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The lack of treatment for poliomyelitis doing that only means of preventing is immunization with live oral polio vaccine (OPV) or/and inactivated polio vaccine (IPV). Poliomyelitis is a very contagious viral infection caused by
*poliovirus*. Children are principally attacked. In this paper, we assess the impact of vaccination in the control of spread of poliomyelitis via a deterministic SVEIR (Susceptible-Vaccinated-Latent-Infectious-Removed) model of infectious disease transmission, where vaccinated individuals are also susceptible, although to a lesser degree. Using Lyapunov-Lasalle methods, we prove the global asymptotic stability of the unique endemic equilibrium whenever
. Numerical simulations, using poliomyelitis data from Cameroon, are conducted to approve analytic results and to show the importance of vaccinate coverage in the control of disease spread.

In the 70s, having noticed that five million children died every year further to an avoidable disease by the vaccination like poliomyelitis, the WHO introduced the Global Immunization Vision and Strategy (GIVS). Poliomyelitis has been eliminated in the most of countries, but recently we observe the upsurge of infectious in some countries [

As part of the necessary multi-disciplinary research approach, mathematical models have been extensively used to provide a framework for understanding of poliomyelitis transmission dynamics and the best strategies to control the spread of infection in the human population. In the literature, considerable work can be found on the mathema- tical modeling of poliomyelitis [

Some SVEIR models are used to assess the potential impact of an imperfect SARS vaccine like SARS vaccine [

In this paper, we study the impact of vaccination in the control of poliomyelitis spread via an SVEIR model of infectious disease transmission. Individuals are classified as one of susceptible

In order to study the stability of a positive endemic equilibrium state, we use Lyapunov’s direct method and LaSalle’s Invariance Principle with a Lyapunov function of the form:

where

The main aim of the present paper is to show that our model has a unique endemic equilibrium which is globally asymptotically stable.

This SVEIR model could be used to assess the potential impact of an extended vaccination program (such as for the monovalent serogroup A conjugate MenVacAfric, an anti-meningococcal vaccine introduced in 2011 in Sub-saharan Africa), in order to compare with the impact of a pulse vaccination program.

In the next section, we present our SVEIR epidemic model. Section 3 presents some basic properties like the computation of the basic reproduction ration,

We divide the entire population into 5 sub-populations of epidemiological significance: susceptible, vaccinated, exposed, infective, and removed compartments with respective sizes

We assume mass action incidence

The average duration of latency in class

Our model consists of the following system of ordinary differential equations:

with initial conditions which satisfy

Since

with initial conditions which satisfy

In order that the model be well-posed, it is necessary that the state variables

where

Lemma 1. The compact set

Proof. For each of the variables

Let

Consequently,

Similarly,

Let

for

This holds for all

W

Since

It is easy to see that the model system (3) has a disease-free equilibrium

Additionally, an endemic equilibrium

Using the method of the references [

Replacing

When there is no vaccination (

From Equation (10), we claim the following result.

Proposition 1.

Proof. It follows from (11) that

Thus,

from which the result follows. W

The value of

Theorem 1. If

(See Appendix for proof).

For local stability of the disease-free equilibrium, we claim the following:

Theorem 2. If

Proof. The Jacobian matrix of model (3) evaluate at the disease-free equilibrium is given by

The eigenvalues of

The characteristic polynomial of

It clear that the roots of

The following result is proven in ( [

Theorem 3. If

If

Our main result is the following theorem.

Theorem 4. If

Proof. Consider the following candidate Lyapunov function

Differentiating

Since

and,

Since arithmetical mean is greater than geometrical mean, we have the following inequalities

Therefore

It remain to prove that

set

it’s clear that;

Backing to the above relations, we have the following implications.

If we set

Finally we have,

At the endemic equilibrium, we have

Replacing

If we compare relation (26) with the last equation of (25), then we have:

Consequently:

Finally

Thus, the largest invariant set contained in

Then the global stability of

In this section we show via numerical simulations that when

Most of parameters values are from Cameroon, like natural rate of mortality. We assume that the natural rates of mortality of susceptible, recovered, exposed are the same. Value of vaccine efficacy, recovery rate and rate of apparition of clinical symptoms are coming from WHO. For vaccination coverage, we take different values in order to explore different situations. The recruitment rate of susceptible humans,

In

Parameter | Description | Based line value or range |
---|---|---|

Recruitment rate of susceptible | 2.5 | |

Effective contact rate | 0.1 | |

Vaccine efficacy | 0.8 | |

Rate of development of clinical symptoms | 0.05 - 0.5 | |

Recovery rate | 0.05 | |

Vaccination coverage rate | ||

Natural mortality rate of susceptible | 0.0551 | |

Natural mortality rate of vaccinated | 0.0551 | |

Natural mortality rate of exposed | 0.0551 | |

Mortality rate of infectious | 0.08 | |

Natural mortality rate of recovered | 0.0551 |

In

We are in front of paralytic polio. We assume

1) even if the vaccine is perfect and nobody is vaccinated; the infection is and remains high in the population

2) The vaccination is made; even if the coverage is low infection decreases and reaches a an equilibrium point

3) The last and not realistic situation is that infection is eradicated after one year, and when we have perfect vaccine and maximal vaccination coverage

We highlighted in this article the importance of vaccination in the control of the propagation of the poliomyelitis. We relied on the compartmentalized SVEIR model that characterizes the infectious diseases. We computed

Using data from AHALA (district of Yaound in Cameroon), we simulated the three different forms of polio namely the minor illness, the meningitis form and the paralytic form. In the case of minor illness of polio, we assumed that

The first author acknowledges with thanks the High teacher Training College of Yaounde. H. A. would like to thank the Direction of the University Institute of Technology of Ngaoundere for their financial assistance in the context of research missions of September 13, 2016.

Nkamba, L.N., Nta- ganda, J.M., Abboubakar, H., Kamgang, J.C. and Castelli, L. (2017) Global Stability of a SVEIR Epidemic Model: Application to Po- liomyelitis Transmission Dynamics. Open Journal of Modelling and Simulation, 5, 98- 112. http://dx.doi.org/10.4236/ojmsi.2017.51008

Proof. In order to determine the existence of possible endemic equilibrium, that is, equilibrium with all positive components which we denote by

we have to look for the solution of the algebraic system of equations obtained by equating the right hand sides of system (3) to zero. In this way we obtain the implicit system of equations,

where

with

and

Note that coefficient

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