### Abstract

This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss-Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens' non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.

Original language | English (US) |
---|---|

Pages (from-to) | 131-150 |

Number of pages | 20 |

Journal | Mathematical and Computer Modelling |

Volume | 53 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- Epidemic model
- Equilibria
- Non-standard finite difference scheme
- Reproduction number
- Stability

### ASJC Scopus subject areas

- Computer Science Applications
- Modeling and Simulation

### Cite this

*Mathematical and Computer Modelling*,

*53*(1-2), 131-150. https://doi.org/10.1016/j.mcm.2010.07.026

**Dynamically-consistent non-standard finite difference method for an epidemic model.** / Garba, S. M.; Gumel, Abba; Lubuma, J. M S.

Research output: Contribution to journal › Article

*Mathematical and Computer Modelling*, vol. 53, no. 1-2, pp. 131-150. https://doi.org/10.1016/j.mcm.2010.07.026

}

TY - JOUR

T1 - Dynamically-consistent non-standard finite difference method for an epidemic model

AU - Garba, S. M.

AU - Gumel, Abba

AU - Lubuma, J. M S

PY - 2011/1

Y1 - 2011/1

N2 - This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss-Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens' non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.

AB - This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss-Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens' non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.

KW - Epidemic model

KW - Equilibria

KW - Non-standard finite difference scheme

KW - Reproduction number

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=77958483922&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958483922&partnerID=8YFLogxK

U2 - 10.1016/j.mcm.2010.07.026

DO - 10.1016/j.mcm.2010.07.026

M3 - Article

AN - SCOPUS:77958483922

VL - 53

SP - 131

EP - 150

JO - Mathematical and Computer Modelling

JF - Mathematical and Computer Modelling

SN - 0895-7177

IS - 1-2

ER -