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Modeling the effect of discrete delay in SIR epidemic model with logistic growth

  • Pankaj Kumar , Raminder Pal Singh and Dipesh
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Advances in Mathematical and Computational Sciences
This chapter is in the book Advances in Mathematical and Computational Sciences

Abstract

In this work, a discrete delayed SIR epidemic model with logistic growth is analyzed. A threshold value is obtained, which depicts the dynamics and outcomes of the disease. A basic reproductive number R0 is defined with this a disease-free and non-zero endemic equilibrium is discussed. A disease-free equilibrium is shown to be asymptotically stable for a given value of the delay parameter τ. The disease disappears when the basic fundamental reproductive number R0 < 1. For a certain value of the delay parameter τ, it is shown that the equilibrium E* is asymptotically stable and devoid of diseases.When R0 > 1, the disease persists, The stability switches are seen for τ τǂ 0 and Hopf bifurcation occurred when τ going beyond some critical point value. The delayed system for DFE (disease-free equilibrium) at R0 = 1, has also been shown to be linearly neutrally stable. We also analyze the sensitivity of the state variables with respect to the fundamental reproductive number R0. We conclude with some MATLAB-based numerical simulations to demonstrate the analytical findings.

Abstract

In this work, a discrete delayed SIR epidemic model with logistic growth is analyzed. A threshold value is obtained, which depicts the dynamics and outcomes of the disease. A basic reproductive number R0 is defined with this a disease-free and non-zero endemic equilibrium is discussed. A disease-free equilibrium is shown to be asymptotically stable for a given value of the delay parameter τ. The disease disappears when the basic fundamental reproductive number R0 < 1. For a certain value of the delay parameter τ, it is shown that the equilibrium E* is asymptotically stable and devoid of diseases.When R0 > 1, the disease persists, The stability switches are seen for τ τǂ 0 and Hopf bifurcation occurred when τ going beyond some critical point value. The delayed system for DFE (disease-free equilibrium) at R0 = 1, has also been shown to be linearly neutrally stable. We also analyze the sensitivity of the state variables with respect to the fundamental reproductive number R0. We conclude with some MATLAB-based numerical simulations to demonstrate the analytical findings.

Chapters in this book

  1. Frontmatter I
  2. About the book VII
  3. Preface IX
  4. Contents XI
  5. About the Editors XV
  6. Logarithmic third Hankel determinant for a subclass of starlike functions 1
  7. Effect of gyrotactic microorganisms in rotating magnetic nanofluids in porous media 17
  8. Some applications of graph spectrum in post quantum cryptography 31
  9. Finite-time stability analysis of neutral type delay Cohen–Grossberg neural network with discontinuous activation function 47
  10. Influence of temperature modulation over heat/mass transport of Rivlin–Ericksen viscoelastic nanoliquid in Hele-Shaw cell 57
  11. On an operator preserving polynomial inequality of Paul Turán 75
  12. Effect of interaction between two finite faults on ground deformation 85
  13. On the number of bounded regions generated by a class of paths in the complex plane 111
  14. Binomial and de Moivre revisited through differential equations 125
  15. Modified projection method for second kind Fredholm integral equations with trigonometric polynomials 131
  16. Effect of heat and mass transfer in binary nanofluids (for silver nanoparticles) saturated in porous medium under the influence of LTNE model and different gravity modulation 145
  17. A comparative analysis of standard Knapsack and Legendre–Knapsack cryptosystem 169
  18. Study of creep in composite functionally graded disc with non-linear thickness 179
  19. On the h(hv)-torsion tensor and the v-curvature tensor of Matsumoto change with an h-vector 189
  20. Software reliability growth model considering power-law testing effort using optimization on hybrid PSO-GWO algorithm 199
  21. On near-perfect numbers with five prime factors 207
  22. Eulerian integrals associated with the product of multivariable H-function and generalized M-series 221
  23. Upper bounds for the complex growth rates in the Darcy–Brinkman convection in a binary viscoelastic fluid in a saturated porous layer 233
  24. Some result of fixed-point theorems in complete M-fuzzy metric space 243
  25. FR-labeling on hybrid partition shell graphs 261
  26. On essential M-Artinian modules 269
  27. Properties of reductions of multiplication modules 275
  28. Computational analysis of fractional vibrational equations 285
  29. Analysis of modified fractional order Lotka–Volterra model 307
  30. Topological indices of two chemical compounds used for drugs in the treatment of H1N1 335
  31. On the Gaussian higher-order Mersenne numbers 355
  32. Petri net: modeling, properties and analysis techniques 367
  33. An inventory system with time-dependent linear demand for deteriorating items under uncertainty with a completely backlogged 379
  34. On ∗-reverse derivations of involutive rings 389
  35. More functions via soft pre∗-generalized closed sets 397
  36. Modeling of piston problem for generalized Chaplygin gas 409
  37. A new estimation of the degree of approximation of functions belonging to Lipschitz class by Borel–Euler summability method of Fourier series 421
  38. Modeling the effect of discrete delay in SIR epidemic model with logistic growth 429
  39. Simple harmonic motion in view of quantum calculus 455
  40. On Bernstein and Turán-type inequalities for polynomials 461
  41. A weak and strong form of Ng ∗ b-continuous function in nano topological space 473
  42. Securing networks with graph theory: an approach to cyber security 483
https://doi.org/10.1515/9783111313634-033

Chapters in this book

  1. Frontmatter I
  2. About the book VII
  3. Preface IX
  4. Contents XI
  5. About the Editors XV
  6. Logarithmic third Hankel determinant for a subclass of starlike functions 1
  7. Effect of gyrotactic microorganisms in rotating magnetic nanofluids in porous media 17
  8. Some applications of graph spectrum in post quantum cryptography 31
  9. Finite-time stability analysis of neutral type delay Cohen–Grossberg neural network with discontinuous activation function 47
  10. Influence of temperature modulation over heat/mass transport of Rivlin–Ericksen viscoelastic nanoliquid in Hele-Shaw cell 57
  11. On an operator preserving polynomial inequality of Paul Turán 75
  12. Effect of interaction between two finite faults on ground deformation 85
  13. On the number of bounded regions generated by a class of paths in the complex plane 111
  14. Binomial and de Moivre revisited through differential equations 125
  15. Modified projection method for second kind Fredholm integral equations with trigonometric polynomials 131
  16. Effect of heat and mass transfer in binary nanofluids (for silver nanoparticles) saturated in porous medium under the influence of LTNE model and different gravity modulation 145
  17. A comparative analysis of standard Knapsack and Legendre–Knapsack cryptosystem 169
  18. Study of creep in composite functionally graded disc with non-linear thickness 179
  19. On the h(hv)-torsion tensor and the v-curvature tensor of Matsumoto change with an h-vector 189
  20. Software reliability growth model considering power-law testing effort using optimization on hybrid PSO-GWO algorithm 199
  21. On near-perfect numbers with five prime factors 207
  22. Eulerian integrals associated with the product of multivariable H-function and generalized M-series 221
  23. Upper bounds for the complex growth rates in the Darcy–Brinkman convection in a binary viscoelastic fluid in a saturated porous layer 233
  24. Some result of fixed-point theorems in complete M-fuzzy metric space 243
  25. FR-labeling on hybrid partition shell graphs 261
  26. On essential M-Artinian modules 269
  27. Properties of reductions of multiplication modules 275
  28. Computational analysis of fractional vibrational equations 285
  29. Analysis of modified fractional order Lotka–Volterra model 307
  30. Topological indices of two chemical compounds used for drugs in the treatment of H1N1 335
  31. On the Gaussian higher-order Mersenne numbers 355
  32. Petri net: modeling, properties and analysis techniques 367
  33. An inventory system with time-dependent linear demand for deteriorating items under uncertainty with a completely backlogged 379
  34. On ∗-reverse derivations of involutive rings 389
  35. More functions via soft pre∗-generalized closed sets 397
  36. Modeling of piston problem for generalized Chaplygin gas 409
  37. A new estimation of the degree of approximation of functions belonging to Lipschitz class by Borel–Euler summability method of Fourier series 421
  38. Modeling the effect of discrete delay in SIR epidemic model with logistic growth 429
  39. Simple harmonic motion in view of quantum calculus 455
  40. On Bernstein and Turán-type inequalities for polynomials 461
  41. A weak and strong form of Ng ∗ b-continuous function in nano topological space 473
  42. Securing networks with graph theory: an approach to cyber security 483
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