key: cord-0732852-nxuj8v0j authors: Das, Kalyan; Kumar G, Ranjith; Reddy K, Madhusudhan; K, Lakshminarayan title: Sensitivity and elasticity analysis of novel corona virus transmission model: A mathematical approach date: 2021-03-10 journal: Sensors international DOI: 10.1016/j.sintl.2021.100088 sha: 8712aa2343af6d169ecef4f5c11c5dd60893845f doc_id: 732852 cord_uid: nxuj8v0j The deadly corona virus continues to pound the globe mercilessly compelling mathematical models and computational simulations which might prove effective tools to enable global efforts to estimate key transmission parameters involved in the system. We propose a mathematical model using a set of non-linear differential equations to account for the spread of the COVID-19 infection with special compartment class isolation or quarantine and estimate the model parameters by fitting the model with reported data of the ongoing pandemic situation in India. The basic reproduction number is defined and local stability analysis is carried out at each equilibrium point in terms of the reproduction number R 0 . The model is fitted mathematically and makes the data India specific. Additionally, we examined sensitivity analysis of the model. These outcomes recommend how to control the spread of corona, keeping in mind contact and recovery rate. Also we have investigated the elasticity of the basic reproduction number as a measure of control parameters of the dynamical system. Numerical simulations were also done to show that the proposed model is valid for the type and spread of the outbreak which happened in India. Mathematical models relating to infectious disease transmission dynamics have assumed a degree of pervasiveness. The epic COVID-19 which started in recent times in Wuhan province of China has unleashed its tentacles across the globe. The World Health Organization declared it a pandemic. In the absence of a vaccine, social distancing has emerged as the most widely endorsed strategy for mitigation and control [1] . In the context of all that has been mentioned, the mathematical model proposed in the paper comes into its own is a very useful and important tool for the analysis of infectious diseases because this analysis enables prediction of future outbreaks while also detailing strategies to control the epidemic. Mathematical modeling based on a system of differential nonlinear equations may give a sensible tool to explain the elements of COVID-19 transmission. The global problem of outbreak of COVID-19 has attracted the interest of researchers in different disciplines. Okhuese et al. [5] [13] discussed current treatment protocol for COVID-19 in India and Aayushi Kundu et al. [14] discussed some factors such as dietary habits, vaccination, climate conditions which could be the explanation for the contrasting impact of COVID-19 in India and other developed nations. Some authors estimated the value of reproduction number, which helped to predict the outbreak of the disease. In [15] Read et al. calculated basic reproduction number using an assumption of poission-distributed daily time increments. Some authors [16, 17] estimated the mean reproduction number for COVID-19 in the early phase of outbreak, which is slightly higher than that for SARS-CoV. Still other works where the basic reproduction number is estimated for different countries can be found in [18] [19] [20] [21] [22] [23] [24] [25] [26] . The human to human contact is the likely reason for episodes of COVID-19. Therefore, isolation of the infected human overall can reduce the risk of future COVID-19 spread. In order to do this, we divided the total population into five compartments, classifying them as susceptible, exposed, infected, isolated and recovered from the disease. The remainder of our study is organized as follows. In section 2, we have proposed a mathematical model specific to COVID-19. In section 3, defines the basic reproduction In this work we proposed a deterministic model for COVID-19 using a set of ordinary nonlinear differential equations. The population has been divided into five compartments that is , , , S E I Q and R . Many authors have carried out studies on COVID-19 with different compartments, but in our model we take into account asymptomatic cases, implying cases involving those who are unaware of infections or the limited capacity for testing. For Individuals without any symptoms, it is very difficult to detect the presence of the virus. Hence asymptomatic transmission is the most challenging one in COVID-19. Alongside this we consider another compartment i.e., isolated population which represents home quarantine and the hospitalized. This model assumes a completely susceptible population with homogeneous mixing. The disease has an incubation period and after that the exposed individuals becomes infectious at the rate β . The exposed individuals are removed and added to the infectious at the rateα . Generally the infected individuals are isolated when the susceptible individuals are not infected. To factor this we added a compartment Q representing the quarantine/isolated. The isolated or quarantine individuals are removed from I and added to the compartment Q at the rate q . The recovered individuals from I and Q are added to the compartment R at a rate and γ θ respectively. µ represents the natural death rate and the disease death rate. We assumed the total population to be constant. From the all above assumptions we consider the following SEIQR mathematical model for We can omit without generality the last equation because first four equations are independent of ( ) R t and the initial populations are defined in the following manners It measures the disease spread in the population and it is defined as the average number of J o u r n a l P r e -p r o o f cases produced if one infected person or individual introduced in the population. Here we defined this 0 R by using the next generation matrix approach [27] . The system (1) has unique disease free equilibrium i.e., corona virus free equilibrium 0 ,0, The following matrices F and V represents nonlinear terms of new infection and out flow terms respectively which are given by Therefore, the linearized matrices at free corona virus equilibrium point 0 E is given by and The basic reproduction number 0 R is given by the dominant eigen value of Therefore, The Jacobian matrix of the system (1) at the free corona virus equilibrium is given by The characterstic equation of (6) is given by It is easy to see that 3 0 P > then which is equivalent to 0 The characteristic equation corresponding to (9) is given by The numerical simulations are performed for the proposed model and the outcomes are contrasted and the genuine information pertaining to India which were acquired from numerous reports distributed by the WHO and World meter [28, 29] . We classified the outcomes into two sections-lockdown period and unlock period and our Figure2. Confirmed cases in India (lockdown period). Dots represent the real data and the line is corresponding to the real data which was obtained by solving the system (1) numerically. J o u r n a l P r e -p r o o f Figure3. Confirmed cases in India (unlock period). Dots represent the real data and the line is corresponding to the real data which was obtained by solving the system (1) numerically. The definition of sensitivity is local because the sensitivity is computed while all parameters, including parameter k , are kept at their estimated values. However, this method does no longer absolutely explore the input space; in view that it does not take into account the simultaneous variation of input parameters. Another disadvantage is that it depends strongly on the magnitude of k and the quantityW . In this respect, a much greater useful concept is elasticity. The Elasticity of quantity W with respect to the parameter k is given , which means the percentage change in W with respect to the percentage change in the parameter k . Now we are computing the elasticity of 0 R with respect to the It is easy to show that the elasticity of 0 R with respect to the parameter β is 1, i.e., Computing the elasticity of reproduction number with respect to the γ is This gives the result that 1% of increase in γ will produce 0.6% decrease in 0 R . From these results the elasticity's suggest that the recovery rate γ is more effect on 0 R , if we compare with transmission rate β . From these results we conclude that our model supports real data for the parameter values (Table1) and the virus spread depend on transmission rate or contact rate and recovery rate from infection. can say that the best factor on spreading corona virus is contact rate and recovery rate. Additionally we examined the elasticity of basic reproduction number with respect to the transmission rate and recovery rate. These results indicate that the recovery rate highly impacts basic reproduction number. From the results, we observed that if the rate of transmission of disease is expanded, the disease will spread rapidly, and to control COVID-19 spread, we need to focus on contact rate. To control contact rate, we carefully follow physical distancing, wearing cover or face shield and so forth, in any case no control is possible. Among infected people, without isolation, tracking contacts is an important factor in fighting the epidemic. It is helpful to have a moderate analysis in the follow-up and isolation, J o u r n a l P r e -p r o o f as quickly as can be expected under the circumstances, so that those who may have been infected with the virus have not transmitted the disease. All the information utilized in this work has been obtained from authentic sources. By and large the numerical models have certain constraints, while our mathematical model performs rather well. Our model is based on data available on public platforms until 8 th September 2020. Future models can include greater granularity as more data become available, and dynamics of the COVID-19 virus becomes better known. The authors declare no conflict of interest. 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