{"title":"Global well-posedness of infectious disease models without life-time immunity: the cases of cholera and avian influenza.","authors":"Kazuo Yamazaki","doi":"10.1093/imammb/dqx016","DOIUrl":null,"url":null,"abstract":"<p><p>We study the systems of partial differential equations with diffusion that model the dynamics of infectious diseases without life-time immunity, in particular the cases of cholera from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261) and avian influenza from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848). In both works, similarly to all others in the literature on various models of infectious diseases and more, it had to be assumed for a technical reason that the diffusivity coefficients of the susceptible, infected and recovered individuals, humans or birds, had to be identical in order to prove the existence of their unique solutions for all time. Considering that such uniform diffusivity strengths among the susceptible, infected and recovered hosts may not always be plausible in real world, we investigate the global well-posedness issue when such conditions are relaxed. In particular for the cholera model from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261), we prove the global well-posedness with no condition on the diffusivity coefficients at all. For the avian influenza model from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848), we prove the global well-posedness with no condition on the diffusivity coefficients if the spatial dimension is one, and under a partial condition that the diffusivity coefficients of the susceptible and the infected hosts are same otherwise.</p>","PeriodicalId":49863,"journal":{"name":"Mathematical Medicine and Biology-A Journal of the Ima","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqx016","citationCount":"13","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Medicine and Biology-A Journal of the Ima","FirstCategoryId":"99","ListUrlMain":"https://doi.org/10.1093/imammb/dqx016","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"BIOLOGY","Score":null,"Total":0}
引用次数: 13
Abstract
We study the systems of partial differential equations with diffusion that model the dynamics of infectious diseases without life-time immunity, in particular the cases of cholera from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261) and avian influenza from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848). In both works, similarly to all others in the literature on various models of infectious diseases and more, it had to be assumed for a technical reason that the diffusivity coefficients of the susceptible, infected and recovered individuals, humans or birds, had to be identical in order to prove the existence of their unique solutions for all time. Considering that such uniform diffusivity strengths among the susceptible, infected and recovered hosts may not always be plausible in real world, we investigate the global well-posedness issue when such conditions are relaxed. In particular for the cholera model from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261), we prove the global well-posedness with no condition on the diffusivity coefficients at all. For the avian influenza model from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848), we prove the global well-posedness with no condition on the diffusivity coefficients if the spatial dimension is one, and under a partial condition that the diffusivity coefficients of the susceptible and the infected hosts are same otherwise.
我们研究了具有扩散的偏微分方程系统,该系统模拟了没有终身免疫的传染病的动力学,特别是Wang & Wang (2015, J. Biol.)的霍乱病例。来自Vaidya等人的禽流感(2012,离散连续性)。直流发电机系统。爵士。[j] .农业工程学报,2017,28(2):429 - 448。在这两本著作中,与所有其他关于各种传染病模型的文献一样,出于技术原因,必须假设易感、受感染和康复个体(人类或鸟类)的扩散系数必须相同,以便证明它们始终存在唯一的解决方案。考虑到这种在易感、感染和恢复宿主之间的均匀扩散强度在现实世界中可能并不总是可信的,我们研究了放宽这些条件时的全局适定性问题。特别是Wang & Wang (2015, J. Biol.)的霍乱模型。Dyn., 9, 233-261),我们证明了扩散系数不设任何条件的全局适定性。对于Vaidya等人(2012,Discrete Contin)的禽流感模型。直流发电机系统。爵士。B, 17, 2829-2848),我们证明了在空间维数为1时扩散系数不设条件的全局适定性,以及在易感宿主和感染宿主的扩散系数相同的部分条件下。
期刊介绍:
Formerly the IMA Journal of Mathematics Applied in Medicine and Biology.
Mathematical Medicine and Biology publishes original articles with a significant mathematical content addressing topics in medicine and biology. Papers exploiting modern developments in applied mathematics are particularly welcome. The biomedical relevance of mathematical models should be demonstrated clearly and validation by comparison against experiment is strongly encouraged.
The journal welcomes contributions relevant to any area of the life sciences including:
-biomechanics-
biophysics-
cell biology-
developmental biology-
ecology and the environment-
epidemiology-
immunology-
infectious diseases-
neuroscience-
pharmacology-
physiology-
population biology