Modeling the effects of the contaminated environments on COVID-19 transmission in India

COVID-19 is an infectious disease caused by the SARS-CoV-2 virus that caused an outbreak of typical pneumonia first in Wuhan and then globally. Although researchers focus on the human-to-human transmission of this virus but not much research is done on the dynamics of the virus in the environment and the role humans play by releasing the virus into the environment. In this paper, a novel nonlinear mathematical model of the COVID-19 epidemic is proposed and analyzed under the effects of the environmental virus on the transmission patterns. The model consists of seven population compartments with the inclusion of contaminated environments means there is a chance to get infected by the virus in the environment. We also calculated the threshold quantity R0 to know the disease status and provide conditions that guarantee the local and global asymptotic stability of the equilibria using Volterra-type Lyapunov functions, LaSalle’s invariance principle, and the Routh–Hurwitz criterion. Furthermore, the sensitivity analysis is performed for the proposed model that determines the relative importance of the disease transmission parameters. Numerical experiments are performed to illustrate the effectiveness of the obtained theoretical results.


Introduction
COVID-19 disease is caused by SARS-COV-2 that represents a causative agent of a potentially fatal disease of great global public health concern. It is a disease that became a pandemic in a short period due to its accessible transmission routes as it transmits via respiratory droplets exhaled during sneezing, talking, or coughing. This deadly outbreak started in India on March 2, 2020, and is still uncontrolled. There are almost 30,585,229 confirmed cases and 402,758 deaths reported in India as of July 01, 2021. The virus belongs to a family of viruses that can cause illnesses such as the common cold, severe acute respiratory syndrome (SARS), Middle East respiratory syndrome (MERS), and even if left untreated can cause lung damage [1,2]. Some people may have only a few signs, and some people may have no symptoms at all. Fever, cough, and tiredness are common symptoms of infected people. However, loss of taste or even smell can also be found in such patients. Although the novel coronavirus that causes COVID-19 disease first appeared in mainland China's Wuhan city in late 2019 before it rapidly spread globally, still the biological origin of the virus is not known. The symptoms of coronavirus disease from the people who got infected may appear 2-14 days after exposure. The time the Covid-19 pandemic may be partially suppressed with temperature and humidity increases in India. Kumar and Kumar [18] proposed a correlation study between meteorological parameters and the COVID-19 pandemic in Mumbai, India. In their study, they found that the relative humidity and pressure parameters had the most influencing effect out of all other significant parameters (obtained from Spearman's method) on the active number of COVID-19 cases. Raza et al. [19], in their paper, investigated the impact of meteorological indicators (temperature, rainfall, and humidity) on total COVID-19 cases in Pakistan, its provinces, and administrative units from March 10, 2020, to August 25, 2020. The correlation analysis of the study showed that COVID-19 cases and temperature are positively correlated. Zu et al. [20] constructed a compartmental model a data-and model-driven approach for the study of transmission patterns of COVID-19 in the mainland of China. In their work, they have studied the efficacy of different control strategies and estimated the model parameters to the real reported data collected from January 10-February 17, 2020 from the National Health Commission of PR China. Rafiq et al. [21] developed a mathematical model to describe the spreading of epidemic disease in a human population with the emphasis on the study of the propagation of the coronavirus disease . They investigated the model analytically as well as numerically. Rohith and Devika [22] modeled COVID-19 dynamics using a susceptible-exposed-infectious-removed model with a nonlinear incidence rate. In their study, they performed the bifurcation analysis and studied the effect of varying reproduction numbers on the COVID-19 transmission. Shen et al. [23] developed a dynamic compartmental model of COVID-19 transmission in New York City to assess the effect of the executive order on face masks used on infections and deaths due to COVID-19 in the city. In their study, they provided the importance of implementing face mask policies in local areas as early as possible to control the spread of COVID-19 and reduce mortality. Bherwani et al. [24] explored the dependence of COVID-19 on environmental factors and spread prediction in India. In their work, the impact of environmental factors like temperature and relative humidity (RH) using statistical methods, including Response Surface Methodology (RSM) and Pearson's correlation, is also studied on numbers of COVID-19 cases per day. Although, there is a good number of studies proposed by many researchers so far [2,25,26], there is still a need to develop more accurate and feasible models for the prevention and control of COVID-19 disease.
In this work, we develop and analyze a novel non-linear epidemic model for the propagation of COVID-19 incorporating the effects of contaminated environments on transmission dynamics in India as the virus can survive for an extended period outside the host in suitable conditions. Hence contaminated environments may play an important role in COVID-19 infection. The model consists of seven population compartments represented by a set of non-linear coupled ordinary differential equations in which various real parameters are included with the inclusion of virus in the environment as a new compartment which is the density of pathogen of the contaminated environments including door handles, towels, handkerchiefs, toys, utensils, bed and toilet seat, car and bus seats, bathroom washbasin tap lever, bathroom ceilingexhaust louvre, etc. The use of such a model is based on the current understanding of the mechanisms of propagation of COVID-19. In particular, the consideration of the subpopulations of susceptible, exposed, asymptomatic, symptomatic, confirmed, hospitalized, and recovered individuals are strongly supported by both the effects of COVID-19 in human health and the measures adopted by the governments of all countries affected [21,27]. Various assumptions have been imposed on the problem considered in this work and obtained a system of ordinary differential equations from those hypotheses. We provide local as well as global stability results of the equilibrium points of the system and the nature of the local stability of those equilibria will be thoroughly investigated with respect to the basic reproduction number.
The rest of the paper is designed as after the introduction in Section "Introduction", Section "Model formulation", gives the development of the proposed COVID-19 model and its well-posedness. In Section "Analysis of the model", we discuss the mathematical analysis of the proposed model along with the equilibrium points and the stability of equilibrium points. Section "Fitting of the model to the real statistical data for India", gives the model fitting to the real statistical data for India. In Section "Sensitivity analysis", sensitivity analysis of the parameters involved in threshold parameter through PRCC to identify the key factors and the relative importance of the model parameters is discussed. Section "Numerical results and discussion" gives the numerical simulations of the proposed model to validate the analytical studies. The behavior of the obtained solutions is also discussed in this section. Finally, Section "Concluding remarks" concludes all the major findings of the present research study.

Model formulation
There are many classes of mathematical models used within epidemiology. Deterministic compartmental models divide the population into groups defined, at a minimum, by the possible disease states that one could be in over time. They are the foundation of mathematical epidemiology and provide a straightforward introduction to how models are built. Here, to develop a mathematical model for the transmission dynamics of the COVID-19 epidemic, we proposed a novel deterministic compartmental model contains the following seven classes of the population, namely susceptible class denoted by ( ); Exposed class denoted by ( ); Asymptomatic class denoted by ( ); Symptomatic infected class denoted by ( ); Confirmed class denoted by ( ); Hospitalized and treated class denoted by ( ); Recovered class denoted by ( ) and Virus in the environment denoted by ( ) at any time . Hence, the total population ( ) is given by ( ) = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ). To better understand the dynamic process, the population is mixing and interacting homogeneously means no lockdown, no quarantine, and no restrictions. Also, infection is spread due to the interaction of susceptible individuals with asymptomatic, symptomatic, and virus in the environment. Therefore, the proposed model describes the dynamics of COVID-19 transmission among different population classes can be represented through the following ordinary differential equations: (1) subject to non-negative initial conditions The recruitment rate for the susceptible population is denoted by . The susceptible individuals become infected, due to effective contact with the exposed class at a rate ; asymptomatic class at a rate ; symptomatic class at a rate and with the virus in the environment at a rate . is the natural death rate; is the rate at which exposed individuals become infectious; is the rate at which exposed individuals become asymptomatic by a proportion ; 1 is the recovery rate of asymptomatic individuals due to natural immunity; is the COVID-19 induced death rate; is the rate at which infected individuals P.A. Naik et al.  Virus released rate via the exposed individuals is the rate at which confirmed join the hospitalized class; 2 is the recovery rate of confirmed individuals; 3 is the recovery rate by treatment; 1 , 2 , and 3 are the virus released rates via the asymptomatic, exposed and symptomatic class respectively and is the virus clearance rate. The transfer relationships between the eight compartments is shown in Fig.1 (see Fig. 1).

Analysis of the model
The stability analysis of an epidemic model determines its behavior in disease dynamics. In this section, the positivity and boundedness of the solution for the proposed model (1) are given. After that, the existence conditions and the stability results for the equilibria are provided.

Positivity and boundedness
From the system (1), we get the total differential equation as:  Proof. In order to explain the non-negative region  8 + , is positively invariant region for model (1), we proceed same as in the works [28-34] and get Under the same ideas followed by [31][32][33][34] guarantees that the state variables remain positive during the entire scope of the study i.e., the domain  8 + is a positively invariant set. □ In the next theorem, we will show the boundedness of the solution to the proposed model (1). Naik et al. Proof. To prove this, we have from first equation of (2),

Theorem 2. The model (1) has a bounded solution with non-negative set of initial values
Now, let us take to be a solution which is unique to the initial value problem which on simplification yields Therefore, it follows from the comparison principle [35], that ≤ . Then, we have from the second equation of (2), Now, let to be a solution which is unique to the initial value problem which on simplification yields Therefore, it follows from the comparison principle [35], that Therefore, from (6) and (9) [35], the solution exist and is defined for all ≥ 0. Furthermore, for → +∞, we get This shows that the total population ( ), i.e., the subpopulations ( ), ( ), ( ), ( ), ( ), ( ), ( ) and ( ), are bounded in making the model both mathematically and epidemiologically well posed. This proves the boundedness of the solution of system (1). □ Therefore, the biologically feasible region for the system (1) is

Equilibria and their stability
For the equilibrium points, we set the right-hand side of system (1) equal to zero first to get disease free equilibrium point as 0 = ( , 0, 0, 0, 0, 0, 0, 0) at this point the system is free from the disease. Now for the endemic equilibrium point, we solve the system with an assumption that ( , the point at which virus persists in the population. The solution of system (10) yields the endemic equilibrium point 1 = ( * , * , * , * , * , * , * , * ), where * = ( + ) Here  0 denotes the basic reproduction number which is given in the next section.

Basic reproduction number
Let ( ) be the input rate of newly infected individuals and ( ) be the rate of transfer of individuals [36][37][38][39], then among the infected classes ( , , , , , ), we have After computing for the eigenvalues of the matrix = FV −1 , we have the expression for  0 as given in Box I. Here,  0 indicates the average number of secondary infections generated by a single infected individual introduced into a completely susceptible population directly during their life cycle.  0 indicates the average number of secondary infections generated by the virus that is released into the environment during their life cycle.
Using the possible values of the reproductive number, we will devote our next efforts to establish conditions that guarantee the local asymptotic stability of the equilibrium points obtained above. Beforehand, notice that the general Jacobian matrix associated to system (1) is given by Proof. An easy substitution of the point 0 , in the general Jacobian matrix readily yields the matrix of the form As per Routh-Hurwitz criterion, for  0 < 1 the disease-free equilibrium 0 of the proposed model (1) is locally asymptotically stable if all the eigenvalues , = 1, 2, … , 8 of the matrix ( 0 ) are negative numbers or have negative real parts. We can evaluate these eigenvalues from the following characteristic polynomial wherêis an identity matrix of order eight and is the eigenvalue. Therefore, we get a characteristic polynomial of the form ( 2 + ( 4 + 5 ) + ( 4 5 ) )( 2 + ( 6 + 7 ) + ( 6 7 ) ) .

Proof.
In order to obtain the global stability of the point 0 , we consider the Volterra-type Lyapunov functional approach to define a function £ 1 ( ) ∶ ( ) → , given by After simplification using the disease-free steady state condition of model (1), we have from Eq. (14) as Therefore, It can be noticed that if  0 < 1, then the right-hand side of Eq. (15) is non-positive and it is equal to zero if = 0 , = = = = = = = 0. Therefore, the maximum invariant set for { (  , , , , , , , is the singleton set 0 . This means that the only trajectory of the system on which £ 1 = 0 is 0 . According to the LaSalle's invariance principle [40][41][42][43], we know that all solutions in 0 converge to 0 . Therefore, the disease-free steady state of model (1) is globally asymptotically stable when  0 < 1. This completes the proof of Theorem 5. □ This further implies on substitutions from Eq. (1) ( 1 + 2 + 3 − ) On simplification using the endemic state condition of model (1), we have from Eq. (16) as Therefore, It can be noticed that if  0 > 1, then the right-hand side of Eq. (17) is non-positive and it is equal to zero if = * , = * , = * , = * , = * , = * , = * , = * . Therefore, the maximum invariant set for { (  , , , , , , , | (1) = 0} is the singleton set 1 . This means that the only trajectory of the system on which £ 2 = 0 is 1 . According to the LaSalle's invariance principle [40][41][42][43], we know that all solutions in * converge to 1 . Therefore, the endemic state of the model (1) is globally asymptotically stable when  0 > 1. This completes the proof of Theorem 6. □

Fitting of the model to the real statistical data for India
India is a country with a 1.3 billion population. There were some restrictions and lockdown phases in the initial stage of the virus, but it is not possible for the government to keep all populations under lockdown for long. It has been a year since this virus in India, but still, it is not controlled. Here, we attempt to fit and estimate our model parameters to the actual statistical data from India. To avoid the possibility of pitfalls described by the authors in [44], we do not fit the model to a cumulative number of cases or a cumulative number of deaths. We convert this data to a fraction of the population by taking total population data from The World Bank Group (about 60297396, in 2019). For the model fitting, we use updated data from Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) (https://github.com/CSSEGISandData/COVID-19) for India. We use recent data from the last six months of this virus in India, i.e., from 22nd October 2020 to 31st March 2021. The estimated parameter values are based on the data about the number of currently infected individuals that can be observed and roughly corresponding to ( ( ) + ( ) + ( )). Parameters except for recruitment rate of susceptible ( ) and natural death rate ( ) have been estimated from fitting the model to the data as is shown in Fig. 2. In Fig. 2, the real COVID-19 cases for India are shown by blue circles, whereas the best-fitted curve of the model is shown by the red solid line. The summary of the biological parameters involved in the model is listed in Table 1 and their best-estimated values obtained via the non-linear least-squares method are listed in Table 2. The estimated values of the parameters for the real COVID-19 cases in India have produced the value of basic reproduction number  0 = 3.0438 from October 22, 2020, to March 31, 2021, as the epidemic is still on its peak in India and is uncontrolled. The obtained results are comparable with findings in [45].
The ordinary differential equation (ODE) system was solved using LSODA [46,47]. We use lmfit python package [44] for non-linear leastsquares and minimize the sum of the squares of the errors using the trust region reflective method and obtaining goodness of fit measure of 2 = 6.2555 −08 for India. The problem to minimize error is shown in the following equation for a fitting parameter set : where are observations and is the model output.

Sensitivity analysis
Sensitivity analysis can aid in identifying influential model parameters and optimizing model structure. In the proposed COVID-19 epidemic model, due to uncertainties associated with the estimation of certain parameter values, it is useful to carry out the sensitivity analysis to determine the model's robustness as the parameter values changes. Having analytical expression for the reproduction number  0 , it is reasonable to employ the normalized forward sensitivity index of a variable  0 , that depends on a parameter , that is defined as [48,49] where is one of the parameters whose sensitivity on  0 is sought. This index implies that the higher the value in its magnitude, the more sensitive  0 is to the parameter. Also, the positive (or negative) sign indicates that  0 increases (or decreases) as increases. For our proposed COVID-19 epidemic model (1), all eight state variables and eighteen parameters were uncertain. Following the detailed analysis as done by Marino et al. [50], we use Latin-hypercube sampling-based method to quantify the uncertainty and sensitivity of all the model parameters. A positive partial rank correlation coefficient (PRCC) value in Fig. 3 indicates an increase in the parameter leads to an increase in  0 , while a negative value shows increasing the parameter decreases  0 . Among these parameters, , , , , , and 1 have a positive influence on  0 , while , 1 , , and have a negative influence on  0 . Thus, the sensitivity analysis results show , 1 , , and are the most influential parameters for the proposed model (1), as the increase in the value of these parameters decreases the value of  0 .

Numerical results and discussion
In this section, we discuss the behavior of obtained numerical results. The model parameters used in simulations are described in Table 2. Fig. 4 shows the temporal variations of different populations classes. It can be seen from Fig. 4 that as time increases, the number of infected individuals also increases. Also, Fig. 4 shows that the environmental virus goes on increasing by the movement of the asymptomatic as well     Fig. 6 that for the higher concentration of the virus, the infected population is higher than the lower values of the virus. Also, it can be seen that these parameters have a positive effect on the value of  0 , i.e., the increase in these parameters increases the value of  0 means they are directly proportional to the value of  0 , thus causes disease spread. Further, it can be seen from Fig. 6 that the transmission coefficient of virus in the environment to the susceptible is more significant than other transmission rates as an increase in its value makes the value of  0 very high.      Figs. 7-10 shows the 3D dynamics of  0 against different model parameters. It can be seen from the figures that each parameter contributes towards the dynamics of  0 as the changes in the values of these parameters significantly affect the value of  0 . In Fig. 7, the variation of  0 for 1 and is presented. It can be seen from Fig. 7 that 1 contributes towards the increasing value of  0 , which is the virus releasing rate via asymptomatic individuals. This indicates freely movement of individuals without any restriction can spread this disease. In Fig. 8, the variation of  0 for and is shown. From  Fig. 8, it is clear that environmental viruses can make the infection spread in the population. In contrast, the spread can be controlled by decreasing the virus in the environment by restricting the free movement of individuals. Similar behavior is observed from Figs. 9-10 for the parameters and , respectively.
While mathematical models do not provide a cure for a given infectious disease, they are, however, used to replicate possible scenarios of the dynamic at hand. In the modeling of COVID-19 disease, mathematical models may help to explore the transmission dynamics, understanding the trajectory of the epidemic, prediction, and design effective control measures for the spread of this fatal disease. However, any study is not ultimate in science, there must be some limitations and assumptions which are deviated from actual reality. There were some limitations to this study that must be considered. First, we ignored the effect of uneven population distribution and assumed that the total population was homogeneously distributed. Second, we ignored P.A. Naik et al. the differences in individual susceptibility and we assumed that infection susceptibility for all individuals in the free environment was the same; whereas, in actuality, adults and older people are more likely to be infected by SARS-CoV-2. Third, we did not take into account the limitation of medical resources, such as health care workers and medical protective equipment. Furthermore, since the study was constructed from reported data and some parameters were calculated based on preliminary studies, these data came from heterogeneous sources, which may have introduced biases. It was important to note that when we predicted the epidemic trend of COVID-19 without any control measures, due to the small amount of reported data, the estimated parameters might have certain errors and the predicted results might represent an over-prediction. The current study is designed for the country India however the model can perform well if applied to other countries with high COVID-19 cases like Brazil or the USA and get results to depend on the reported cases for these countries. Many studies have already been reported in the literature [51][52][53][54][55][56][57] that studied the effect of environmental parameters like temperature, humidity, pollution, etc on the transmission dynamics of COVID-19. From these studies, it is clear that environmental parameters play important role in the dynamic process of COVID-19 disease.

Concluding remarks
In this paper, we have proposed and investigated a novel mathematical model for the transmission of COVID-19 under the influence of contaminated environments. This model has been used to describe the diverse transmission passages in the infection dynamics and affirms the role of the environmental reservoir in the transmission and outbreak of COVID-19 disease. The model is numerically simulated to signify the application of the study in India. We have presented a detailed analysis of the proposed model including, the derivation of equilibrium points, endemic and disease-free, the reproductive number  0 , and the positiveness of the model solutions. By using the threshold quantity  0 , the location and the existing conditions of the disease-free and endemic equilibrium point have been determined. The diseasefree equilibrium point is proved to be globally asymptotically stable if  0 < 1. Whereas, if  0 > 1, the endemic equilibrium point exists and is locally asymptotically stable. In the segment of sensitivity analysis the importance of various parameters has been determined that highly affect the reproduction number and it signifies that to reduce the transmission rates, combined efforts towards the detection of undetected individuals, virus in the environment, and the treatment of infectives are required. The obtained analytical results explore that the proposed model provides a better fit to the real statistical data. The statistical results of this work may help the government and other proxies to reconfigure their strategies according to the expected situation.
From numerical simulations, it is observed that to control the disease, the reproduction number must be decreased below one. To reduce the reproduction number below unity, a continuous reduction in the transmission rate of COVID-19 is required by self-quarantine, isolation of infectives, and initiation of treatment for the infected individuals as early as possible. Based on the scientific evidence, the premature pulled out of these strategies influences the infection prevalence and implies that incautious decisions have been taken as can be visualized from the numerical simulations. Thus, we have concluded that if the reproduction number for the COVID-19 disease is reduced below unity by decreasing the transmission rates and increasing the detection rate, then the epidemic can be eradicated from the population.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Availability of data
The data used for the simulations and data fitting is collected from the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) (https://github.com/CSSEGISandData/COVID-19).