Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Abstract

In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs). The basic reproduction number R0 of the resulting fractional model is computed and it is shown that if R0 is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for R0 and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and α-stability of the endemic equilibrium points. The proposed model is applied on Pertussis disease and the fractional nonlinear system of the model is solved by applying multi-step generalized differential transform method (MSGDTM). Our results show that historical effects play an important role on the disease spreading.

A fractional order SIR epidemic model with nonlinear incidence rate

“…In addition, the model converges rapidly to its steady state when the value of α is very small. This result was also observed in [20,29].…”

“…Here, we solve the nonlinear fractional system (2) by applying the numerical method presented in [25]. System (2) can be solved by other numerical methods for fractional differential equations [26][27][28][29].…”

Solutions of a disease model with fractional white noise

“…Now, we solve the system (14) numerically employing an Euler's type of numerical solution technique as follows:…”

“…These systems were generated by taking into account some facts such as duration of disease, availability and resistance against vaccination, immune systems of individuals in the population and so on. There are mathematical models employing deterministic [4][5][6][7] , stochastic [1][2][3] , fractional-order [8][9][10][11][12][13][14][15] system of differential equations. Almost each of these models were generated by compartmental models considering each compartment as individuals of susceptible (denoted by S), infected (I), exposed (E), and recovered (R) ones.…”

New aspects of poor nutrition in the life cycle within the fractional calculus

“…In [32], a new fractional epidemic model with vaccination was introduced by using the Caputo fractional derivative. Some other noticeable efforts have also been made in [33,34].…”