Limiting Behaviors of Stochastic Spread Models Using Branching Processes
In this paper, we introduce a spread model using multi-type branching processes to investigate the evolution of the population during a pandemic in which individuals are classified into different types. We study some limiting behaviors of the population including the growth rate of the population an...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-06-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/7/652 |
| _version_ | 1797590308502896640 |
|---|---|
| author | Jyy-I Hong |
| author_facet | Jyy-I Hong |
| author_sort | Jyy-I Hong |
| collection | DOAJ |
| description | In this paper, we introduce a spread model using multi-type branching processes to investigate the evolution of the population during a pandemic in which individuals are classified into different types. We study some limiting behaviors of the population including the growth rate of the population and the spread rate of each type. In particular, the work in this paper focuses on the cases where the offspring mean matrices are non-primitive but can be decomposed into two primitive components, <i>A</i> and <i>B</i>, with maximal eigenvalues <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>A</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>B</mi></msub></semantics></math></inline-formula>, respectively. It is shown that the growth rate and the spread rate heavily depend on the conditions of these two maximal eigenvalues and are related to the corresponding eigenvectors. In particular, we find the spread rates for the case with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>B</mi></msub><mo>></mo><msub><mi>ρ</mi><mi>A</mi></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and the case with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>A</mi></msub><mo>></mo><msub><mi>ρ</mi><mi>B</mi></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. In addition, some numerical examples and simulations are also provided to support the theoretical results. |
| first_indexed | 2024-03-11T01:18:44Z |
| format | Article |
| id | doaj.art-8e3db977a2504f5d91678a82c103a5d0 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T01:18:44Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-8e3db977a2504f5d91678a82c103a5d02023-11-18T18:17:33ZengMDPI AGAxioms2075-16802023-06-0112765210.3390/axioms12070652Limiting Behaviors of Stochastic Spread Models Using Branching ProcessesJyy-I Hong0Department of Mathematical Sciences, National Chengchi University, Taipei 11605, TaiwanIn this paper, we introduce a spread model using multi-type branching processes to investigate the evolution of the population during a pandemic in which individuals are classified into different types. We study some limiting behaviors of the population including the growth rate of the population and the spread rate of each type. In particular, the work in this paper focuses on the cases where the offspring mean matrices are non-primitive but can be decomposed into two primitive components, <i>A</i> and <i>B</i>, with maximal eigenvalues <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>A</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>B</mi></msub></semantics></math></inline-formula>, respectively. It is shown that the growth rate and the spread rate heavily depend on the conditions of these two maximal eigenvalues and are related to the corresponding eigenvectors. In particular, we find the spread rates for the case with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>B</mi></msub><mo>></mo><msub><mi>ρ</mi><mi>A</mi></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and the case with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>A</mi></msub><mo>></mo><msub><mi>ρ</mi><mi>B</mi></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. In addition, some numerical examples and simulations are also provided to support the theoretical results.https://www.mdpi.com/2075-1680/12/7/652spread rategrowth ratestochastic spread modelbranching process |
| spellingShingle | Jyy-I Hong Limiting Behaviors of Stochastic Spread Models Using Branching Processes Axioms spread rate growth rate stochastic spread model branching process |
| title | Limiting Behaviors of Stochastic Spread Models Using Branching Processes |
| title_full | Limiting Behaviors of Stochastic Spread Models Using Branching Processes |
| title_fullStr | Limiting Behaviors of Stochastic Spread Models Using Branching Processes |
| title_full_unstemmed | Limiting Behaviors of Stochastic Spread Models Using Branching Processes |
| title_short | Limiting Behaviors of Stochastic Spread Models Using Branching Processes |
| title_sort | limiting behaviors of stochastic spread models using branching processes |
| topic | spread rate growth rate stochastic spread model branching process |
| url | https://www.mdpi.com/2075-1680/12/7/652 |
| work_keys_str_mv | AT jyyihong limitingbehaviorsofstochasticspreadmodelsusingbranchingprocesses |