**3.1 Case study of India**

In this research work we have discussed about the COVID 19 disease. also know about how many people got sick in India due to pandemic disease COVID19. In this article, we have given a mathematical model to COVID 19 with the help of SIR model [4–6]. We have taken data of how many people had become ill in India by COVID 19 on 20 January 2021 and using this data , we have created a mathematical model of COVID 19 with the help of SIR model. We have created three differential equations by taking the original data and solving those equations by the Homotopy Perturbation method (HPM ).We got out the numerical solutions and made a table, and with the help of that table, we tried to tell what is the position of COVID 19 in India by making the graphs. There are a lot of methods which solves the differential equations [7–9], but we have used the HPM method. This method solves the biggest *SIR Model with Homotopy to Predict Corona Cases DOI: http://dx.doi.org/10.5772/intechopen.96481*

and most difficult equations very easily and with less of calculations.We will solve the Differential Equations of COVID 19, which is made with the help of SIR model. In these equations, we took the data of 20 January 2021 COVID 19 people of India who were caught by COVID 19 epidemics.

Total confirmed cases on 20 January 2021 in India is 10611728. Death 152907 Recovered is 102656163 Active cases are 189245 Susceptible persons are 189347782 So, we take the

> *s*ð Þ¼ 0 18*:*9347782 *i*ð Þ¼ 0 1*:*0611728

$$r\left(0\right) = 10265163 + 152907 = 1.0418070$$

*<sup>g</sup>* <sup>¼</sup> *active cases of india on* <sup>20</sup> *january* <sup>2021</sup> *for COVID* <sup>19</sup> *susceptible people of india on* 20 *january* 2021 *for COVID* 19

$$g = \frac{189245}{18.9347782} = 0.00999457$$

$$f = \frac{1}{14} = 0.0714$$

$$\frac{ds}{dt} = -gsi \tag{8}$$

$$\frac{di}{dt} = \text{gsi} \, \, -\text{fi} \,\, \tag{9}$$

$$\frac{dr}{dt} = f\mathbf{i} \tag{10}$$

where *t* is the independent variable *s*, *i*,*r* is dependent variables i.e.

*s* is denote the susceptible person at the time *t*

*i* is denote the infected person at the time *t*

*r* is denote the recovered person at the time *t*

*g* is transmission coefficient

*f* is recovery

Now we will solve these equations with the help of HPM method. By the homotopy method we get,

ð Þ <sup>1</sup> � *<sup>p</sup> dS dt* <sup>þ</sup> *<sup>p</sup>*(*dS dt* þ *si*Þ

$$\frac{d\mathbf{S}}{dt} = p(-0.00999457si)$$

$$p(1-p)\frac{di}{dt} + p\left(\frac{di}{dt} - 0.00999457si + 0.0714i\right) \tag{11}$$

$$\frac{di}{dt} = p(0.009999457si - 0.0714i)\tag{12}$$

$$\frac{dr}{dt} = p0.0714i$$

## *A Collection of Papers on Chaos Theory and Its Applications*

$$\mathfrak{s} = \mathfrak{s}\_0 + p^1 \mathfrak{s}\_1 + p^2 \mathfrak{s}\_2 + \dots \tag{13}$$

$$i = i\_0 + p^1 i\_1 + p^2 i\_2 + \dots \tag{14}$$

$$r = r\_0 + p^1 r\_1 + p^2 r\_2 + \dots \tag{15}$$

Putting value *s*, *i*,*r* we get,

$$\frac{ds}{dt} = p(-0.00999457\{s\_0 + p^1\varepsilon\_1 + p^2\varepsilon\_2 + \dots\}\{i\_0 + p^1i\_1 + p^2i\_2 + \dots\})\tag{16}$$

$$\frac{di}{dt} = p(\ 0.00999457[\{\ s\_0 + p^1 \mathbf{s}\_1 + p^2 \mathbf{s}\_2 + \dots \} \{\ i\_0 + p^1 \mathbf{i}\_1 + p^2 \mathbf{i}\_2 + \dots \} ]}{-0.0714 \{\ i\_0 + \ \ p^1 \mathbf{i}\_1 + p^2 \mathbf{i}\_2 + \dots \})}\tag{17}$$

$$\frac{dr}{dt} = p \ 0.0714 \{i\_0 + p^1 i\_1 + p^2 i\_2 \tag{18}$$

Both side comparing the coefficient of p we get

$$s\_0 = 18.9347782$$

$$i\_0 = 1.0611728$$

$$r\_0 = 1.0418070$$

$$\frac{ds\_1}{dt} = -0.00999457(18.9347782)(1.0611728)$$

$$s\_1 = -0.2008216t$$

$$\frac{di\_1}{dt} = 0.00999457(18.9347782)(1.0611728) - 0.0714(1.0611728)$$

$$i\_1 = 0.1250538t$$

$$\frac{dr\_1}{dt} = 0.0714\{1.0611728\}$$

$$r\_1 = 0.0757677t$$

So, by the HPM we get the solution:

$$s(t) = 18.9347782 - 0.2008216t + \dots$$

$$i(t) = 1.0611728 + 0.1250538t + \dots$$

$$r(t) = \begin{array}{c} 1.0418070 + 0.0757677t + \dots \end{array}$$

We have the table


*SIR Model with Homotopy to Predict Corona Cases DOI: http://dx.doi.org/10.5772/intechopen.96481*


From the above table we can predict that infected cases of corona on 31 January which is almost same as actual cases on 31 January. The current COVID-19 pandemic is unprecedented, but the global response draws on the lessons learned from other disease outbreaks over the past several decades.

World scientists on COVID-19 then met at the World Health Organization's Geneva headquarters on 11–12 February 2020 to assess the current level of

**Figure 1.** *SIR chart depicting no of people susceptible, infected and recovered.*

**Figure 2.** *SIR chart depicting no of people susceptible.*

knowledge about the new virus, agree on critical research questions that need to be answered urgently, and to find ways to work together to accelerate and fund priority research to curtail this outbreak and prepare for those in the future see **Figures 1**–**3** for reference.
