Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan
This paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that a person may lose the res...
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| Format: | Article |
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MDPI AG
2023-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/2/130 |
| _version_ | 1827757308400107520 |
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| author | Bing Guo Asad Khan Anwarud Din |
| author_facet | Bing Guo Asad Khan Anwarud Din |
| author_sort | Bing Guo |
| collection | DOAJ |
| description | This paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that a person may lose the resistance after vaccination, implying that vaccination might create temporary protection against the disease. Initially, the deterministic model is formulated, and then it has been expanded to a stochastic system, and it is well-founded that the stochastic model is both theoretically and practically viable by demonstrating that the model has a global solution, which is positive and stochastically confined. Next, we infer adequate criteria for the disease’s elimination and permanence. Furthermore, the presence of a stationary distribution is examined by developing an appropriate Lyapunov function, wherein we noticed that the disease will persist for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">R</mi><mrow><mn>0</mn></mrow><mrow><mi>s</mi><mo>></mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula> and that the illness will vanish from the community when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">R</mi><mrow><mn>0</mn></mrow><mrow><mi>s</mi><mo><</mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula>. We tested the model against the accessible data of measles in Pakistan during the first ten months of 2019, using the conventional curve fitting methods and the values of the parameters were calculated accordingly. The values obtained were employed in running the model, and the conceptual findings of the research were evaluated by simulations and conclusions were made. Simulations imply that, in order to fully understand the dynamic behavior of measles epidemic, time-delay must be included in such analyses, and that advancements in every vaccine campaign are inevitable for the control of the disease. |
| first_indexed | 2024-03-11T08:49:11Z |
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| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T08:49:11Z |
| publishDate | 2023-01-01 |
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| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-b85b9de2c2df43a891ac71f318ff322d2023-11-16T20:36:24ZengMDPI AGFractal and Fractional2504-31102023-01-017213010.3390/fractalfract7020130Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in PakistanBing Guo0Asad Khan1Anwarud Din2School of Mathematic and Statistics, Jishou University, Jishou 416000, ChinaSchool of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou 510006, ChinaDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaThis paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that a person may lose the resistance after vaccination, implying that vaccination might create temporary protection against the disease. Initially, the deterministic model is formulated, and then it has been expanded to a stochastic system, and it is well-founded that the stochastic model is both theoretically and practically viable by demonstrating that the model has a global solution, which is positive and stochastically confined. Next, we infer adequate criteria for the disease’s elimination and permanence. Furthermore, the presence of a stationary distribution is examined by developing an appropriate Lyapunov function, wherein we noticed that the disease will persist for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">R</mi><mrow><mn>0</mn></mrow><mrow><mi>s</mi><mo>></mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula> and that the illness will vanish from the community when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">R</mi><mrow><mn>0</mn></mrow><mrow><mi>s</mi><mo><</mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula>. We tested the model against the accessible data of measles in Pakistan during the first ten months of 2019, using the conventional curve fitting methods and the values of the parameters were calculated accordingly. The values obtained were employed in running the model, and the conceptual findings of the research were evaluated by simulations and conclusions were made. Simulations imply that, in order to fully understand the dynamic behavior of measles epidemic, time-delay must be included in such analyses, and that advancements in every vaccine campaign are inevitable for the control of the disease.https://www.mdpi.com/2504-3110/7/2/130stochastic measles epidemic modelstationary distributionparameter estimationreal dataPakistan measles outbreak |
| spellingShingle | Bing Guo Asad Khan Anwarud Din Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan Fractal and Fractional stochastic measles epidemic model stationary distribution parameter estimation real data Pakistan measles outbreak |
| title | Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan |
| title_full | Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan |
| title_fullStr | Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan |
| title_full_unstemmed | Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan |
| title_short | Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan |
| title_sort | numerical simulation of nonlinear stochastic analysis for measles transmission a case study of a measles epidemic in pakistan |
| topic | stochastic measles epidemic model stationary distribution parameter estimation real data Pakistan measles outbreak |
| url | https://www.mdpi.com/2504-3110/7/2/130 |
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