# Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation of the SVIR Model

## 3. Equilibria

## 4. Basic Reproduction Number

## 5. Local Stability

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

## 6. Global Stability

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

## 7. The Well-Known Runge–Kutta Method of Order Four

## 8. Numerical Results

#### 8.1. The Euler’s Method

#### 8.2. The Modified Euler Method

#### 8.3. Comparison between the Results of Euler, Modified Euler, and RK4 Method

## 9. Conclusions and Future Recommendations

- The concentration of $S\left(t\right)$ decreases, while $V\left(t\right),I\left(t\right)$, and $R\left(t\right)$ increase with an increasing rate of $\psi $.
- Increasing the values of ${\lambda}_{2}$, the population dynamics of $S\left(t\right),V\left(t\right)$ and $I\left(t\right),R\left(t\right)$ is observed to decrease and increase, respectively.
- The rate of saturation constant ${\xi}_{2}$ results in a decrease in the density of $S\left(t\right),I\left(t\right)$ while an increase in the density of $I\left(t\right)$ and $R\left(t\right)$.
- Decreasing the initial conditions has a decreasing effect on the population dynamics of $S\left(t\right)$, $V\left(t\right),I\left(t\right)$, and $R\left(t\right)$.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Palaniappan, R.U.M.; Ramanujam, S.; Chang, Y. Leptospirosis: Pathogenesis, immunity and diagnosis. J. Curr. Opin. Infect. Dis.
**2007**, 20, 284–292. [Google Scholar] [CrossRef] [PubMed] - Sanhueza, J.M.; Baker, M.G.; Benschop, J.; Collins-Emerson, J.M.; Wilson, P.R.; Heuer, C. Estimation of the burden of leptospirosis in New Zealand. Zoonoses Public Health
**2020**, 67, 167–176. [Google Scholar] [CrossRef] [PubMed] - Chadsuthi, S.; Bicout, D.J.; Wiratsudakul, A.; Suwancharoen, D.; Petkanchanapong, W.; Modchang, C.; Triampo, W.; Ratanakorn, P.; Chalvet-Monfray, K. Investigation on predominant Leptospira serovars and its distribution in humans and livestock in Thailand, 2010–2015. PLoS Neglected Trop. Dis.
**2017**, 11, e0005228. [Google Scholar] [CrossRef] - Inada, R.; Ido, Y. Etiology mode of infection and specific therapy of Weil’s disease. J. Exp. Med.
**1916**, 23, 377–402. [Google Scholar] [CrossRef] [PubMed] - Abdulkader, R.C.; Seguro, A.C.; Malheiro, P.S.; Burdmann, E.A.; Marcondes, M. Peculiar electrolytic and hormonal abnormalities in acute renal failure due to leptospirosis. Am. J. Trop. Med. Hyg.
**1996**, 54, 1–6. [Google Scholar] [CrossRef] - Arean, V.M.; Sarasin, G.; Green, J.H. The pathogenesis of leptospirosis: Toxin production by leptospira icterohaemorrhagiae. Am. J. Vet. Res.
**1964**, 28, 836–843. [Google Scholar] - Arean, V.M. Studies on the pathogenesis of leptospirosis. II, clinicopathologic evaluation of hepatic and renal function in experimental leptospiral infections. J. Lab. Investig.
**1962**, 11, 273–288. [Google Scholar] - Barkay, S.; Garzozi, H. Leptospirosis and uveitis. J. Ann. Ophthalmol.
**1984**, 16, 164–178. [Google Scholar] - Chitnis, N.; Smith, T.; Steketee, R. A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population. J. Biol. Dyn.
**2008**, 2, 259–285. [Google Scholar] [CrossRef] [Green Version] - Derouich, M.; Boutayeb, A. Dengue fever: Mathematical modelling and computer simulation. J. Appl. Math. Comput.
**2006**, 177, 528–544. [Google Scholar] [CrossRef] - Esteva, L.; Vargas, C. A model for dengue disease with variable human population. J. Math. Biol.
**1999**, 38, 220–240. [Google Scholar] [CrossRef] [PubMed] - Pongsuumpun, P.; Miami, T.; Kongnuy, R. Age structural transmission model for leptospirosis. In Proceedings of the 3rd International Symposium on Biomedical Engineering, Changsha, China, 8–10 June 2008; pp. 411–416. [Google Scholar]
- Triampo, W.; Baowan, D.; Tang, I.M.; Nuttavut, N.; Wong-Ekkabut, J.; Doungchawee, G. A simple deterministic model for the spread of leptospirosis in Thailand. Int. J. Biomed. Sci.
**2007**, 2, 22–26. [Google Scholar] - Zaman, G. Dynamical behavior of leptospirosis disease and role of optimal control theory. Int. J. Math. Comput.
**2010**, 7, 80–92. [Google Scholar] - Zaman, G.; Khan, M.A.; Islam, S.; Chohan, M.I.; Jung, I.H. Modeling dynamical interactions between leptospirosis infected vector and human population. J. Appl. Math. Sci.
**2012**, 6, 1287–1302. [Google Scholar] - Lashari, A.A.; Hattaf, K.; Zaman, G.; Li, X. Backward bifurcation and optimal control of a vector borne disease. J. Appl. Math. Inf. Sci.
**2013**, 7, 301–309. [Google Scholar] [CrossRef] - Rafizah, A.A.N.; Aziah, B.D.; Azwanyetal, Y.N. Risk factors of leptospirosis among febrile hospital admissions in northeastern Malaysia. J. Prev. Med.
**2013**, 57, S11–S13. [Google Scholar] [CrossRef] - Hattaf, K.; Lashari, A.A.; Louartassi, Y.; Yousfi, N. A delayed sir epidemic model with general incidence rate. Electron. J. Qual. Theory Differ. Equ.
**2016**, 3, 1–9. [Google Scholar] [CrossRef] - Lashari, A.A.; Hattaf, K.; Zaman, G. A delay differential equation model of a vector borne disease with direct transmission. Int. J. Ecol. Econ. Stat.
**2012**, 27, 25–35. [Google Scholar] - Typhoid Fever. Available online: http://www.en.wikipedia.org/wiki/Typhoid-fever (accessed on 20 August 2022).
- Hartley, D.M.; Morris, J.G.; Smith, D.L. Hyperinfectivity: A Critical Element in the Stability of Vector cholera Cause Epidemics? PLoS Med.
**2006**, 3, 63–69. [Google Scholar] - Crump, J.A.; Luby, S.P.; Mintz, E.D. The Global Burden of Typhoid Fever. Bull. World Health Organ.
**2004**, 82, 346–353. [Google Scholar] - World Health Organization Facts on HIV/AID. Available online: http://www.who.int/feature/factfile/hiv/en (accessed on 19 August 2022).
- Prathumwan, D.; Trachoo, K.; Maiaugree, W.; Chaiya, I. Preventing extinction in Rastrelliger brachysoma using an impulsive mathematical model. AIMS Math.
**2022**, 7, 1–24. [Google Scholar] [CrossRef] - Mahdy, A.M.S.; Lotfy, K.; Ismail, E.A.; El-Bary, A.; Ahmed, M.; El-Dahdouh, A.A. Analytical solutions of time-fractional heat order for a magneto-photothermal semiconductor medium with Thomson effects and initial stress. Results Phys.
**2020**, 18, 103174. [Google Scholar] [CrossRef] - Mahdy, A.M.S.; Lotfy, K.; El-Bary, A.; Sarhan, H.H. Effect of rotation and magnetic field on a numerical-refined heat conduction in a semiconductor medium during photo-excitation processes. Eur. Phys. J. Plus
**2021**, 136, 553. [Google Scholar] [CrossRef] - Alharbi, A.R.; Almatrafi, M.B.; Lotfy, K. Constructions of solitary travelling wave solutions for Ito integro-differential equation arising in plasma physics. Results Phys.
**2020**, 19, 103533. [Google Scholar] [CrossRef] - Mahdy, A.M.S.; Mohamed, M.S.; Lotfy, K.; Alhazmi, M.; El-Bary, A.A.; Raddadi, M.H. Numerical solution and dynamical behaviors for solving fractional nonlinear Rubella ailment disease model. Results Phys.
**2021**, 24, 104091. [Google Scholar] [CrossRef] - Mahdy, A.M.S.; Gepreel, K.A.; Lotfy, K.; El-Bary, A.A. A numerical method for solving the Rubella ailment disease model. Int. J. Mod. Phys. C
**2021**, 32, 2150097. [Google Scholar] [CrossRef] - Chaiya, I.; Trachoo, K.; Nonlaopon, K.; Prathumwan, D. The mathematical model for streptococcus suis infection in pig-human population with humidity effect. Comput. Mater. Contin.
**2022**, 71, 2981–2998. [Google Scholar] [CrossRef] - Chuchard, P.; Prathumwan, D.; Trachoo, K.; Maiaugree, W.; Chaiya, I. The SLI-SC Mathematical Model of African Swine Fever Transmission among Swine Farms: The Effect of Contaminated Human Vector. Axioms
**2022**, 11, 329. [Google Scholar] [CrossRef] - Prathumwan, D.; Chaiya, I.; Trachoo, K. Study of Transmission Dynamics of Streptococcus suis Infection Mathematical Model between Pig and Human under ABC Fractional Order Derivative. Symmetry
**2022**, 14, 2112. [Google Scholar] [CrossRef] - Dhar, J.; Sharma, A. The Role of the Incubation Period in a Disease Model. J. Appl. Math.
**2009**, 9, 146–153. [Google Scholar] - Dhar, J.; Sharma, A. The Role of Viral Infection in Phytoplankton Dynamics with the Inclusion of Incubation Class. J. Non-Linear Anal. Hybrid Syst.
**2010**, 4, 9–15. [Google Scholar] [CrossRef] - Sahu, G.P.; Dhar, J. Analysis of an SVEIS Epidemic Model with Partial Temporary Immunity and Saturation Incidence Rate. J. Appl. Math. Model
**2012**, 36, 908–923. [Google Scholar] [CrossRef] - Zhou, X.; Cui, J. Analysis of Stability and Bifurcation for an SEIV Epidemic Model with Vaccination and Non-Linear Incidence Rate. Non-Linear Dyn.
**2011**, 63, 639–653. [Google Scholar] [CrossRef] - Tharakaraman, K.; Jayaraman, A.; Raman, R.; Viswanathan, K.; Stebbins, N.; Johnson, D.; Shriver, Z.; Sasisekharan, V.; Sasisekharan, R. Glycan Receptor Binding of the Influenza a Virus H7N9 Hemagglutinin. Cell
**2013**, 153, 1486–1493. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tharakaraman, K.; Raman, R.; Viswanathan, K.; Stebbins, N.; Jayaraman, A.; Krishnan, A.; Sasisekharan, V.; Sasisekharan, R. Structural Determinants for Naturally Evolving H5N1 Hemagglutinin to Switch its Receptor Specicity. Cell
**2013**, 153, 1475–1485. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shim, E. A Note on Epidemic Models with Infective Immigrants and Vaccination. J. Math. Biosci.
**2006**, 3, 557–566. [Google Scholar] - WHO. Shaping the Future, the World Health Report; WHO: Geneva, Switzerland, 2003. [Google Scholar]
- WHO. Diet, Nutrition and the Prevention of Chronic Diseases; WHO Technical Report Series 916; WHO: Geneva, Switzerland, 2003. [Google Scholar]
- WHO–IARC. Biennial Report (2002–2003); IARC: Lyon, France, 2003. [Google Scholar]
- Boutayeb, A.; Boutayeb, S. The burden of noncommunicable diseases in developing countries. Int. J. Equity Health
**2005**, 4, 2. [Google Scholar] [CrossRef] - Parkin, D.M. Global cancer statistics. J. Clin. Cancer
**1999**, 3364, 49. [Google Scholar] [CrossRef] - Aron, J.L.; May, R.M. The population dynamics of Malaria. In Population Dynamics of Infectious Diseases; Chapman and Hall: London, UK, 1982; pp. 139–179. [Google Scholar]
- World Health Organization. Dengue. 2010. Available online: http://www.who.int/topics/dengue/en/ (accessed on 19 August 2022).
- Hopp, M.J.; Foley, J.A. Global-scale relationships between climate and the dengue fever vector, Aedes Aegypti. Clim. Chang.
**2001**, 48, 441–463. [Google Scholar] [CrossRef] - Elbasha, E.H.; Galvani, A.P. Vaccination against multiple HPV types. J. Math. Biosci.
**2005**, 197, 88–117. [Google Scholar] [CrossRef] - World Health Organization. Dengue and Severe Dengue. 2010. Available online: https://www.who.int/news-room/fact-sheets/detail/dengue-and-severe-dengue (accessed on 19 August 2022).
- Rodrigues, H.S.; Teresa, M.; Monteiro, T.; Torres, D.F.M. Dengue in Cape Verde: Vector control and vaccination. J. Math. Popul. Stud.
**2013**, 20, 208–223. [Google Scholar] [CrossRef] - Anderson, M.; May, M.; Anderson, B. Infectious Diseases of Humans: Dynamics and Control; Wiley Online Library: Hoboken, NJ, USA, 1992; Volume 28. [Google Scholar]
- Kermack, W.O.; McKendrick, A.G. Contributions to the mathematical theory of epidemics-II, the problem of endemicity. Proc. R. Soc. Lond. Ser.-A
**1932**, 138, 55–83. [Google Scholar] - Attaullah; Tufail Khan, M.; Alyobi, S.; Yassen, M.F.; Prathumwan, D. A Computational Approach to a Model for HIV and the Immune System Interaction. Axioms
**2022**, 11, 578. [Google Scholar] [CrossRef] - Amin, R.; Yuzbasi, S.; Nazir, S. Efficient Numerical Scheme for the Solution of HIV Infection CD4 (+) T-Cells Using Haar Wavelet Technique. CMES-Comput. Model. Eng. Sci.
**2022**, 131, 639–653. [Google Scholar] [CrossRef] - Laarabi, H.; Abta, A.; Hattaf, K. Optimal control of delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor.
**2014**, 63, 87–97. [Google Scholar] [CrossRef] [PubMed] - Bell, S.K.; Mcmickens, C.L.; Selby, K.J. Biographies of Disease: AIDS; Greenwood Press: Westport, CT, USA, 2011. [Google Scholar]
- Zhao, Y.; Jiang, D. The behavior of an SVIR epidemic model with stochastic perturbation. Abstr. Appl. Anal.
**2014**, 2014, 742730. [Google Scholar] [CrossRef] [Green Version] - Zhao, K.; Ma, S. Qualitative analysis of a two-group SVIR epidemic model with random effect. Adv. Differ. Equ.
**2021**, 2021, 172. [Google Scholar] [CrossRef] - Djilali, S.; Bentout, S. Global dynamics of SVIR epidemic model with distributed delay and imperfect vaccine. Results Phys.
**2021**, 25, 104245. [Google Scholar] [CrossRef] - Wang, C.; Fan, D.; Xia, L.; Yi, X. Global stability for a multi-group SVIR model with age of vaccination. Int. J. Biomath.
**2018**, 11, 1850068. [Google Scholar] [CrossRef] - Xing, Y.; Li, H.X. Almost periodic solutions for a SVIR epidemic model with relapse. Math. Biosci. Eng.
**2021**, 18, 7191–7217. [Google Scholar] [CrossRef] - Wang, K. A Simple Approach for the Fractal Riccati Differential Equation. J. Appl. Comput. Mech.
**2021**, 7, 177–181. [Google Scholar] [CrossRef] - He, J.-H. Taylor series solution for a third order boundary value problem arising in Architectural Engineering. Ain Shams Eng. J.
**2020**, 11, 1411–1414. [Google Scholar] [CrossRef]

**Figure 2.**The population dynamics of (

**a**) $S\left(t\right),$ (

**b**) $V\left(t\right),$ (

**c**) $I\left(t\right)$, and (

**d**) $R\left(t\right)$ for different values of ‘$\psi $’ and (

**e**) $S\left(t\right),$ (

**f**) $V\left(t\right),$ with different values of ‘${\lambda}_{2}$’.

**Figure 3.**Numerical simulations of the model variables (

**a**) $S\left(t\right),$ (

**b**) $V\left(t\right),$ (

**c**) $I\left(t\right)$, and (

**d**) $R\left(t\right)$ for different values of ‘${\lambda}_{2}$’ and (

**e**) $S\left(t\right),$ (

**f**) $V\left(t\right),$ with different values of ‘${\xi}_{2}$’.

**Figure 4.**The population dynamics of (

**a**) $S\left(t\right),$ (

**b**) $V\left(t\right),$ (

**c**) $I\left(t\right)$, and (

**d**) $R\left(t\right)$ for different values of ‘${\xi}_{2}$’ and (

**e**) different values of the initial condition.

**Figure 5.**Graphical comparison of RK4-method and Euler method for (

**a**) $S\left(t\right),$ (

**b**) $V\left(t\right),$ (

**c**) $I\left(t\right)$, and (

**d**) $R\left(t\right)$. Numerical comparison of RK-4 method and modified Euler method for (

**e**) $S\left(t\right),$ (

**f**) $V\left(t\right)$, (

**g**) $I\left(t\right)$, and (

**h**) $R\left(t\right)$.

Parameters | Explanation | Values |
---|---|---|

$S\left(0\right)$ | Susceptible individuals who can contract the disease | 90 per day |

$V\left(0\right)$ | Vaccinated individuals who are vaccinated | 25 per day |

$I\left(0\right)$ | Infected individuals that have capacity to spread sickness to others | 30 per day |

$R\left(0\right)$ | Recovered individuals who have acquired immunity | 18 per day |

$\alpha \beta $ | Population recruitment rate | $0.00018$ per day |

$\mu $ | The fraction of individuals to be vaccinated | 0 per day |

$\alpha $ | Natural death rate | $0.09$ per day |

${\lambda}_{1}$ | The disease contact rate | $0.0002$ per day |

${\lambda}_{2}$ | The interaction between vaccinated and infected | $0.09$ per day |

$\gamma $ | Waning of vaccine | $0.01$ per day |

$\varphi $ | Recovery rate | $0.01$ per day |

$\eta $ | The individuals who needs vaccination | $0.009$ per day |

$\psi $ | The disease included death rate | $0.02$ per day |

${\xi}_{1}$ | Reflects the effect of vaccine reducing the infection rate | $0.05$ per day |

${\xi}_{2}$ | The saturation constant | $0.3$ per day |

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $90.000000000000000$ | $90.000000000000000$ | $0.000000000000000$ |

$0.1$ | $89.034762948033872$ | $89.029617999999999$ | $0.005144948033873$ |

$0.2$ | $88.079745470036670$ | $88.069561229557578$ | $0.010184240479092$ |

$0.3$ | $87.134842868778534$ | $87.119723407963733$ | $0.015119460814802$ |

$0.4$ | $86.199951484273271$ | $86.179999316198234$ | $0.019952168075037$ |

$0.5$ | $85.274968681238036$ | $85.250284783712715$ | $0.024683897525321$ |

$0.6$ | $84.359792836886470$ | $84.330476675573507$ | $0.029316161312963$ |

$0.7$ | $83.454323329042992$ | $83.420472879951788$ | $0.033850449091204$ |

$0.8$ | $82.558460524566343$ | $82.520172295948527$ | $0.038288228617816$ |

$0.9$ | $81.672105768070509$ | $81.629474821741852$ | $0.042630946328657$ |

$1.0$ | $80.795161370931382$ | $80.748281343044667$ | $0.046880027886715$ |

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $25.000000000000000$ | $25.000000000000000$ | $0.000000000000000$ |

$0.1$ | $24.596395992509390$ | $24.592500000000001$ | $0.003895992509388$ |

$0.2$ | $24.200511635620451$ | $24.192824725287501$ | $0.007686910332950$ |

$0.3$ | $23.812237223402949$ | $23.800865148540041$ | $0.011372074862908$ |

$0.4$ | $23.431461615664698$ | $23.416510567635029$ | $0.014951048029669$ |

$0.5$ | $23.058072446901463$ | $23.039648827631819$ | $0.018423619269644$ |

$0.6$ | $22.691956326423597$ | $22.670166533866979$ | $0.021789792556618$ |

$0.7$ | $22.332999029641439$ | $22.307949256089849$ | $0.025049773551590$ |

$0.8$ | $21.981085680526895$ | $21.952881723606147$ | $0.028203956920748$ |

$0.9$ | $21.636100925301140$ | $21.604848011435209$ | $0.031252913865931$ |

$1.0$ | $21.297929097427900$ | $21.263731717520841$ | $0.034197379907059$ |

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $30.000000000000000$ | $30.000000000000000$ | $0.000000000000000$ |

$0.1$ | $29.980146014399079$ | $29.982900000000001$ | $0.002753985600922$ |

$0.2$ | $29.954862429909038$ | $29.960256183154929$ | $0.005393753245890$ |

$0.3$ | $29.924267607139907$ | $29.932187033704771$ | $0.007919426564865$ |

$0.4$ | $29.888480809334528$ | $29.898812164962287$ | $0.010331355627759$ |

$0.5$ | $29.847621999039390$ | $29.860252102516966$ | $0.012630103477576$ |

$0.6$ | $29.801811643920956$ | $29.816628076688705$ | $0.014816432767748$ |

$0.7$ | $29.751170531724419$ | $29.768061824282324$ | $0.016891292557904$ |

$0.8$ | $29.695819594336569$ | $29.714675399653530$ | $0.018855805316960$ |

$0.9$ | $29.635879740882331$ | $29.656590995059311$ | $0.020711254176980$ |

$1.0$ | $29.571471699755335$ | $29.593930770231555$ | $0.022459070476220$ |

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $18.000000000000000$ | $18.000000000000000$ | $0.000000000000000$ |

$0.1$ | $17.868582780067243$ | $17.867999999999999$ | $0.000582780067244$ |

$0.2$ | $17.738320527288252$ | $17.737170899999999$ | $0.001149627288253$ |

$0.3$ | $17.609197547275553$ | $17.607496618083154$ | $0.001700929192399$ |

$0.4$ | $17.481198404554231$ | $17.478961335554111$ | $0.002237069000120$ |

$0.5$ | $17.354307921038636$ | $17.351549495699086$ | $0.002758425339550$ |

$0.6$ | $17.228511174324886$ | $17.225245802340311$ | $0.003265371984575$ |

$0.7$ | $17.103793495809906$ | $17.100035218195938$ | $0.003758277613969$ |

$0.8$ | $16.980140468647647$ | $16.975902963056456$ | $0.004237505591192$ |

$0.9$ | $16.857537925552993$ | $16.852834511788600$ | $0.004703413764393$ |

$1.0$ | $16.735971946463639$ | $16.730815592177564$ | $0.005156354286076$ |

**Table 6.**Comparison between the results of the modified Euler method and RK4 method for $S\left(t\right)$.

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $90.000000000000000$ | $90.000000000000000$ | $0.000000000000000$ |

$0.1$ | $89.034762948033872$ | $89.034780614778782$ | $0.000017666744910$ |

$0.2$ | $88.079745470036670$ | $88.079780449716438$ | $0.000034979679768$ |

$0.3$ | $87.134842868778534$ | $87.134894813323911$ | $0.000051944545376$ |

$0.4$ | $86.199951484273271$ | $86.200020051221500$ | $0.000068566948229$ |

$0.5$ | $85.274968681238036$ | $85.275053533603440$ | $0.000084852365404$ |

$0.6$ | $84.359792836886470$ | $84.359893643035804$ | $0.000100806149334$ |

$0.7$ | $83.454323329042992$ | $83.454439762575433$ | $0.000116433532440$ |

$0.8$ | $82.558460524566343$ | $82.558592264197998$ | $0.000131739631655$ |

$0.9$ | $81.672105768070509$ | $81.672252497523374$ | $0.000146729452865$ |

$1.0$ | $80.795161370931382$ | $80.795322778826517$ | $0.000161407895135$ |

**Table 7.**Comparison between the results of the modified Euler method and RK4 method for $V\left(t\right)$.

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $25.000000000000000$ | $25.000000000000000$ | $0.000000000000000$ |

$0.1$ | $24.596395992509390$ | $24.596412362643751$ | $0.000016370134361$ |

$0.2$ | $24.200511635620451$ | $24.200544044586529$ | $0.000032408966078$ |

$0.3$ | $23.812237223402949$ | $23.812285327914147$ | $0.000048104511198$ |

$0.4$ | $23.431461615664698$ | $23.431525061732632$ | $0.000063446067934$ |

$0.5$ | $23.058072446901463$ | $23.058150871082471$ | $0.000078424181009$ |

$0.6$ | $22.691956326423597$ | $22.692049357026683$ | $0.000093030603086$ |

$0.7$ | $22.332999029641439$ | $22.333106287895113$ | $0.000107258253674$ |

$0.8$ | $21.981085680526895$ | $21.981206781702834$ | $0.000121101175939$ |

$0.9$ | $21.636100925301140$ | $21.636235479792926$ | $0.000134554491787$ |

$1.0$ | $21.297929097427900$ | $21.298076711783498$ | $0.000147614355598$ |

**Table 8.**Comparison between the results of the modified Euler method and RK4 method for $I\left(t\right)$.

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $30.000000000000000$ | $30.000000000000000$ | $0.000000000000000$ |

$0.1$ | $29.980146014399079$ | $29.980128091577463$ | $0.000017922821616$ |

$0.2$ | $29.954862429909038$ | $29.954827092314897$ | $0.000035337594142$ |

$0.3$ | $29.924267607139907$ | $29.924215370430119$ | $0.000052236709788$ |

$0.4$ | $29.888480809334528$ | $29.888412195475798$ | $0.000068613858730$ |

$0.5$ | $29.847621999039390$ | $29.847537535051817$ | $0.000084463987573$ |

$0.6$ | $29.801811643920956$ | $29.801711860665900$ | $0.000099783255056$ |

$0.7$ | $29.751170531724419$ | $29.751055962739020$ | $0.000114568985399$ |

$0.8$ | $29.695819594336569$ | $29.695690774716830$ | $0.000128819619739$ |

$0.9$ | $29.635879740882331$ | $29.635737206216305$ | $0.000142534666026$ |

$1.0$ | $29.571471699755335$ | $29.571315985107560$ | $0.000155714647775$ |

**Table 9.**Comparison between the results of the modified Euler method and RK4 method for $R\left(t\right)$.

${\mathit{t}}_{\mathit{i}}$ | Euiler Method | RK4 Method | Absolute Errors |
---|---|---|---|

$0.0$ | $18.000000000000000$ | $18.000000000000000$ | $0.000000000000000$ |

$0.1$ | $17.868582780067243$ | $17.868585450000001$ | $0.000002669932758$ |

$0.2$ | $17.738320527288252$ | $17.738325782082203$ | $0.000005254793951$ |

$0.3$ | $17.609197547275553$ | $17.609205303340765$ | $0.000007756065212$ |

$0.4$ | $17.481198404554231$ | $17.481208579809305$ | $0.000010175255074$ |

$0.5$ | $17.354307921038636$ | $17.354320434934280$ | $0.000012513895644$ |

$0.6$ | $17.228511174324886$ | $17.228525947864242$ | $0.000014773539355$ |

$0.7$ | $17.103793495809906$ | $17.103810451565728$ | $0.000016955755822$ |

$0.8$ | $16.980140468647647$ | $16.980159530776440$ | $0.000019062128793$ |

$0.9$ | $16.857537925552993$ | $16.857559019806192$ | $0.000021094253199$ |

$1.0$ | $16.735971946463639$ | $16.735995000195960$ | $0.000023053732320$ |

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Attaullah; Khurshaid, A.; Zeeshan; Alyobi, S.; Yassen, M.F.; Prathumwan, D.
Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate. *Axioms* **2022**, *11*, 651.
https://doi.org/10.3390/axioms11110651

**AMA Style**

Attaullah, Khurshaid A, Zeeshan, Alyobi S, Yassen MF, Prathumwan D.
Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate. *Axioms*. 2022; 11(11):651.
https://doi.org/10.3390/axioms11110651

**Chicago/Turabian Style**

Attaullah, Adil Khurshaid, Zeeshan, Sultan Alyobi, Mansour F. Yassen, and Din Prathumwan.
2022. "Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate" *Axioms* 11, no. 11: 651.
https://doi.org/10.3390/axioms11110651