DYNAMICS OF ENVIRONMENTAL POLLUTIONTHROUGH VEHICLES POLLUTIONTHROUGH VEHICLES.”

: In today’s time as air pollution is increasing day by day the use of non-polluted has to be increased in almost all nooks and corner of the countries. In this paper a mathematical model is developed to analyse environmental pollution through polluted and non-polluted vehicles. Basic reproduction number has been calculated which will the decide the behavior of the system. Stability analysis has been carried out at equilibrium points. Numerical simulation is done to analyse the result for various compartments .


Introduction
Pollution is one of the biggest problem the world faces today [3]. It occurs when harmful materials enter into air, water or soil. Various kinds of pollution prevail in the environment such as air pollution, water pollution, noise pollution, soil pollution, radioactive pollution, etc. Air pollution makes it difficult for plants, animals and human to survive as the air becomes dirty due to emission of harm full gases, dust, smoke in the atmosphere. Excessive air pollution is due to the vehicles.
Vehicles can be classified in to different categories as road vehicles, highway vehicles, own vehicles. Heavy duty vehicles and passenger vehicles are major contributors for creating pollution as they emit nitrogen oxides, carbon monoxide, hydrocarbon, etc [3] [4]. These vehicles pollute many cities in India like Delhi, Kolkata, Mumbai, Chennai [6] and many more but WHO said that Delhi is the most polluted city [7]. Even more, DNA observed there were only few days in the year 2017 which had good air quality. Supreme court ordered Delhi government to reduce air pollution by [2], i. Pollution check by transport department. ii. Public awareness. iii. Catalytic converters. iv.
Mass rapid transport vehicles. v. Pashing out of old commercial vehicles. vi.
Tightening of mass ambition standard for new vehicles. [39] In this paper, we are targeting own vehicles which are classified into polluted and non-polluted. Non-polluted vehicles are those which are purchased recently or has work for 3-4 years. After that usage of some vehicles has been defined as polluted vehicles because the efficiency of its engine and other motor capacity. Conventional motor vehicles have high impact on environment pollution. Vehicles which are using gasoline, diesel or CNG are main source of causing air pollution [5].
In this paper, a mathematical model is formulated for the analysis of environmental pollution through vehicles. The notations, mathematical model and basic reproduction number are given in section 2. Section 3 discusses local and global stability which are evaluated at equilibrium points of the model. Sensitivity analysis and numerical simulation are discussed in section 4 and 5 to support the analytical result.

Mathematical Model
Here, we formulate a mathematical model for vehicles which are polluted or non-polluted. The model is developed with the notations defined in Table 1.   Now, from figure 1 we have the following system of differential equation describing movement of vehicles from one compartment to the other Adding all the above equations, we have This gives, Therefore, the feasible region for Thus, the equilibrium point 0 , 0, 0 Next, the basic reproduction number 0 R can be calculated using the next generation matrix [1]. Hence, the basic reproduction number 0 R is the spectral radius of matrix On solving system (1), we get an endemic equilibrium point

Equilibrium
In this section, the local and global stability of environmental pollution through vehicles is discussed.

Global Stability
In this section, we will discuss the global stability of the proposed model.   . al., Vol.5 (Iss.4)

Sensitivity Analysis
In this section, the sensitivity analysis for all parameters is deliberated in Table 2.
The normalized sensitivity index of the parameters is calculated by using the following formula: where  denotes the model parameter.

Numerical Simulation
In this section, we will study the numerical results of the model.     Now, from figure 4 it is observed that if one will increase the rate of vehicles used for less than 5 years creating pollution ( ) 2  , the number of non-polluted vehicles decreases which is not advisable.   on V P Figure 6 specifies that if we increase the value from 0.06% to 0.6% initially. There is no increase or decrease in the number of polluted vehicles but with the passage of time they start to decrease. So, on increasing this rate the number of polluted vehicles decreases which is beneficial for the society.

Conclusions & Recommendations
A nonlinear mathematical model for environmental pollution through vehicles is developed. Local and global stability has been proved at equilibrium point 0 E . We conclude that increase in polluted vehicles cause more pollution which is harmful for environment and society. Instead of using polluted vehicles one should start using non-polluted vehicles or electric vehicles.
For that, some remedial steps can be taken such as: 1) Proper maintenance of car and truck emission control system. 2) Alternative fuels such as CNG and LPG.
Using parametric values given in the table 1. We computed that 35% of non-polluted vehicles exist in the society.The aim of the preparing model is to reduce environmental pollution through vehicles.