**2. Analysis of SIR model**

SIR model is first introduced by W.O. Kermach and A.G Mckendrick in 1927. SIR model is a best model of an infectious disease. This model divided the population into the three groups. The groups name is.


This model is constructing the ordinary differential equations in this model time t is the independent variable and S, I, R is the dependent variables. These groups have taken the number of people on every day. Yet, the data is transitions with time, as human being act from one group to another group. Illustration, human being in group **S** will act to the group **I**, that is the infected. Furthermore, infected person, **I** will act to the recovered **R** group that is they are recover or die from the disease. This method has been used successfully many times before in spreading

disease like yellow fever, plague, fever, influenza, avian influenza, etc. Therefore, we have made the differential equations of COVID 19 using this method. This method is very helpful in giving a mathematical model to COVID.

$$\frac{ds}{dt} = -\text{gsi.}\tag{1}$$

$$\frac{di}{dt} = \text{g}si - fi \tag{2}$$

$$\frac{dr}{dt} = f\hat{r} \tag{3}$$

where *t* is the independent variable *s*, *i*,*r* is dependent variables i.e. *s* is denote the susceptible person at the time *t*


*f* is recovery

If *s*> 0, *i*> 0 then

$$\frac{ds}{dt} = -\text{g}s i < 0, \forall s > 0, i > 0\tag{4}$$

$$\frac{\text{di}}{\text{dt}} = \text{i}(\text{gs} - \text{f})\tag{5}$$

$$\text{g}\text{s} - \text{f} = \text{0}$$

$$
\mathbf{s} = \frac{f}{\mathbf{g}}
$$

$$
\mathbf{s} - \frac{f}{\mathbf{g}} = \mathbf{0} \tag{6}
$$

$$\text{So } \frac{di}{dt} < 0 \text{ if } s < \frac{f}{\mathcal{g}} \tag{7}$$

*di dt* <sup>&</sup>gt; 0 if *<sup>s</sup>*<sup>&</sup>gt; *<sup>f</sup> <sup>g</sup>* is defined the direction diagram of trajectories We find the trajectories

$$\frac{di}{ds} = \frac{\frac{di}{dt}}{\frac{ds}{dt}} = \frac{g\dot{s} - f\ddot{s}}{-g\dot{s}}$$

$$\frac{di}{ds} = -\mathbf{1} + \frac{f}{g\mathbf{s}}$$

$$di = -\mathbf{1}ds + \frac{f}{g\mathbf{s}}\,ds$$

$$i = -s + \frac{f}{g}\log s + c$$

Initial conditions

$$s(\mathbf{0}) = s\_0$$

$$i(\mathbf{0}) = i\_0$$

(If)

$$s \to \mathbf{0}, i \to -\infty$$

$$s \to \infty, i \to -\infty$$

1. It is impossible for the disease to infect all the susceptible person.

$$\text{2. } s\_0 > \frac{f}{g} \text{ for an equivalence to occur.}$$

3. *s*<sup>0</sup> < *<sup>f</sup> <sup>g</sup>* disease dig out.

4. *gs*<sup>0</sup> *<sup>f</sup>* >1 then number of infected is increase.

5. *s* þ *i* þ *r* is the total population.
