here p, , q, and are nonnegative constants.

The model can be described as below:

a' = a(p - qαb - fa) (1.7)

b' = kc - b(r + m + p - fa) (1.8)

c' = u(t) - hc - ack + ab(r + p + αb) (1.9)

With the conditions taken initially at t=0, a, b, and c are all non-negative constants.

represents the virus’s rate of reduction in the community during their production.

denotes the virus’s rate of reduction in the community during the process of metabolization.

h represents the virus’s rate of reduction in the atmosphere.

u(t) is considered as the extrinsic virus input rate, which is continuous and bounded nonnegative function of .

k is the virus’s rate of reduction in the atmosphere due to their presence in the population [13].

Population of India is N and it can be divided up into three parts as:

population I' is the number of confirmed cases

S' number of susceptible cases

R' number of recovery cases

Note that the recovery category consists of those who have died or are immune to the disease naturally.

Therefore it is obvious that

N = I' + S' + R' (1.10)

Where N, I', S', R' all depend on the time t.

It is seen in the 3 compartment model which is having the motive that the I' compartment should be minimized under the existing conditions:

The probable situation when it is considered that the virus has not evolved:

The virus transmission from one person to several in the diagram susceptible person is recognized by infected if the symptoms are positive, then the infected person gets to die or recover. If the immune power is strong, the person can avoid the infection or if they are kept in isolation, the effect of COVID-19 can be reduced. Hence, social distancing is very essential for the effect of the virus to work, so the lockdown position is being adopted in the country.

When it is considered that the virus has evolved:

The virus is constantly expanding. If the symptoms appear in the suspected person after the test appears positive, it is considered to be confirmed, i.e., the person is now infected, quarantined, and treated. Since the infection in the atmosphere and in people is increasing, a person can become infected with the virus even after being healthy. Also, if somebody has been infected and has died, their bodies can also increase the infection, so once infected, they can be re-infected, i.e., a cyclic process.

Obviously, if we talk about the confirmed cases, the per capita rate of increment of the confirmed cases has a direct proportionality law with the number of susceptible confirm cases zone. Thus, the first compartment has a total intake (σS')I' where σ is the transmission rate, meaning the probability of a person catching the virus from an infected person.

The differential equation model [S’I’R’-Model] [1] is as follows:

dI'/dt = σS'I' - γI' = I'(σS' - γ) (1.11)

dS'/dt = -σS'I' (1.12)

dR'/dt = γI' (1.13)

Here, γ is the recovery rate from the infection. γ is the rate at which the infected become healthy or achieve death, so it can be said that they cannot be infected again.

Conclusion

On studying the data and graphs displayed in this paper, it is evident that covid-19 is increasing wildly in cases of infection, and it is also evident that physical immunity, non-exit of the house, adherence to social distancing and enhancing the physical immune system are factors that can reduce the infection of taxes. That is, all these things have to be followed in our lives to level the curve.

The situation in the country has been locked down, but the most important services of life are constantly reaching the masses. The vaccine will have to be discovered as soon as possible, with financial funding, infrastructure and adequate facilities. We have to maintain our faith in science and scientists. Being optimistic we have to defeat covid-19 in battle so that our lives can be back on track.

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