Pub Date : 2019-01-01DOI: 10.1080/23737867.2019.1656115
Z. Alqahtani, M. El-shahed, N. Mottram
In this paper, the static and dynamic behaviour of a fractional-order predator–prey model are studied, where the nonlinear interactions between the two species lead to multiple stable states. As has been found in many previous systems, the stability of such states can be dependent on the fractional order of the time derivative, which is included as a phenomenological model of memory-effects in the predator and prey species. However, what is less well understood is the transient behaviour and dependence of the observed domains of attraction for each stable state on the order of the fractional time derivative. These dependencies are investigated using analytical (for the stability of equilibria) and numerical (for the observed domains of attraction) techniques. Results reveal far richer dynamics compared to the integer-order model. We conclude that, as well as the species and controllable parameters, the memory effect of the species will play a role in the observed behaviour of the system.
{"title":"Derivative-order-dependent stability and transient behaviour in a predator–prey system of fractional differential equations","authors":"Z. Alqahtani, M. El-shahed, N. Mottram","doi":"10.1080/23737867.2019.1656115","DOIUrl":"https://doi.org/10.1080/23737867.2019.1656115","url":null,"abstract":"In this paper, the static and dynamic behaviour of a fractional-order predator–prey model are studied, where the nonlinear interactions between the two species lead to multiple stable states. As has been found in many previous systems, the stability of such states can be dependent on the fractional order of the time derivative, which is included as a phenomenological model of memory-effects in the predator and prey species. However, what is less well understood is the transient behaviour and dependence of the observed domains of attraction for each stable state on the order of the fractional time derivative. These dependencies are investigated using analytical (for the stability of equilibria) and numerical (for the observed domains of attraction) techniques. Results reveal far richer dynamics compared to the integer-order model. We conclude that, as well as the species and controllable parameters, the memory effect of the species will play a role in the observed behaviour of the system.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"6 1","pages":"32 - 49"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2019.1656115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43789093","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 : 2019-01-01DOI: 10.1080/23737867.2019.1655497
Maitraya Ghatak, J. Urcuyo, P. Wise, R. T. Trout Fryxell, S. Lenhart
ABSTRACT La Crosse Virus (LACV) is an arbovirus found in Eastern Appalachia and can cause pediatric encephalitis in prepubescent children. To assess the risk and transmission of this disease, it is particularly important to understand the average population of Aedes mosquitoes, which are the vectors of this virus. We use a deterministic compartmental model to study the effects of environmental factors on the population dynamics of Aedes mosquitoes in the Knox County area. We use locally-collected mosquito population data to adjust our model outputs and find that model transitions are heavily dependent on the fluctuations of both temperature and accumulated precipitation. These findings should be considered for mosquito management in Southern Appalachia, as well as in other regions with slight modifications to our model.
{"title":"Modeling the average population of La Crosse vectors in Knox County, Tennessee","authors":"Maitraya Ghatak, J. Urcuyo, P. Wise, R. T. Trout Fryxell, S. Lenhart","doi":"10.1080/23737867.2019.1655497","DOIUrl":"https://doi.org/10.1080/23737867.2019.1655497","url":null,"abstract":"ABSTRACT La Crosse Virus (LACV) is an arbovirus found in Eastern Appalachia and can cause pediatric encephalitis in prepubescent children. To assess the risk and transmission of this disease, it is particularly important to understand the average population of Aedes mosquitoes, which are the vectors of this virus. We use a deterministic compartmental model to study the effects of environmental factors on the population dynamics of Aedes mosquitoes in the Knox County area. We use locally-collected mosquito population data to adjust our model outputs and find that model transitions are heavily dependent on the fluctuations of both temperature and accumulated precipitation. These findings should be considered for mosquito management in Southern Appalachia, as well as in other regions with slight modifications to our model.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"6 1","pages":"20 - 31"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2019.1655497","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41789261","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 : 2019-01-01DOI: 10.1080/23737867.2019.1624631
Selenne Bañuelos, Marco V. Martinez, C. Mitchell, Alicia Prieto-Langarica
Zika is a vector borne disease for which the latest world wide outbreak inspired a renewed interest in epidemiological modelling of vector borne diseases. However, due to the possibility of sexual transmission and the high proportion of asymptomatic individuals, models for similar diseases, such as dengue or chikungunya, are no longer applicable. It is of interest to study how the existence and behaviour of asymptomatic individuals and the potential of them transmitting the disease affect the overall epidemic dynamics. The model presented here aims to be as simple as possible, while at the same time taking into account the features that make Zika unique among other vector borne diseases. This model allows for the exploration of sexual transmission and how the sexual behaviour of asymptomatic individuals may affect the spread of the disease. In addition, the model was used to determine the basic reproductive number, with and without the effect of sexual transmission as well as to implement a simple version of control using Wolbachia bacterium.
{"title":"Using mathematical modelling to investigate the effect of the sexual behaviour of asymptomatic individuals and vector control measures on Zika","authors":"Selenne Bañuelos, Marco V. Martinez, C. Mitchell, Alicia Prieto-Langarica","doi":"10.1080/23737867.2019.1624631","DOIUrl":"https://doi.org/10.1080/23737867.2019.1624631","url":null,"abstract":"Zika is a vector borne disease for which the latest world wide outbreak inspired a renewed interest in epidemiological modelling of vector borne diseases. However, due to the possibility of sexual transmission and the high proportion of asymptomatic individuals, models for similar diseases, such as dengue or chikungunya, are no longer applicable. It is of interest to study how the existence and behaviour of asymptomatic individuals and the potential of them transmitting the disease affect the overall epidemic dynamics. The model presented here aims to be as simple as possible, while at the same time taking into account the features that make Zika unique among other vector borne diseases. This model allows for the exploration of sexual transmission and how the sexual behaviour of asymptomatic individuals may affect the spread of the disease. In addition, the model was used to determine the basic reproductive number, with and without the effect of sexual transmission as well as to implement a simple version of control using Wolbachia bacterium.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"6 1","pages":"1 - 19"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2019.1624631","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45986559","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 : 2019-01-01DOI: 10.1080/23737867.2019.1682473
Alexis White, Elsa Schaefer, Chelsea Wright Thompson, Christopher M Kribs, Holly Gaff
A mathematical model for a two-pathogen, one-tick, one-host system is presented and explored. The model system is based on the dynamics of Amblyomma americanum, Rickettsia parkeri, and Rickettsia amblyommatis. The goal of this model is to determine how long an invading pathogen, R. parkeri, persists within a tick population, A. americanum, in which a resident pathogen, R. amblyommatis, is already established. The numerical simulations of the model demonstrate the parameter ranges that allow for coexistence of the two pathogens. Sensitivity analysis highlights the importance of vector-borne, tick-to-host, transmission rates on the invasion reproductive number and persistence of the pathogens over time. The model is then applied to a case study based on a reclaimed swampland field site in south-eastern Virginia using field and laboratory data. The results pinpoint the thresholds required for persistence of both pathogens in the local tick population. However, R. parkeri, is not predicted to persist beyond 3 years. Understanding the persistence and coexistence of tick-borne pathogens will allow public health officials increased insight into tick-borne disease dynamics.
{"title":"Dynamics of two pathogens in a single tick population.","authors":"Alexis White, Elsa Schaefer, Chelsea Wright Thompson, Christopher M Kribs, Holly Gaff","doi":"10.1080/23737867.2019.1682473","DOIUrl":"https://doi.org/10.1080/23737867.2019.1682473","url":null,"abstract":"<p><p>A mathematical model for a two-pathogen, one-tick, one-host system is presented and explored. The model system is based on the dynamics of <i>Amblyomma americanum</i>, <i>Rickettsia parkeri</i>, and <i>Rickettsia amblyommatis</i>. The goal of this model is to determine how long an invading pathogen, <i>R. parkeri</i>, persists within a tick population, <i>A. americanum</i>, in which a resident pathogen, <i>R. amblyommatis</i>, is already established. The numerical simulations of the model demonstrate the parameter ranges that allow for coexistence of the two pathogens. Sensitivity analysis highlights the importance of vector-borne, tick-to-host, transmission rates on the invasion reproductive number and persistence of the pathogens over time. The model is then applied to a case study based on a reclaimed swampland field site in south-eastern Virginia using field and laboratory data. The results pinpoint the thresholds required for persistence of both pathogens in the local tick population. However, <i>R. parkeri</i>, is not predicted to persist beyond 3 years. Understanding the persistence and coexistence of tick-borne pathogens will allow public health officials increased insight into tick-borne disease dynamics.</p>","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"6 1","pages":"50-66"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2019.1682473","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38457216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-14DOI: 10.1080/23737867.2018.1525666
{"title":"Call for Papers","authors":"","doi":"10.1080/23737867.2018.1525666","DOIUrl":"https://doi.org/10.1080/23737867.2018.1525666","url":null,"abstract":"","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"5 1","pages":"252 - 254"},"PeriodicalIF":0.0,"publicationDate":"2018-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2018.1525666","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46113103","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 : 2018-12-14DOI: 10.1080/23737867.2017.1412811
Robyn M. Nadolny, H. Gaff
Abstract Many tick species are invading new areas because of anthropogenic changes in the landscape, shifting climatic variables and increasing populations of suitable host species and tick habitat. However, the relative influences of habitat and hosts in tick dispersal and tick population establishment remain in question. A spatially explicit agent-based model was developed to explore the spatio-temporal dynamics of a generic tick population in the years immediately following the introduction of ticks into a novel environment. The general model was then adapted to investigate a case study of two recent tick species invasions into the Mid-Atlantic United States. The recent simultaneous range expansions of two ixodid tick species, Ixodes affinis and Amblyomma maculatum, provided an opportunity to determine if invasion patterns observed in the field could be replicated in silico on a small scale. The models presented here indicated that for generalist parasites, habitat connectivity is a better indicator than host mobility for spatial and genetic patterns of parasite range expansion. In addition, our results demonstrate the utility of including genetic variables into agent-based models: gene flow functions as a proxy for measuring dispersal, and models can be validated using results from the field.
{"title":"Modelling the effects of habitat and hosts on tick invasions","authors":"Robyn M. Nadolny, H. Gaff","doi":"10.1080/23737867.2017.1412811","DOIUrl":"https://doi.org/10.1080/23737867.2017.1412811","url":null,"abstract":"Abstract Many tick species are invading new areas because of anthropogenic changes in the landscape, shifting climatic variables and increasing populations of suitable host species and tick habitat. However, the relative influences of habitat and hosts in tick dispersal and tick population establishment remain in question. A spatially explicit agent-based model was developed to explore the spatio-temporal dynamics of a generic tick population in the years immediately following the introduction of ticks into a novel environment. The general model was then adapted to investigate a case study of two recent tick species invasions into the Mid-Atlantic United States. The recent simultaneous range expansions of two ixodid tick species, Ixodes affinis and Amblyomma maculatum, provided an opportunity to determine if invasion patterns observed in the field could be replicated in silico on a small scale. The models presented here indicated that for generalist parasites, habitat connectivity is a better indicator than host mobility for spatial and genetic patterns of parasite range expansion. In addition, our results demonstrate the utility of including genetic variables into agent-based models: gene flow functions as a proxy for measuring dispersal, and models can be validated using results from the field.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"5 1","pages":"2 - 29"},"PeriodicalIF":0.0,"publicationDate":"2018-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1412811","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49272824","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 : 2018-12-02DOI: 10.1080/23737867.2018.1551075
I. Laukó, G. Pinter, Rachel Elizabeth TeWinkel
ABSTRACT We consider a system of non-linear differential equations describing the spread of an epidemic in two interacting populations. The model assumes that the epidemic spreads within the first population, which in turn acts as a reservoir of infection for the second population. We explore the conditions under which the epidemic is endemic in both populations and discuss the global asymptotic stability of the endemic equilibrium using a Lyapunov function and results established for asymptotically autonomous systems. We discuss monkeypox as an example of an emerging disease that can be modelled in this way and present some numerical results representing the model and its extensions.
{"title":"Equilibrium analysis for an epidemic model with a reservoir for infection","authors":"I. Laukó, G. Pinter, Rachel Elizabeth TeWinkel","doi":"10.1080/23737867.2018.1551075","DOIUrl":"https://doi.org/10.1080/23737867.2018.1551075","url":null,"abstract":"ABSTRACT We consider a system of non-linear differential equations describing the spread of an epidemic in two interacting populations. The model assumes that the epidemic spreads within the first population, which in turn acts as a reservoir of infection for the second population. We explore the conditions under which the epidemic is endemic in both populations and discuss the global asymptotic stability of the endemic equilibrium using a Lyapunov function and results established for asymptotically autonomous systems. We discuss monkeypox as an example of an emerging disease that can be modelled in this way and present some numerical results representing the model and its extensions.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"5 1","pages":"255 - 274"},"PeriodicalIF":0.0,"publicationDate":"2018-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2018.1551075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45560528","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 : 2018-09-29DOI: 10.1080/23737867.2018.1526132
T. O. Orwa, Rachel Waema Mbogo, L. Luboobi
ABSTRACT Despite the success of the existing malaria control strategies, reported malaria cases are still quite high. In 2016, the WHO reported about 216 million malaria cases; 90% of which occurred in the WHO African Region. In this paper, a mathematical model for the in-host Plasmodium falciparum malaria subject to malaria vaccines is formulated and analysed. An efficacious pre-erythrocytic vaccine is shown to greatly reduce the severity of clinical malaria. Based on the normalized forward sensitivity index technique, the average number of merozoites released per bursting blood schizont is shown to be the most sensitive parameter in the model. Numerical simulation results further suggest that an efficacious blood stage vaccine has the potential to reduce the burst size of the blood schizonts and maximize the rate of activation of CD8+ T cells during malaria infection. Moreover, vaccine combinations that are efficacious might help in achieving a malaria free population by the year 2030. This paper provides useful insights in malaria vaccine control and a unique opportunity to intensify support and funding for malaria vaccine development.
{"title":"Mathematical model for the in-host malaria dynamics subject to malaria vaccines","authors":"T. O. Orwa, Rachel Waema Mbogo, L. Luboobi","doi":"10.1080/23737867.2018.1526132","DOIUrl":"https://doi.org/10.1080/23737867.2018.1526132","url":null,"abstract":"ABSTRACT Despite the success of the existing malaria control strategies, reported malaria cases are still quite high. In 2016, the WHO reported about 216 million malaria cases; 90% of which occurred in the WHO African Region. In this paper, a mathematical model for the in-host Plasmodium falciparum malaria subject to malaria vaccines is formulated and analysed. An efficacious pre-erythrocytic vaccine is shown to greatly reduce the severity of clinical malaria. Based on the normalized forward sensitivity index technique, the average number of merozoites released per bursting blood schizont is shown to be the most sensitive parameter in the model. Numerical simulation results further suggest that an efficacious blood stage vaccine has the potential to reduce the burst size of the blood schizonts and maximize the rate of activation of CD8+ T cells during malaria infection. Moreover, vaccine combinations that are efficacious might help in achieving a malaria free population by the year 2030. This paper provides useful insights in malaria vaccine control and a unique opportunity to intensify support and funding for malaria vaccine development.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"5 1","pages":"222 - 251"},"PeriodicalIF":0.0,"publicationDate":"2018-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2018.1526132","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48485186","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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