Stability and control in a stochastic model of malaria population dynamics
This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the de...
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| Published in | Advances in continuous and discrete models Vol. 2023; no. 1; p. 45 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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Cham
Springer International Publishing
15.11.2023
Springer Nature B.V |
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| ISSN | 2731-4235 1687-1839 2731-4235 1687-1847 |
| DOI | 10.1186/s13662-023-03791-3 |
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| Abstract | This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the deterministic model, stability of the disease-free equilibrium holds more generally for the stochastic model. The optimal control theory is applied to the stochastic model, revealing some important new insights. Theoretical results are illustrated by way of simulations. |
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| AbstractList | This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the deterministic model, stability of the disease-free equilibrium holds more generally for the stochastic model. The optimal control theory is applied to the stochastic model, revealing some important new insights. Theoretical results are illustrated by way of simulations. |
| ArticleNumber | 45 |
| Author | Maku Vyambwera, Sibaliwe Witbooi, Peter J. Muller, Grant E. van Schalkwyk, Garth J. |
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| Keywords | SDE model Basic reproduction number 92D30 Malaria Elimination 34H05 Exponential stability 34K20 |
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| References | Witbooi, Abiodun, van Schalkwyk, Ahmed (CR20) 2020; 2020 Otieno, Koske, Mutiso (CR15) 2016; 2016 Fatmawati, Tasman, Purwati, Herdicho, Chukwu (CR3) 2021; 2021 Ishikawa (CR7) 2012; 6 Mahrouf, Lotfi, Hattaf, Yousfi (CR13) 2023; 31 Lashari, Aly, Hattaf, Zaman, Jung, Li (CR10) 2012; 2012 Liu, Ikram, Khan, Din (CR11) 2022 Gray, Greenhalgh, Hu, Mao, Pan (CR5) 2011; 71 Liu, Jiang, Hayat, Alsaedi (CR12) 2018; 355 Witbooi (CR19) 2017; 2017 Mao (CR14) 1997 Jovanovic, Krstic (CR8) 2012; 36 Keno, Dano, Ganati (CR9) 2022; 2022 Vyambwera, Witbooi (CR17) 2018; 2018 El Fatini, Sekkak, Taki, El Guendouz (CR2) 2021; 29 Abiodun, Witbooi, Okosun (CR1) 2017; 10 Tumwiine, Mugisha, Luboobi (CR16) 2007; 189 Wang, Teng, Ji, Feng, Wang (CR18) 2019; 521 Fleming, Soner (CR4) 2006 Handari, Vitra, Ahya, Nadya S., Aldila (CR6) 2019; 2019 Witbooi, Muller, Van Schalkwyk (CR22) 2015; 2015 Witbooi, Muller, Ongansie, Ahmed, Okosun (CR21) 2022; 15 P. Liu (3791_CR11) 2022 X. Mao (3791_CR14) 1997 B.D. Handari (3791_CR6) 2019; 2019 M. Jovanovic (3791_CR8) 2012; 36 M. El Fatini (3791_CR2) 2021; 29 P.J. Witbooi (3791_CR22) 2015; 2015 M. Mahrouf (3791_CR13) 2023; 31 A. Gray (3791_CR5) 2011; 71 A.A. Lashari (3791_CR10) 2012; 2012 Q. Liu (3791_CR12) 2018; 355 G.J. Abiodun (3791_CR1) 2017; 10 P.J. Witbooi (3791_CR21) 2022; 15 G. Otieno (3791_CR15) 2016; 2016 S.M. Vyambwera (3791_CR17) 2018; 2018 Fatmawati (3791_CR3) 2021; 2021 W.H. Fleming (3791_CR4) 2006 M. Ishikawa (3791_CR7) 2012; 6 T.D. Keno (3791_CR9) 2022; 2022 L. Wang (3791_CR18) 2019; 521 P.J. Witbooi (3791_CR20) 2020; 2020 P.J. Witbooi (3791_CR19) 2017; 2017 J. Tumwiine (3791_CR16) 2007; 189 |
| References_xml | – volume: 2017 year: 2017 ident: CR19 article-title: An SEIRS epidemic model with stochastic transmission publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-017-1166-6 – volume: 2021 year: 2021 ident: CR3 article-title: An optimal control problem of malaria model with seasonality effect using real data publication-title: Commun. Math. Biol. Neurosci. – volume: 2015 year: 2015 ident: CR22 article-title: Vaccination control in a stochastic SVIR epidemic model publication-title: Comput. Math. Methods Med. doi: 10.1155/2015/271654 – year: 2022 ident: CR11 article-title: The measles epidemic model assessment under real statistics: an application of stochastic optimal control theory publication-title: Comput. Methods Biomech. Biomed. Eng. doi: 10.1080/10255842.2022.2050222 – volume: 2020 year: 2020 ident: CR20 article-title: Stochastic modeling of a mosquito-borne disease publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-020-02803-w – volume: 71 start-page: 876 issue: 3 year: 2011 end-page: 902 ident: CR5 article-title: A stochastic differential equation SIS epidemic models publication-title: J. Appl. Math. – volume: 31 start-page: 93 year: 2023 end-page: 111 ident: CR13 article-title: Non-pharmaceutical interventions and vaccination controls in a stochastic SIVR epidemic model publication-title: Differ. Equ. Dyn. Syst. doi: 10.1007/s12591-020-00538-4 – volume: 15 start-page: 441 issue: 2 year: 2022 end-page: 456 ident: CR21 article-title: A stochastic population model of cholera disease publication-title: Discrete Contin. Dyn. Syst., Ser. S doi: 10.3934/dcdss.2021116 – volume: 355 start-page: 8891 issue: 17 year: 2018 end-page: 8914 ident: CR12 article-title: Stationary distribution and extinction of a stochastic dengue epidemic model publication-title: J. Franklin Inst. doi: 10.1016/j.jfranklin.2018.10.003 – year: 1997 ident: CR14 publication-title: Stochastic Differential Equations and Applications – volume: 2022 year: 2022 ident: CR9 article-title: Optimal control and cost-effectiveness strategies of malaria transmission with impact of climate variability publication-title: J. Math. doi: 10.1155/2022/5924549 – volume: 29 start-page: 713 issue: 3 year: 2021 end-page: 729 ident: CR2 article-title: A control treatment for a stochastic epidemic model with relapse and Crowly-Martin incidence publication-title: J. Anal. doi: 10.1007/s41478-020-00276-4 – volume: 521 start-page: 435 year: 2019 end-page: 454 ident: CR18 article-title: Dynamical behaviors of a stochastic malaria model: a case study for Yunnan, China publication-title: Phys. A, Stat. Mech. Appl. doi: 10.1016/j.physa.2018.12.030 – volume: 2016 year: 2016 ident: CR15 article-title: Transmission dynamics and optimal control of malaria in Kenya publication-title: Discrete Dyn. Nat. Soc. doi: 10.1155/2016/8013574 – volume: 10 issue: 04 year: 2017 ident: CR1 article-title: Modeling and analyzing the impact of temperature and rainfall on mosquito population dynamics over KwaZulu-Natal, South Africa publication-title: Int. J. Biomath. doi: 10.1142/S1793524517500553 – volume: 2012 year: 2012 ident: CR10 article-title: Presentation of malaria epidemics using multiple optimal controls publication-title: J. Appl. Math. doi: 10.1155/2012/946504 – volume: 189 start-page: 1953 year: 2007 end-page: 1965 ident: CR16 article-title: A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity publication-title: Appl. Math. Comput. – volume: 36 start-page: 5214 year: 2012 end-page: 5228 ident: CR8 article-title: Stochastically perturbed vector-borne disease models with direct transmission publication-title: Appl. Math. Model. doi: 10.1016/j.apm.2011.11.087 – volume: 2019 year: 2019 ident: CR6 article-title: Optimal control in a malaria model: intervention of fumigation and bed nets publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-019-2424-6 – volume: 2018 year: 2018 ident: CR17 article-title: A stochastic TB model for a crowded environment publication-title: J. Appl. Math. doi: 10.1155/2018/3420528 – year: 2006 ident: CR4 publication-title: Controlled Markov Processes and Viscosity Solutions – volume: 6 start-page: 2153 issue: 8 year: 2012 end-page: 2160 ident: CR7 article-title: Optimal control problems for the stochastic sir model under vaccination publication-title: ICIC Express Lett., Part B, Appl. – volume: 189 start-page: 1953 year: 2007 ident: 3791_CR16 publication-title: Appl. Math. Comput. – volume: 521 start-page: 435 year: 2019 ident: 3791_CR18 publication-title: Phys. A, Stat. Mech. Appl. doi: 10.1016/j.physa.2018.12.030 – volume: 29 start-page: 713 issue: 3 year: 2021 ident: 3791_CR2 publication-title: J. Anal. doi: 10.1007/s41478-020-00276-4 – volume: 6 start-page: 2153 issue: 8 year: 2012 ident: 3791_CR7 publication-title: ICIC Express Lett., Part B, Appl. – volume: 31 start-page: 93 year: 2023 ident: 3791_CR13 publication-title: Differ. Equ. Dyn. Syst. doi: 10.1007/s12591-020-00538-4 – volume: 2018 year: 2018 ident: 3791_CR17 publication-title: J. Appl. Math. doi: 10.1155/2018/3420528 – volume: 2019 year: 2019 ident: 3791_CR6 publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-019-2424-6 – year: 2022 ident: 3791_CR11 publication-title: Comput. Methods Biomech. Biomed. Eng. doi: 10.1080/10255842.2022.2050222 – volume: 10 issue: 04 year: 2017 ident: 3791_CR1 publication-title: Int. J. Biomath. doi: 10.1142/S1793524517500553 – volume: 355 start-page: 8891 issue: 17 year: 2018 ident: 3791_CR12 publication-title: J. Franklin Inst. doi: 10.1016/j.jfranklin.2018.10.003 – volume: 2016 year: 2016 ident: 3791_CR15 publication-title: Discrete Dyn. Nat. Soc. doi: 10.1155/2016/8013574 – volume: 2012 year: 2012 ident: 3791_CR10 publication-title: J. Appl. Math. doi: 10.1155/2012/946504 – volume-title: Stochastic Differential Equations and Applications year: 1997 ident: 3791_CR14 – volume: 2021 year: 2021 ident: 3791_CR3 publication-title: Commun. Math. Biol. Neurosci. – volume-title: Controlled Markov Processes and Viscosity Solutions year: 2006 ident: 3791_CR4 – volume: 2022 year: 2022 ident: 3791_CR9 publication-title: J. Math. doi: 10.1155/2022/5924549 – volume: 15 start-page: 441 issue: 2 year: 2022 ident: 3791_CR21 publication-title: Discrete Contin. Dyn. Syst., Ser. S doi: 10.3934/dcdss.2021116 – volume: 2020 year: 2020 ident: 3791_CR20 publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-020-02803-w – volume: 2015 year: 2015 ident: 3791_CR22 publication-title: Comput. Math. Methods Med. doi: 10.1155/2015/271654 – volume: 71 start-page: 876 issue: 3 year: 2011 ident: 3791_CR5 publication-title: J. Appl. Math. – volume: 36 start-page: 5214 year: 2012 ident: 3791_CR8 publication-title: Appl. Math. Model. doi: 10.1016/j.apm.2011.11.087 – volume: 2017 year: 2017 ident: 3791_CR19 publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-017-1166-6 |
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| Snippet | This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem... |
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| SubjectTerms | Analysis Birth rate Control stability Control theory Dengue fever Difference and Functional Equations Differential equations Dynamic stability Equilibrium Functional Analysis Infections Malaria Mathematics Mathematics and Statistics Mortality Mosquitoes Optimal control Ordinary Differential Equations Partial Differential Equations Pathogens Per capita Stochastic models Theorems Tropical diseases |
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| Title | Stability and control in a stochastic model of malaria population dynamics |
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