TY  - JOUR
AU  - Younas, Hazrat
AU  - Ahmad, Imtiaz
AU  - Haq, Naseer Ul
AU  - Rahman, Azizur
AU  - Ullah, Hanif
PY  - 2026
DA  - 2026/09/30
TI  - Mathematical Modeling and NSFD Numerical Investigation of Rabies Transmission Dynamics with Immigration Factors
JO  - Journal of Numerical Simulations in Physics and Mathematics
T2  - Journal of Numerical Simulations in Physics and Mathematics
JF  - Journal of Numerical Simulations in Physics and Mathematics
VL  - 3
IS  - 1
SP  - 1
EP  - 18
DO  - 10.62762/JNSPM.2026.292872
UR  - https://www.icck.org/article/abs/JNSPM.2026.292872
KW  - rabies
KW  - stability analysis
KW  - RK-4 vs NSFD method
AB  - In this study, a compartmental mathematical model is developed to study the rabies transmission process from dog to human. The model consists of eight compartments: susceptible, exposed, infected, recovered groups for dogs and humans $(S_D, E_D, I_D, R_D, S_H, E_H, I_H, R_H)$. The basic reproduction number $R_0$ is derived, and sensitivity analysis is performed to identify the most influential epidemiological parameters. Disease-free equilibrium stability analysis is also carried out to identify the conditions for rabies eradication. One of the main goals of this work is to compare the disease dynamics in vaccinated and unvaccinated cases, which is carried out with the support of numerical simulations. Both the fourth-order Runge--Kutta (RK4) and Nonstandard Finite Difference (NSFD) methods are used to solve the system. Numerical simulations demonstrate that vaccination substantially reduces the infected populations and increases the partially immune populations in both species. Furthermore, the importance of the transmission rate $\beta_{DH}$ of the dog to human transmission is explored to gain insight into the role of this transmission route in the incidence of human rabies. In general, the results point to vaccination as a very potent control measure and show that NSFD method maintains the qualitative features of the system and yields results similar to the classical RK4 method.
SN  - 3068-9082
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Younas2026Mathematic,
  author = {Hazrat Younas and Imtiaz Ahmad and Naseer Ul Haq and Azizur Rahman and Hanif Ullah},
  title = {Mathematical Modeling and NSFD Numerical Investigation of Rabies Transmission Dynamics with Immigration Factors},
  journal = {Journal of Numerical Simulations in Physics and Mathematics},
  year = {2026},
  volume = {3},
  number = {1},
  pages = {1-18},
  doi = {10.62762/JNSPM.2026.292872},
  url = {https://www.icck.org/article/abs/JNSPM.2026.292872},
  abstract = {In this study, a compartmental mathematical model is developed to study the rabies transmission process from dog to human. The model consists of eight compartments: susceptible, exposed, infected, recovered groups for dogs and humans \$(S\_D, E\_D, I\_D, R\_D, S\_H, E\_H, I\_H, R\_H)\$. The basic reproduction number \$R\_0\$ is derived, and sensitivity analysis is performed to identify the most influential epidemiological parameters. Disease-free equilibrium stability analysis is also carried out to identify the conditions for rabies eradication. One of the main goals of this work is to compare the disease dynamics in vaccinated and unvaccinated cases, which is carried out with the support of numerical simulations. Both the fourth-order Runge--Kutta (RK4) and Nonstandard Finite Difference (NSFD) methods are used to solve the system. Numerical simulations demonstrate that vaccination substantially reduces the infected populations and increases the partially immune populations in both species. Furthermore, the importance of the transmission rate \$\beta\_{DH}\$ of the dog to human transmission is explored to gain insight into the role of this transmission route in the incidence of human rabies. In general, the results point to vaccination as a very potent control measure and show that NSFD method maintains the qualitative features of the system and yields results similar to the classical RK4 method.},
  keywords = {rabies, stability analysis, RK-4 vs NSFD method},
  issn = {3068-9082},
  publisher = {Institute of Central Computation and Knowledge}
}