Through the study, the nonlinear delayed modelling has vital significance in the different field of allied sciences like computational biology, computational chemistry, computational physics, computational economics and many more. Polio is a contagious viral illness that in its most severe form causes nerve injury leading to paralysis, difficulty breathing and sometimes death. In recent years, developing regions like Asia, Africa and sub-continents facing a dreadful situation of poliovirus. That is the reason we focus on the treatment of the polio epidemic model with different delay strategies in this article. Polio delayed epidemic model is categorized into four compartments like susceptible, exposed, infective and vaccinated classes. The equilibria, positivity, boundedness, and reproduction number are investigated. Also, the sensitivity of the parameters is analyzed. Well, known results like the Routh Hurwitz criterion and Lyapunov function stabilities are investigated for polio delayed epidemic model in the sense of local and global respectively. Furthermore, the computer simulations are presented with different traditions in the support of the analytical analysis of the polio delayed epidemic model.

Poliomyelitis is a disease which is most commonly known as Polio, this virus comes from the family of Picornaviridae. Polio is an extremely contagious disease that can incapacitate a person and can be a life-threatening disease. It’s a type of virus that can affect a person at any age but most victims are children who fall under the age of three. This virus feast through the mouth and attacks the nervous system of a person, instigating irreversible paralysis within no time. The initial symptom includes vomiting, headache, fever, fatigue, limb pain, and stiff neck. Poliovirus has three wild types (WPV)-known as type 1, type 2 and type 3. Among these three types, type 1 (WPV1) is still circulating causing permanent paralysis to the victims in Pakistan, Afghanistan and Nigeria. The best defence against this disease is Polio Vaccine which can protect people from getting infected. The usage of these vaccines is prone to its cost as Oral Poliovirus Vaccine (OPV) is economical as compared to Inactivated Poliovirus Vaccine (IPV), it multiplies in the gut of the legatee of exorbitant paramount intestinal immunity than IPV thus more worthwhile in preventing the wild virus from refraining Mission. Recently, due to its serious consequences, significant consideration has been made to observing the vaccine-derived poliovirus (VDPV). Due to the silence and frequent circulation of the vaccine-derived poliovirus (VDPV), poliovirus (PV) surveillance is one of the problematic situations with poor vaccination coverage. Regardless of the aforementioned continual intestinal and ridiculous exceptions, VDPV is natively unstable and can reappear to wild-type wireless according to several research and studies, the vaccine can inflict vaccine-associated flaccid tetraplegia. Substantially, it can produce long-term replicas in conjunction with suboptimal vaccinations. The reciprocity of PV and CD155 receptors accelerate its penetration. After that, the viral RNA is emancipated. The host cell translates the genome attached to the viral particle; it is used as mRNA. For about 20 years, there are no reported cases of Serotype2 wild poliovirus in the world. The last documented case attendant was in 1999 with the naturally happening of serotype (WPV2), with a certificate of global extinction in 2015. For stereotype 3 virus (WPV3), the last reported case was documented in 2012. In April 2016, GPEI executed a globally compatible switch with the trivial poliovirus vaccine. The polio vaccine will not be passed indefinitely in the post-vaccination period. The committee has made a significant difference in our understanding of the strain transfer of oral polio vaccines and their ability to survive in populations with low immunity. As the global polio eradication initiative (GPEI) moves towards the obliteration of the ferocious poliovirus, civil and worldwide health leaders should motionless actively appraise options for supervising the menace of the poliovirus, additionally the use of oral polio vaccine [

In order to analyze the spread of the polio disease more naturally polio delayed epidemic model is presented. In which we divided the whole population

The others parameters of the model are defined as,

Subject to non-negative initial conditions S =

In this section, we will discuss the positivity and boundedness of the polio delayed epidemic model. For the sake of significant analysis of the model, all the variables

For positivity and boundedness, we will use the following results.

Which is the desired result.

By the Gronwall’s inequality, we get

Which shows that the solution of the system (1–4) is bounded and lies in the feasible region

Equilibria of the polio delayed model will be discussed in this section like polio trivial equilibrium point

The most important factor of the model is the reproduction number denoted by

where,

Consider,

By next generation matrix method, it can be concluded that the spectrum radius of

Parameters used in the model have substantial role in the dynamics of the disease. In this section we present the sensitivity analysis of the parameters as follows:

From the above calculations it can be concluded that

In this section local stability of the polio delayed epidemic model will be checked at the equilibrium points of the model using the following well-known results.

The Jacobean matrix evaluated at polio trivial equilibrium point (PTE−

By evaluating

where

After solving the above determinant, the following Eigen values are found

where

And

So, by Routh-Hurwitz Criterion for 2^{nd} degree polynomial, both fixed values of

Otherwise, if

where

So, by Routh-Hurwitz Criterion for 3^{rd} degree polynomial, the following constraint has been verified if,

Hence, the polio existing equilibria (PEE-

Well known results are presented for the stability of polio delayed epidemic model in the sense of globally as follows:

From the above calculations, it can be observed that

It can be concluded from the above equation that

In this section, the numerical treatments of polio delayed epidemic model are presented by using the values of the parameters as discussed in

Parameters | Values/days |
---|---|

0.5 | |

1.002 (PFE) | |

2.002 (DEE) | |

0.6 | |

0.6 | |

0.01 | |

0.9 | |

0.05 | |

0.5 |

In this section, we taken the effect of the system (1–4), at the diarrhea existing equilibrium of the model with effective use of artificial delay tactics.

Let,

For the different values of τ, exhibits that infective class of peoples reduces even completely die out from population. Subsequently,

In this section, we plotted two-dimensional behavior of the model with the effective uses of artificial delay techniques. Even though, theses simulations are the interaction of susceptible class with remaining compartments of the model.

In this article, we investigated the dynamics of the polio delayed epidemic model with the effective use of delay strategies. In which we divided the entire population into four different groups which are susceptible, exposed, infected and vaccinated. We have calculated the reproduction number for the polio delayed epidemic model and also gave the sensitivity analysis of the parameters involved in reproduction number. We have also presented the local and global stabilities of the model at the equilibrium of the model, that is, at polio-free equilibrium and at polio existing equilibrium by using well-known results of mathematics. In addition, we carefully studied the effect of delay factor on infected class, exposed class and reproduction number and we have come to the conclusion that we can control the dynamics of polio by using different effected techniques of delay like vaccination is proved to be the ultimate solution to avoid the spread of this disease. Immunization is much important and is recommended for children’s who fall under the age group of 2 to 18 months. Moreover, Booster doses are recommended for children’s up to 12 years of age to make sure of immunization. And most importantly a good hygiene system at a personal level is much needed to adapt, which can play a vital role to reduce the spread of the poliovirus.