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Introduction

Firstly, this pandemic COVID-19 has been identified in Wuhan city situated in China in the month of December 2019 and is an infectious disease. Yang and Wang [1] developed a mathematical model for the novel coronavirus epidemic in Wuhan city, China. Day by day, this pandemic spread due to the high transmission rate because no vaccine at present available. Some renowned researchers have dedicated nice work based on COVID-19 disease. COVID-19 disease came in the state of Rajasthan on 2nd March and identified in the SMS hospital which is situated in Jaipur city. Initially, the pandemic was slow increasing due to slow transmission rate. Nowadays, this disease is rapidly distributing to all over Rajasthan. The government organizations have taken some nice primary decision and tried to control it with the implementation of lockdown as well as manage the social distancing. Guckenheimer and Holmes [2] proposed a mathematical model along with nonlinear oscillation, dynamical systems and bifurcations of vector field. Driessche and Watmough [3] presented a mathematical model with the help of reproduction factor and endemic equilibriums for disease transmission. Mizumoto and Chowell [4] have explained a mathematical model based on potential of novel corona virus. Rothan and Byrareddy [5] have presented a mathematical model concern with the epidemiology and pathogenesis coronavirus disease (COVID-19). Sohrabi et al. [6] developed a mathematical model based on COVID-19 disease using data of World Health Organization. They reviewed the latest situation. World Health Organization declared the highly infectious COVID-19 as a global pandemic in March 2020. Thevarajan et al. [7] explained the symptoms of COVID-19 which includes high fever, cough and difficulties in inhalation. Riou and Althaus [8] studied a mathematical model in which they discussed that people may get infected while breathing in an infected environment or touching any infected surface or by coming in contact of an infected person. Some authors like Khot and Nadkar [9], Wang [10], have contributed their work wherein they studied that aged people, children and adults with low persons immunity are prone to catch infection and may be severely affected from this disease, even lead to death. Some experts like Sahin et al. [11], Cheng and Shan [12] considered the risk factor in their study as people are unaware of reason behind the spreading of virus.

All mathematical models discussed above are predictive model and have not given an exact solution. The above said references motivate us to develop a mathematical model under different strategies to discuss COVID-19 pandemic situation in Rajasthan. The present mathematical model explains the transmission of COVID-19 disease in the cold season and sensitivity analysis has been studied with the help of modelling graph under different cases. Moreover, a comparative analysis of results obtained by predictive mathematical model has been developed with the exact data plotting by curve fitting as obtained from Rajasthan government website. We have taken some input data from the site of Rajasthan, India (as shown in Annexures 5.1, 5.2, 5.3 and 5.4).

The spread of COVID-19 is found to be pandemic in nature for the state of Rajasthan. From the graphical plotting of the last four months date (June, July, August and September), it is noted that there is drastic change in variation of infection, recovery and mortality observed. Here, the input data used for the analysis is taken from the Rajasthan government website and as reflected in Annexures 5.1, 5.2, 5.3 and 5.4.

From the graphical plotting as shown in Figs. 5.1, 5.2 and 5.3 for the month of June 2020 in the state of Rajasthan, it is clearly noted that there is rapid growth in the rate of infection but corresponding recovery rate also found significant but rate of deaths increases as time increases. The commutative infection reaches to 18,000 approximately whereas total recovery observed close to 14,000 and commutative deaths touch 400 till end of the June month.

Fig. 5.1
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Variation of infection versus time for the month of June 2020.

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Fig. 5.2
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Variation of recovery versus time for the month of June 2020.

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Fig. 5.3
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Variation of death versus time for the month of June 2020.

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From Figs. 5.4, 5.5 and 5.6, it is observed, for the month of July 2020 for the state of Rajasthan, growth in the rate of infection is double as compare to June month but there is small decay in corresponding recovery rate. Also, infection outbreak is noted near the end of the month as well as death rate increases with increase in time. The commutative infection reaches more than 40,000 whereas total recovery observed close to 29,000 and commutative deaths touch 680 till end of the July month.

Fig. 5.4
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Variation of infection versus time for the month of July 2020.

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Fig. 5.5
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Variation of recovery versus time for the month of July 2020.

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Fig. 5.6
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Variation of death versus time for the month of July 2020.

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From Figs. 5.7, 5.8, 5.9, 5.10, 5.11 and 5.12, a similar behaviour is noted for the overall variation of infection, recovery and mortality in the month of August and September, respectively. Infection touches 80,000 in August, which is double as happened in July month. Close 110,000 infected people noted in September, but corresponding recovery is not significantly increasing, and death rates become high.

Fig. 5.7
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Variation of infection versus time for the month of August 2020.

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Fig. 5.8
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Variation of death versus time for the month of August 2020.

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Fig. 5.9
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Variation of recovered versus time for the month of August 2020.

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Fig. 5.10
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Variation of infection versus time for the month of September (up to 19th September 2020).

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Fig. 5.11
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Variation of death versus time for the month of September (up to 19 September 2020).

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Fig. 5.12
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Variation of recovered versus time for the month of September (up to 19 September 2020).

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Mathematical Modelling of COVID-19

By looking the overall scenario, here we construct a mathematical modelling of COVID-19 for the state of Rajasthan by making use of the parameters described in Table 5.1. We divide the total population of Rajasthan into mainly five compartments, the susceptible (denoted by S), the exposed (denoted by E), the infected (denoted by I), and the recovered (denoted by R) and mortality (denoted by M). In Fig. 5.13, we describe the flowchart presentation of model. 

Fig. 5.13
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Mathematical representation of model.

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Here it is assumed that infected individual class has fully developed disease symptoms and has the capacity to infect others. Also, exposed class individuals are those who are in incubation period and they do not show symptoms but still capable to pass infection to others. In this model, E and I compartment can be interpreted as they contain asymptomatic and symptomatic infected individuals.

Table 5.1 Notation and its description
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The mathematical model suits the current COVID-19 situation to describe the transmission dynamics represented as:

$$\frac{{{\text{d}}S}}{{{\text{d}}t}} = \Lambda - C_{1} SE - C_{2} SI$$
(5.1)
$$\frac{{{\text{d}}E}}{{{\text{d}}t}} = C_{1} SE + C_{2} SI - \alpha E$$
(5.2)
$$\frac{{{\text{d}}I}}{{{\text{d}}t}} = \alpha E - \left( {m + r} \right)I$$
(5.3)
$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = rI$$
(5.4)
$$\frac{{{\text{d}}M}}{{{\text{d}}t}} = mI$$
(5.5)

Here, \(C_{1}\) and \(C_{2}\) are assumed as non-increasing functions, given that higher values of E and I would motivate stronger control measures that could reduce the transmission rates.

All the initial conditions of the system are assumed non negative as \(S\left( 0 \right) \ge 0,\) \(E\left( 0 \right) \ge 0,\) \(I\left( 0 \right) \ge 0,\) \(R \ge 0\) and \(M\left( 0 \right) \ge 0\).

Numerical Analysis

The basic input parameters of this model are physical in nature and taken by looking the current situation as per the data shown in annexure.

Figures 5.14, 5.15 and 5.16 indicate the variation of infection versus time for the month of June, July and August, respectively. From the figures, it is observed that infection goes on increasing with increase in time. Also, growth of infection found triple in the month of July and five times in the month of August as compared to infection in June. This variation of infection indicates that the virus growth as discussed above by taking exact data using curve fitting (Figs. 5.1 and 5.4) is nearly similar to the analysis done by our mathematical model. Hence, we can say that our developed mathematical model is efficient and works parallel to the exact situation of spread of infection as recorded in Rajasthan state.

Fig. 5.14
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Variation of infection versus time in the month of June 2020.

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Fig. 5.15
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Variation of infection versus time in the month of July 2020.

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Fig. 5.16
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Variation of infection versus time in the month of August 2020.

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Figures 5.17, 5.18 and 5.19 show the variation of recovery versus time for the month of June, July and August, respectively. It is observed that recovery goes on increasing with increase in time and found significant in the month of June but there is small decay noted in the month of July and August as compared to corresponding rate of infection. The same behaviour of recovery rate was observed by curve fitting as shown in Figs. 5.2 and 5.5 for the exact data, hence our mathematical model work exactly parallel in case of rate of recovery also and gives efficient results.

Fig. 5.17
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Variation of recovery versus time in the month of June 2020.

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Fig. 5.18
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Variation of recovery versus time in the month of July 2020.

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Fig. 5.19
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Variation of recovery versus time in the month of August 2020.

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From Figs. 5.20, 5.21 and 5.22, it is clear that rate of mortality goes on increasing with increase in time but at a steady rate, also as compare to growth in infection the rate of mortality is very low. A similar characteristic of mortality was found during curve fitting for the exact recorded data.

Fig. 5.20
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Variation of mortality versus time in the month of June 2020.

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Fig. 5.21
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Variation of mortality versus time in the month of July 2020.

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Fig. 5.22
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Variation of mortality versus time in the month of August 2020.

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Figures 5.23, 5.24 and 5.25 show the variation of exposed versus time. It is noted that with the passage of time, number of exposed goes on increasing. In the June month, number of exposed was nearly 500,000 but in month of July and August, it increases from 800,000 to 1,000,000. It shows that rate of exposed increases rapidly day by day.

Fig. 5.23
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Exposed versus time in the month of June 2020.

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Fig. 5.24
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Exposed versus time in the month of July 2020.

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Fig. 5.25
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Exposed versus time in the month of August 2020.

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Figures 5.26, 5.27 and 5.28 clearly agree with the condition that with the increase in infection corresponding decay is susceptible observed. Variation of susceptible corresponding to different months shown versus time and completely agrees with current scenario of COVID-19 in Rajasthan state.

Fig. 5.26
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Susceptible versus time in the month of June 2020.

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Fig. 5.27
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Susceptible versus time in the month of July 2020.

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Fig. 5.28
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Susceptible versus time in the month of August 2020.

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Figures 5.29, 5.30, 5.31, 5.32 and 5.33 show the overall variation of infection, mortality, recovery, exposed and susceptibles versus for the June to December month. After observing overall plotting, it is clearly visible that the impact of corona virus will increase significantly in the winter months especially in December 2020. It is forecast that till the end of December month infected population will touches 178,000 and becomes steady, mortality may nearly 3500; recovery may approximately 120,000 (which is insignificant variation as compared to infection), number of exposed may increases 200,000 (double as in August month). The impact can be reduced only if the guideline provided by the Government of India should be strictly followed by entire population with the arrangement of proper vaccination.

Fig. 5.29
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Overall variation of infection versus time from June to December 2020.

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Fig. 5.30
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Overall variation of mortality versus time from June to December 2020.

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Fig. 5.31
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Overall variation of recovery versus time from June to December 2020.

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Fig. 5.32
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Overall variation of susceptible versus time from June to December 2020.

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Fig. 5.33
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Overall variation of Exposed versus time from June to December 2020.

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Conclusion

In the present study, a comparison model has been constructed to examine the transmission of dynamic COVID-19 for the state of Rajasthan, India. Numerical results obtained by the mathematical modelling which have been compared with exact curve fitting method, where data is taken from government website of Rajasthan state. Numerical plotting in mathematical and curve fitting shows same characteristic in the case of infection, recovery, mortality, susceptible and exposed. The presented mathematical model is hypothetical in nature but for some particular functions and parameters, it is feasible for the real-world problems. Finally, on the basis of the presented model, we can predict that the situation will become worse in the month of December due to unavailability of proper vaccine. Hence, to control the spreading of virus infection, it is very much essential to follow all the restrictions like lockdown, social distancing, contact tracing, mask cover, etc., as laid down by Government of Rajasthan and India.