COVID-19 acts as a serious challenge to the whole world. Epidemiological data of COVID-19 is collected through media and web sources to analyze and investigate a system of nonlinear ordinary differential equation to understand the outbreaks of this epidemic disease. We analyze the diseases free and endemic equilibrium point including stability of the model. The certain threshold value of the basic reproduction number _{0} is found to observe whether population is in disease free state or endemic state. Moreover, the epidemic peak has been obtained and we expect a considerable number of cases. Finally, some numerical results are presented which show the effect of parameters estimation and different step size on our obtained solutions at the real data of some countries to check the actual behavior of the COVID-19 at different countries.

Epidemiological research plays a significant role in understanding the effects of community-based infectious disease. In mathematical modeling, we investigate model building models, perform parameter estimation, test model sensitivity by varying parameters and compute their numerical simulations. Such research helps to understand the ratio of disease spread within the population and infection of disease [

Severe Acute Respiratory Syndrome (SARS) is caused by a coronavirus and plays important role for its investigation [

In 2019, China faced a major outbreak of coronavirus disease 2019 (COVID-19) and this outbreak has the potential to become a worldwide pandemic [

The goal of this article is to expand on mathematical ways of obtaining _{0} for ODE disease models in a population, taking into account the epidemiological significance of _{0}, and to show how this reproduction numbers can be utilized to direct control strategies that address disease rates and impacts in different countries. Stability and qualitative analysis of the model are discussed in Section 2. In Sections 3 and 4 actual data of different countries are formulated, also find the infected, death and recovered rate. At the end numerical simulation design which support the result and predict the disease stage in society.

Kermack et al. [

With initial condition

Here hard immunity and precaution level is more important at endemic stage to control the virus. The modified form of coronavirus model after adding new parakeets in the model which is given as

where

To evaluate the equilibrium point, we take left hand side equal to zero of

Thus the endemic equilibrium state is given as

_{0} is locally asymptotically stable if _{0} > 1 and otherwise unstable

_{0} > 1, so this is in endemic state. Consider the Jacobean matrix (J) as

Were

By using the relation ^{−1} and got the Eigen value

_{0} is locally asymptotically stable if

The equation which is given above is called characteristic equation. But recall that E is given as

i.e.,

if

Therefore, _{0} is locally asymptotically stable. This proves the proposition.

Many COVID-19-patients are adults. Of the 44,672 confirmed infection patients in China, 2.1% were below the age of 20. Fever, dry cough, and shortness of breath were the most commonly reported symptoms, and most patients (80%) experienced mild illness. Approximately 14% experienced severe disease and 5% were critically ill. The latest outbreak of coronavirus 2019 noted 12 March 12, 2020, as noted in [

Country | Population | Total cases | Total death | Total recovered |
---|---|---|---|---|

USA | 330370141 | 18473716 | 326772 | 10802496 |

Italy | 60461826 | 1964054 | 69214 | 1281258 |

Germany | 83969900 | 1534116 | 27297 | 1115400 |

France | 65273511 | 2479151 | 60900 | 184464 |

China | 1438070898 | 86899 | 4634 | 81950 |

Iran | 83639890 | 1164535 | 53816 | 894366 |

Turkey | 84130947 | 2043704 | 18351 | 1834705 |

Belgium | 11578239 | 632321 | 18939 | 43513 |

Switzerland | 8640059 | 428197 | 7067 | 311500 |

Canada | 37665739 | 528354 | 14597 | 438452 |

Brazil | 212216052 | 7366677 | 189264 | 6405356 |

Portugal | 10203379 | 383258 | 6343 | 308446 |

Austria | 8994761 | 347204 | 5745 | 315952 |

South Korea | 51259193 | 53533 | 756 | 37425 |

Russia | 145920447 | 2963688 | 53096 | 2370857 |

Pakistan | 219887256 | 465070 | 9668 | 417134 |

Country | Death rate | Infected rate | Recovered rate |
---|---|---|---|

USA | 0.017688482 | 0.055918237 | 0.584749489 |

Italy | 0.035240375 | 0.032484199 | 0.652353754 |

Germany | 0.017793309 | 0.018269832 | 0.727063664 |

France | 0.024564861 | 0.037980966 | 0.074406117 |

China | 0.053326275 | 6.04275E-05 | 0.943048827 |

Iran | 0.046212437 | 0.013923201 | 0.768002679 |

Turkey | 0.008979285 | 0.024291941 | 0.897735191 |

Belgium | 0.029951559 | 0.054612882 | 0.068814732 |

Switzerland | 0.016504086 | 0.0495595 | 0.727468899 |

Canada | 0.02762731 | 0.014027443 | 0.829845142 |

Brazil | 0.02569191 | 0.0347131 | 0.86950412 |

Portugal | 0.016550209 | 0.03756187 | 0.8047999 |

Austria | 0.016546468 | 0.038600692 | 0.909989516 |

South Korea | 0.01412213 | 0.001044359 | 0.699101489 |

Russia | 0.017915516 | 0.0203103 | 0.799968485 |

Pakistan | 0.020788268 | 0.002115038 | 0.896927344 |

Before the migration to and from Wuhan was cut off, the basic reproduction number in China was 5.6015. From 23 January to 26 January 2020, the basic reproduction number in China was 6.6037. From 27 January to 11 February 2020, the basic reproduction number outside Hubei province dropped below 1, but that in Hubei province remained 3.7732. Because of stricter controlling measures, especially after the initiation of the large-scale case-screening, the epidemic rampancy in Hubei has also been contained. The average basic reproduction number in Hubei province was 3.4094 as of 25 February 2020. The two uses of the _{0} are, to assess the ability of an infectious disease to invade the community (when the R0 of a disease is greater than 1, the infection will spread, as it indicates that one infected individual will spread the disease to more than one individual) and to determine the fraction of the community which should be vaccinated in order to prevent the growth of the epidemic [

The above results are based on some data collected by the 24th December 2020, but our results are expected by different countries to increase or decrease the rate of infection and recover which help to study the outbreak of COVID-19. Countries properly follow the precautionary measure and operate on it. They are many countries which contain the coronavirus, but most of countries increased the recovered rate efficiently but the risk of infection either fluctuates or stable after some period. The model chosen is easy but it gives the ton of valuable disease-related knowledge that will help prepare and handle the disease in society to reduce the risk of death.

In this paper, we developed an epidemic model analytical solution scheme and formulated the reproductive number in order to observe the state of the coronavirus. The well-known Susceptible-Infected-Recovered (SIR) epidemic model is considered with and without demographic results. It is found that infection and recovered rate play a key role for an epidemic to occur, and that vaccination will regulate the epidemic. The effect of parameter estimation on our obtained solutions is expressed by figures. The numerical simulations show the actual behavior of the COVID-19 in a time t which will be helpful to analyze and control this pandemic disease. This model will assist the public health planar in framing a COVID-19 control policy. In addition, we will expand the model incorporating deterministic and stochastic model comparisons with fractional technique, as well as using optimal control theory for new outcomes.