MATHEMATICAL APPROACH ON HOUSEHOLD WASTE CAUSING ENVIRONMENTAL POLLUTANTS DUE TO LANDFILL AND TREATMENTS APPROACH ON HOUSEHOLD WASTE CAUSING ENVIRONMENTAL POLLUTANTS DUE

: Household solid waste is the solid waste mixture of garbage and rubbish which comes during the use of various products in daily life. It also called as domestic waste or residential waste. It may fall into two categories either hazardous or non-hazardous which are stored and forsaken directly to the landfills. This is how household solid waste plays vital role in spreading environmental pollutants. For reduction of the pollution, treatment plant is constructed for hazardous solid waste and compost plant is organized for non-hazardous solid waste. In this paper, we have developed a system of non-linear differential equations to analyse the household solid waste storage. In order of preventive measures, five various controls are given to the different compartments. The basic reproduction number and the stability are derived to check the endurance of the model. The numerical simulation is also done using validated data.


Introduction 1.
A solid waste is material which can be rejected if: Abandoned, Inherently waste-like, A discarded military munition or recycled in certain ways (https://www.epa.gov/hw/criteriadefinition-solid-waste-and-solid-and-hazardous-waste-exclusions). Solid waste can be classified depending on their sources: (1) Household waste (2) Industrial waste (3) Biomedical waste. In the proposed paper, the prevention to environmental pollutants from household solid waste is found. Household solid waste is of bottles, cans, clothing, compost, disposable food packaging, food scraps, newspaper, magazines and yard trimmings which are released from private homes or apartments. In other words, household solid waste is refuse engendered only from households. Even though it does not contain huge number of germs but it entices flies, mosquitoes and rats, and allows them to breed which causes risk to human health. If household solid waste is kept even only for few hours, flies, mosquitoes and rats can be attracted. This may encourage the [267] spread of diarrhoeal diseases as well dengue fever, yellow fever, bancroftian filariasis and bubonic plague. From all these wastes, certain types of household solid wastes are hazardous because they contain toxic elements. Household solid waste which can be categorised as hazardous solid waste covers batteries, shoe polish, paint tins, old medicines and medicine bottles. Rest of all waste are not consisting of toxic like paper, wood, fruits, vegetable peels, food and sludge considered as non-hazardous solid waste. Understanding how this waste is created is a useful factor of defining whether the waste is considered as hazardous or not. Everything leaves some waste after usage but it is required to take some special care when it is deposited. This is how proper waste management is an important fragment of the society. For large amount of waste, dumping in landfill is very common method. Some waste created from treatment plant and compost plant is also dumped in landfill. Treatment plant is the plant where hazardous solid waste is given treatment and compost plant is where non-hazardous solid waste is provided composting. These plants, landfill and storage of household solid waste make environment polluted. Hence, sprouting waste increase environmental pollutants. Therefore, waste disposal is becoming major problem of the society.
In 1999, Chongwoo and Iain examined an economic analysis of household waste management. Asa et al. studied life cycle assessment of energy from solid wastepart 2: landfilling compared to other treatment methods in 2005. Some researchers settled model on household waste. In 2005, forecasting municipal solid waste generation in a fast-growing urban region with system dynamics modeling is formed by Brian and Chang. Udaya and Achyuth in 2015 formulated mathematical modeling of household wastewater treatment by duckweed batch reactor. In 2008, Sara et al. developed mathematical modeling to predict residential solid waste generation. To revive nature some researchers have constructed mathematical models related to nature. In 2017, Nita et al. have established optimum control for spread of pollutants through forest resources and also expressed optimal control on depletion of green belt due to industries.
In this paper, a mathematical model is formulated in section 2. In section 3, local and global stability is defined. Optimal control is derived in section 4. Using given data, numerical simulation is considered in section 5.

Mathematical Modeling 2.
We live in the society where the consumer market has grown quickly over the last few years. Everyone in the society uses the products being placed in cans, aluminum foils, plastic and other non-biodegradable items though they cause multitudinous damage to the environment. Also, some of these are hazardous wastes which are given treatment for betterment but it also causes harm to the environment. And non-hazardous items are provided the treatment as composting also cause the same problem. control rate on non-hazardous solid waste to provide composting, 4 u as the control rate for environment pollutants discharging from treatment plant and 5 u as the control rate for environment pollutants discharging from compost plant to diminish environmental pollutants. Using above parameters and some necessary assumptions, the mathematical model is constructed. The transmission diagram of household solid waste model is as figure 1.  The system of non-linear differential equation of transmission of household solid waste is given below: Adding above differential equations of system (1), we get Therefore, the feasible region of the model is   8 , , , , , , , : Next, we will compute the basic reproduction number noted as 0 R using the next generation matrix method.
Let us take   ' , , , , , , , ' where   fX is the rate of appearance of new individual in component and   vX is the rate of transfer of household solid waste. They are given by Here, above matrix V is non-singular matrix. Therefore, the expression of basic reproduction number 0 R is as following In the next section, equilibria of the household solid waste model will be conversed.

Equilibrium
In the current section, the equilibrium of local stability and global stability is established for the household solid waste model.

Local Stability
Here, we have proposed the local stability of the household solid waste model at equilibrium point * E . Now, the Jacobian matrix J at the equilibrium point * E is given by The characteristics polynomial of the Jacobian matrix J at * E is of degree 8 with coefficients   11  22  33  44  55  66  77 , , , , , ,

A A A A A A A and 88
A (file can be sent on request). Here, all the coefficients are positive and satisfy the condition of Routh-Hurwitz criterion (Routh E.J. 1877).

Theorem 1:
The unique positive equilibrium point * E is locally asymptotically stable with the condition that 88 0 A  if and only if

Global Stability
Here, we have studied the global stability of the household solid waste model at equilibrium point * E .

Theorem 2:
The unique positive equilibrium point * E is globally asymptotically stable.

Optimal Control
The objective of the household solid waste model is to minimize the environmental pollutants by maximizing treatments. To achieve the objective the controls are considered. For the mathematical model the objective function of household solid waste in the system along with the optimal controls is given as

J u A H A S A H A N A T A C A L
where  signifies the set of all compartmental variables, 1  , , , , w w w w w are weight constants for control variables 1 2 3 4 5 , , , , u u u u u respectively. Since, 1 2 3 4 5 , , , , w w w w w are constants for control rate of solid waste dumping, control rate on hazardous solid waste to give treatment, control rate on non-hazardous solid waste to provide composting, control rate for environment pollutants discharging from treatment plant and control rate for environment pollutants discharging from compost plant, respectively. 14 , uuand 5 u are the control variables to minimize the environmental pollutants comes from storage, treatment plant and compost plant. 2 u and 3 u are the control variables for maximizing the treatment to hazardous or non-hazardous solid waste from treatment plant or compost plant.  , i The partial derivative of the Langragian function with respect to each variable of the compartment gives the adjoint equation variable        1  2  3  4  5  6  7  8 , , , , , , , i A          corresponding to the system which is as follows:  3  3  3 1  3 2  3 2  3  5 1  5 2  7 2   3  3  5  1  3  7  2  3  5  2  3   2     2 The necessary condition for Langragian function L to be optimal for controls is Solving the above equations (13) -(17), we found the values of 1

Numerical Simulation
In this section, we detected the numerical simulation using the parametric values mentioned in Table 1.  Next four figures show the effect of control onstorage, treatment plant, compost plant, landfill the compartments.
From figure 2, it is observed that treatment plant causes less pollution in comparison to other sources. So, one can apply less control on it. Therefore, after applying control, treatment plant rises with smaller rate which can be noticed from figure 3.      To control the environment pollutants, we have applied some controls. The effect of these controls on environment pollutants can be seen from figure 6. Initially environment pollutants are increasing and then after control is given they start decreasing. Here, environment pollutants are decreased by 84.15 to 76.76 unit in 10 years which indicates that if controls are exist then we can revive environment by 8.76%.
[280]  Only 5 ucontrol rate for environment pollutants discharging from plantis applied. i.e.  As controls are good for saving the environment, figure 8 suggests that the highest control is required for 1 u on storage by 51% in 1.8 years and then can be decreased. For 2 u and 3 u corresponding 14% and 10% controls will be required in the beginning and can be decreased. Similarly, 23% and 46% control will be needed for 4 u and 5 u . For the betterment of the environment we should apply appropriate controls rates as mentioned above.

6.
In this paper, a mathematical model of household solid waste is formulated with the aim of preventing environmental pollutants. A mathematical approach on household causing environmental pollutants due to landfill and treatments is to study how much landfill and treatments pollute the environment. We all know that proper waste disposal is useful to save the nature. We should dump our household waste in appropriate container. Also, we can shop for environmentally friendly natural product or else we can reduce our purchase of products that consists of non-reusable ingredients. We can consider the alternative methods to use products for some common household needs. Also, we can search online for simple recipes that can use to create by own.
Using the parametric values given in the table 1, the basic reproduction number is intended as 0.2734 which indicates that the controls are beneficial to save the environment. This advocates that proper planning of waste disposal is useful to make environment and human healthy.