Pub Date : 2018-04-15DOI: 10.11145/J.BIOMATH.2018.04.127
M. Lachowicz
We discuss a class of mathematical models of biological systems at microscopic level - i.e. at the level of interacting individuals of a population. The class leads to partially integral stochastic semigroups- [5]. We state general conditions guaranteeing the asymptotic stability. In particular under some rather restrictive assumptions we observe that any, even non-factorized, initial probability density tends in the evolution to a factorized equilibrium probability density - [4]. We discuss possible applications of the general theory such as redistribution of individuals - [2], thermal denaturation of DNA [1], and tendon healing process - [3]. [1] M. Debowski, M. Lachowicz, and Z. Szymanska, Microscopic description of DNA thermal denaturation, to appear. [2] M. Dolfin, M. Lachowicz, and A. Schadschneider, A microscopic model of redistribution of individuals inside an 'elevator', In Modern Problems in Applied Analysis, P. Drygas and S. Rogosin (Eds.), Bikhauser, Basel (2018), 77--86; DOI: 10.1007/978--3--319--72640-3. [3] G. Dudziuk, M. Lachowicz, H. Leszczynski, and Z. Szymanska, A simple model of collagen remodeling, to appear. [4] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on--line, DOI: 10.1002/mma.4680 [5] K. Pichor and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Analysis Appl. 249 , 2000, 668--685, DOI: 10.1006/jmaa.2000.6968
{"title":"A class of individual-based models","authors":"M. Lachowicz","doi":"10.11145/J.BIOMATH.2018.04.127","DOIUrl":"https://doi.org/10.11145/J.BIOMATH.2018.04.127","url":null,"abstract":"We discuss a class of mathematical models of biological systems at microscopic level - i.e. at the level of interacting individuals of a population. The class leads to partially integral stochastic semigroups- [5]. We state general conditions guaranteeing the asymptotic stability. In particular under some rather restrictive assumptions we observe that any, even non-factorized, initial probability density tends in the evolution to a factorized equilibrium probability density - [4]. We discuss possible applications of the general theory such as redistribution of individuals - [2], thermal denaturation of DNA [1], and tendon healing process - [3]. [1] M. Debowski, M. Lachowicz, and Z. Szymanska, Microscopic description of DNA thermal denaturation, to appear. [2] M. Dolfin, M. Lachowicz, and A. Schadschneider, A microscopic model of redistribution of individuals inside an 'elevator', In Modern Problems in Applied Analysis, P. Drygas and S. Rogosin (Eds.), Bikhauser, Basel (2018), 77--86; DOI: 10.1007/978--3--319--72640-3. [3] G. Dudziuk, M. Lachowicz, H. Leszczynski, and Z. Szymanska, A simple model of collagen remodeling, to appear. [4] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on--line, DOI: 10.1002/mma.4680 [5] K. Pichor and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Analysis Appl. 249 , 2000, 668--685, DOI: 10.1006/jmaa.2000.6968","PeriodicalId":370233,"journal":{"name":"Biomath Communications Supplement","volume":"29 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114166588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-03-03DOI: 10.11145/TEXTS.2018.02.027
P. Hingley
When making statistical inferences about the means of small samples, the confidence limits for the mean are often calculated assuming a normal distribution. But many biological variables follow the lognormal distribution cite{Johnson}, for example the birth weights of babies (EG data at cite{Iannelli}). Here, sampling distributions (probability density functions) are found for the maximum likelihood estimates (MLE) of sample mean and variance when data are lognormally distributed. They are derived analytically, making some use of the Technique for Estimator Densities (TED) cite{Hingley}, and then checked by using simulations with random numbers. For an I.I.D. sample of size n with lognormal estimation, the sample mean has a lognormal distribution that is conditional on the variance. The distribution of the sample variance of the logarithms follows the usual transform of the central chi-squared distribution. The joint distribution of the sample mean and variance shows the extent to which the mean is affected by the variance. When a normal distribution is wrongly used for estimation on lognormally distributed data, the sample mean still has a lognormal distribution. But the distribution for the MLE of the variance differs. From the distribution for one observation, that for larger sample sizes can be approached by using convolutions. The assumption of a normal estimation model biases the confidence interval for the mean. There is a discussion of the extent to which this is of practical importance when estimating means for small samples of birth weights and other lognormally distributed data sets.
{"title":"Estimating the mean of a small sample under the two parameter lognormal distribution","authors":"P. Hingley","doi":"10.11145/TEXTS.2018.02.027","DOIUrl":"https://doi.org/10.11145/TEXTS.2018.02.027","url":null,"abstract":"When making statistical inferences about the means of small samples, the confidence limits for the mean are often calculated assuming a normal distribution. But many biological variables follow the lognormal distribution cite{Johnson}, for example the birth weights of babies (EG data at cite{Iannelli}). Here, sampling distributions (probability density functions) are found for the maximum likelihood estimates (MLE) of sample mean and variance when data are lognormally distributed. They are derived analytically, making some use of the Technique for Estimator Densities (TED) cite{Hingley}, and then checked by using simulations with random numbers. For an I.I.D. sample of size n with lognormal estimation, the sample mean has a lognormal distribution that is conditional on the variance. The distribution of the sample variance of the logarithms follows the usual transform of the central chi-squared distribution. The joint distribution of the sample mean and variance shows the extent to which the mean is affected by the variance. When a normal distribution is wrongly used for estimation on lognormally distributed data, the sample mean still has a lognormal distribution. But the distribution for the MLE of the variance differs. From the distribution for one observation, that for larger sample sizes can be approached by using convolutions. The assumption of a normal estimation model biases the confidence interval for the mean. There is a discussion of the extent to which this is of practical importance when estimating means for small samples of birth weights and other lognormally distributed data sets.","PeriodicalId":370233,"journal":{"name":"Biomath Communications Supplement","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134531803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-01-19DOI: 10.11145/TEXTS.2018.01.083
Salisu M. Garba, Usman Ahmed Danbaba
One of the major reason for the persistence of Zika and other vector borne diseases has been lack of effective mosquito control techniques. Sterile insect technique (SIT) is a non polluting biological method of mosquito control, where sterile mosquitoes are predominantly non reproductive. We present a new deterministic model for the transmission dynamics of Zika, by incorporating both human and mosquito population, with fraction of mosquitoes being sterilized. We consider both aquatic and non-aquatic stages of mosquitoes, so as to evaluate the effect of mosquito control in the transmission of the disease. We computed the basic reproduction number ($R_{0}$), and theoretically analysed the stability properties of the disease-free equilibrium (DFE) and the endemic-equilibrium (EE). In addition, effect of human-human transmission, and other important parameters were assessed. Numerical simulations to support the results will be presented.
{"title":"Analysis of model for the transmission dynamics of Zika with sterile insect technique","authors":"Salisu M. Garba, Usman Ahmed Danbaba","doi":"10.11145/TEXTS.2018.01.083","DOIUrl":"https://doi.org/10.11145/TEXTS.2018.01.083","url":null,"abstract":"One of the major reason for the persistence of Zika and other vector borne diseases has been lack of effective mosquito control techniques. Sterile insect technique (SIT) is a non polluting biological method of mosquito control, where sterile mosquitoes are predominantly non reproductive. We present a new deterministic model for the transmission dynamics of Zika, by incorporating both human and mosquito population, with fraction of mosquitoes being sterilized. We consider both aquatic and non-aquatic stages of mosquitoes, so as to evaluate the effect of mosquito control in the transmission of the disease. We computed the basic reproduction number ($R_{0}$), and theoretically analysed the stability properties of the disease-free equilibrium (DFE) and the endemic-equilibrium (EE). In addition, effect of human-human transmission, and other important parameters were assessed. Numerical simulations to support the results will be presented.","PeriodicalId":370233,"journal":{"name":"Biomath Communications Supplement","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124630001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-28DOI: 10.11145/TEXTS.2017.12.233
R. Anguelov, S. Stoltz
The theory of pattern formation through local self-activation and long range inhibition has been shown to account for much of the observed pattern forming regulatory phenomena [2]. This mechanism is captured mathematically by considering two species, activator and inhibitor, with different spatial diffusivity, so that the resulting model is a system of reaction diffusion equations. The formation of patterns occurring in such systems under certain conditions was discovered by Alan Turing in 1952. Independently of Turing's work, Gierer and Meinhard derived in 1972 their textit{Theory of Biological Pattern Formation} showing that patterns occur only if local self-enhancing reaction is coupled with an antagonistic reaction of long range [2,4]. The theory was embedded in a model comprising a system of reaction diffusion equations satisfying the Turing conditions. This model is now known as the Gierer-Meinhard model. It is used as a mathematical model for pattern formation in many different settings. For example, the Brusselator model for trimolecular chemical reactions is a particular case of it [1]. In this talk we propose modelling of the activation-inhibition mechanism of pattern formation by using nonlocal integral operators. This approach was pioneered in [3] for modelling of vegetation patterns. It turns out that the short range of the activation and the long range of the inhibition can be adequately represented via the supports of the kernels of the respective integrals. An advantage of using the nonlocal operator model from the point of view of its theoretical and numerical analysis is that it does not require smoothness of the solution with respect to the spatial variable. The applicability of this new approach is demonstrated on several biologically relevant examples. ...
{"title":"Modelling of activator-inhibitor dynamics via nonlocal integral operators","authors":"R. Anguelov, S. Stoltz","doi":"10.11145/TEXTS.2017.12.233","DOIUrl":"https://doi.org/10.11145/TEXTS.2017.12.233","url":null,"abstract":"The theory of pattern formation through local self-activation and long range inhibition has been shown to account for much of the observed pattern forming regulatory phenomena [2]. This mechanism is captured mathematically by considering two species, activator and inhibitor, with different spatial diffusivity, so that the resulting model is a system of reaction diffusion equations. The formation of patterns occurring in such systems under certain conditions was discovered by Alan Turing in 1952. Independently of Turing's work, Gierer and Meinhard derived in 1972 their textit{Theory of Biological Pattern Formation} showing that patterns occur only if local self-enhancing reaction is coupled with an antagonistic reaction of long range [2,4]. The theory was embedded in a model comprising a system of reaction diffusion equations satisfying the Turing conditions. This model is now known as the Gierer-Meinhard model. It is used as a mathematical model for pattern formation in many different settings. For example, the Brusselator model for trimolecular chemical reactions is a particular case of it [1]. In this talk we propose modelling of the activation-inhibition mechanism of pattern formation by using nonlocal integral operators. This approach was pioneered in [3] for modelling of vegetation patterns. It turns out that the short range of the activation and the long range of the inhibition can be adequately represented via the supports of the kernels of the respective integrals. An advantage of using the nonlocal operator model from the point of view of its theoretical and numerical analysis is that it does not require smoothness of the solution with respect to the spatial variable. The applicability of this new approach is demonstrated on several biologically relevant examples. ...","PeriodicalId":370233,"journal":{"name":"Biomath Communications Supplement","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121004049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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