In this paper we describe an empirical forecasting method for epidemic outbreaks. It is an iterative process to find possible parameter values for epidemic models to best fit real data. As a demonstration of principle, we used the logistic model, the simplest model in epidemiology, for an experiment of live forecasting. Short-term forecasts can last to 5 or more days with relative errors consistently kept blow 5%. The method should improve with more realistic models.
{"title":"An empirical forecasting method for epidemic outbreaks with application to Covid-19","authors":"Bo Deng","doi":"10.5206/mase/11101","DOIUrl":"https://doi.org/10.5206/mase/11101","url":null,"abstract":"In this paper we describe an empirical forecasting method for epidemic outbreaks. It is an iterative process to find possible parameter values for epidemic models to best fit real data. As a demonstration of principle, we used the logistic model, the simplest model in epidemiology, for an experiment of live forecasting. Short-term forecasts can last to 5 or more days with relative errors consistently kept blow 5%. The method should improve with more realistic models.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46098863","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}
Dylan Hull-Nye, Bhawna Malik, R. Keshavamurthy, E. J. Schwartz
The prevalence of end stage renal disease (ESRD) is rising among HIV-infected populations in several regions worldwide. We used an ordinary differential equation model of the dynamics of the AIDS and HIV+ ESRD populations to investigate the effect of antiretroviral therapy (ART) on the transient dynamics of the epidemic. We considered ART that blocks the entry to each population, by preventing individuals from joining the AIDS population and by reducing the development from AIDS to HIV+ ESRD, as well as the combined effects together. Numerical simulation of our model revealed that when levels of ART are below 100%, the prevalence of HIV+ ESRD drops, but then increases again due to the recovery in the AIDS population. The effect can be seen with ART acting to block entry into either population. We then examined the dip in HIV+ ESRD seen with ART analytically by calculating the minimum HIV+ ESRD level and the time to achieve this minimum. We also evaluated the length of time to reach the minimum and its dependence on ART parameters, both singly and in combination. We conclude that our model predicts that the drop in HIV+ ESRD prevalence seen after increased ART will be followed by an increase, unless ART is sufficiently high enough to eradicate HIV/AIDS.
{"title":"Transient dynamics of the renal disease epidemic among HIV-infected individuals","authors":"Dylan Hull-Nye, Bhawna Malik, R. Keshavamurthy, E. J. Schwartz","doi":"10.5206/mase/10852","DOIUrl":"https://doi.org/10.5206/mase/10852","url":null,"abstract":"The prevalence of end stage renal disease (ESRD) is rising among HIV-infected populations in several regions worldwide. We used an ordinary differential equation model of the dynamics of the AIDS and HIV+ ESRD populations to investigate the effect of antiretroviral therapy (ART) on the transient dynamics of the epidemic. We considered ART that blocks the entry to each population, by preventing individuals from joining the AIDS population and by reducing the development from AIDS to HIV+ ESRD, as well as the combined effects together. Numerical simulation of our model revealed that when levels of ART are below 100%, the prevalence of HIV+ ESRD drops, but then increases again due to the recovery in the AIDS population. The effect can be seen with ART acting to block entry into either population. We then examined the dip in HIV+ ESRD seen with ART analytically by calculating the minimum HIV+ ESRD level and the time to achieve this minimum. We also evaluated the length of time to reach the minimum and its dependence on ART parameters, both singly and in combination. We conclude that our model predicts that the drop in HIV+ ESRD prevalence seen after increased ART will be followed by an increase, unless ART is sufficiently high enough to eradicate HIV/AIDS.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41507192","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}
T. Bolger, Brydon Eastman, Madeleine Hill, G. Wolkowicz
A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions of the Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibrium implies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable, solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostat predator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling Type II functional response. An extension that applies to other functional rsponses is also given.
{"title":"A Predator-Prey model in the chemostat with Holling Type II response function","authors":"T. Bolger, Brydon Eastman, Madeleine Hill, G. Wolkowicz","doi":"10.5206/mase/10842","DOIUrl":"https://doi.org/10.5206/mase/10842","url":null,"abstract":"A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions of the Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibrium implies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable, solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostat predator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling Type II functional response. An extension that applies to other functional rsponses is also given.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44277612","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}
In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.
{"title":"On the focusing generalized Hartree equation","authors":"A. Arora, S. Roudenko, Kai Yang","doi":"10.5206/mase/10855","DOIUrl":"https://doi.org/10.5206/mase/10855","url":null,"abstract":"In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42080027","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}
In many developing countries, including Nepal, rabies epidemics constitute a serious public health concern, partly because of limited resources for proper implementation of control measures. In this study, we develop an extended model by incorporating various controls into the transmission dynamics model with both dog and jackal vectors. We apply the optimal control theory on the developed model system to identify optimal control strategy for mitigating rabies burden in Nepal with limited resources. Among the potential control strategies, human vaccination, dog vaccination, dog culling, dog sterilization, and jackal vaccination, considered in this study, our results show that a combination of dog vaccination and dog culling is the most effective strategy to control rabies in Nepal. Our optimal control solutions provide the strategy for optimal implementation of these controls to suppress rabies prevalence among dogs and jackals of Nepal using a minimum cost associated with controls. We found that given limited resources, implementing controls in a time-dependent manner with a higher level at the beginning of the outbreaks and reducing them during later part of the epidemics can provide maximum benefits.
{"title":"Controlling rabies epidemics in Nepal with limited resources: optimal control theory approach","authors":"Buddhi Pantha, H. Joshi, N. Vaidya","doi":"10.5206/mase/10847","DOIUrl":"https://doi.org/10.5206/mase/10847","url":null,"abstract":"In many developing countries, including Nepal, rabies epidemics constitute a serious public health concern, partly because of limited resources for proper implementation of control measures. In this study, we develop an extended model by incorporating various controls into the transmission dynamics model with both dog and jackal vectors. We apply the optimal control theory on the developed model system to identify optimal control strategy for mitigating rabies burden in Nepal with limited resources. Among the potential control strategies, human vaccination, dog vaccination, dog culling, dog sterilization, and jackal vaccination, considered in this study, our results show that a combination of dog vaccination and dog culling is the most effective strategy to control rabies in Nepal. Our optimal control solutions provide the strategy for optimal implementation of these controls to suppress rabies prevalence among dogs and jackals of Nepal using a minimum cost associated with controls. We found that given limited resources, implementing controls in a time-dependent manner with a higher level at the beginning of the outbreaks and reducing them during later part of the epidemics can provide maximum benefits.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45918845","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}
Wenjing Zhang, Ramnath Bhagavath, N. Madras, J. Heffernan
The progression of HIV infection to AIDS is unclear and under examined. Many mechanisms have been proposed, including a decline in immune response, increase in replication rate, involution of the thymus, syncytium inducing capacity, activation of the latently infected cell pool, chronic activation of the immune system, and the ability of the virus to infect other immune system cells. The significance of each mechanism in combination has not been studied. We develop a simple HIV viral dynamics model incorporating proposed mechanisms as parameters that are allowed to vary. In the entire parameter space, we derive two formulae for the basic reproduction number (R0) by considering the infection starting with a single infected CD4 T cell and a single virion, respectively. We show that both formulae are equivalent. We derive analytical conditions for the occurrence of backward and forward bifurcations. To investigate the influence of the proposed mechanisms to the HIV progression, we perform uncertainty and sensitivity analysis for all parameters and conduct a bifurcation analysis on all parameters that are shown to be significant, in combination, to explore various HIV/AIDS progression dynamics.
{"title":"Examining HIV progression mechanisms via mathematical approaches","authors":"Wenjing Zhang, Ramnath Bhagavath, N. Madras, J. Heffernan","doi":"10.5206/mase/10774","DOIUrl":"https://doi.org/10.5206/mase/10774","url":null,"abstract":"\u0000 \u0000 \u0000The progression of HIV infection to AIDS is unclear and under examined. Many mechanisms have been proposed, including a decline in immune response, increase in replication rate, involution of the thymus, syncytium inducing capacity, activation of the latently infected cell pool, chronic activation of the immune system, and the ability of the virus to infect other immune system cells. The significance of each mechanism in combination has not been studied. We develop a simple HIV viral dynamics model incorporating proposed mechanisms as parameters that are allowed to vary. In the entire parameter space, we derive two formulae for the basic reproduction number (R0) by considering the infection starting with a single infected CD4 T cell and a single virion, respectively. We show that both formulae are equivalent. We derive analytical conditions for the occurrence of backward and forward bifurcations. To investigate the influence of the proposed mechanisms to the HIV progression, we perform uncertainty and sensitivity analysis for all parameters and conduct a bifurcation analysis on all parameters that are shown to be significant, in combination, to explore various HIV/AIDS progression dynamics. \u0000 \u0000 \u0000","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43109779","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}
We will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of laplacian by using a method analogous to the direct method of calculus of variations. More precisely, we will discuss the existence of a minimizer of a suitable energy type functional whose Euler-Lagrange equation is the given semilinear pseudo-differential equation, and also discuss the regularity of such a minimizer so that it will be a solution to the semilinear equation.
{"title":"A solution to a fractional order semilinear equation using variational method","authors":"R. Karki","doi":"10.5206/mase/9413","DOIUrl":"https://doi.org/10.5206/mase/9413","url":null,"abstract":"We will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of laplacian by using a method analogous to the direct method of calculus of variations. More precisely, we will discuss the existence of a minimizer of a suitable energy type functional whose Euler-Lagrange equation is the given semilinear pseudo-differential equation, and also discuss the regularity of such a minimizer so that it will be a solution to the semilinear equation.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41394478","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}
We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy.
{"title":"A family of quasi-variable meshes high-resolution compact operator scheme for Burger's-Huxley, and Burger's-Fisher equation","authors":"Navnit Jha, Madhav Wagley","doi":"10.5206/mase/10837","DOIUrl":"https://doi.org/10.5206/mase/10837","url":null,"abstract":"We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45366410","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}
Oncolytic virus (OV) therapy is a promising treatment for cancer due to the OVs selective ability to infect and replicate inside cancer cells, thus killing them, without harming healthy cells. In this work, we present a new non-local multiscale moving boundary model for the spatio-temporal cancer-OV interactions. This model explores an important double feedback loop that links the macro-scale dynamics of cancer-virus interactions and the micro-scale dynamics of proteolytic activity taking place at the tumour interface. The cancer cell-cell and cell-matrix interactions are assumed to be nonlocal, while the cell-virus interactions are assumed local. With the help of this model we investigate computationally various cancer treatment scenarios involving oncolytic viruses (i.e., the effect of injecting the OV inside the tumour, or outside it). Moreover, we investigate the effect of different cell-cell and cell-matrix interaction strengths on the success of OV spreading throughout the tumour, and the effect of constant or density-dependent virus diffusion coefficients on viral spread.
{"title":"Non-local multiscale approaches for tumour-oncolytic viruses interactions","authors":"Abdulhamed Alsisi, R. Eftimie, D. Trucu","doi":"10.5206/MASE/10773","DOIUrl":"https://doi.org/10.5206/MASE/10773","url":null,"abstract":"Oncolytic virus (OV) therapy is a promising treatment for cancer due to the OVs selective ability to infect and replicate inside cancer cells, thus killing them, without harming healthy cells. In this work, we present a new non-local multiscale moving boundary model for the spatio-temporal cancer-OV interactions. This model explores an important double feedback loop that links the macro-scale dynamics of cancer-virus interactions and the micro-scale dynamics of proteolytic activity taking place at the tumour interface. The cancer cell-cell and cell-matrix interactions are assumed to be nonlocal, while the cell-virus interactions are assumed local. With the help of this model we investigate computationally various cancer treatment scenarios involving oncolytic viruses (i.e., the effect of injecting the OV inside the tumour, or outside it). Moreover, we investigate the effect of different cell-cell and cell-matrix interaction strengths on the success of OV spreading throughout the tumour, and the effect of constant or density-dependent virus diffusion coefficients on viral spread.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43720291","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}
This paper deals with a fractional-order three-dimensional compartmental model with fear effect. We have investigated whether fear can play an important role or not to spread and control the infectious diseases like COVID-19, SARS etc. in a bounded region. The basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. Stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity greater than a certain threshold value.We have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. Numerical simulation suggests that the fear factor is an important role which is observed through Hopf-bifurcation.
{"title":"Dynamical analysis of a fractional order model incorporating fear in the disease transmission rate of COVID-19","authors":"D. Mukherjee, C. Maji","doi":"10.5206/mase/10745","DOIUrl":"https://doi.org/10.5206/mase/10745","url":null,"abstract":"This paper deals with a fractional-order three-dimensional compartmental model with fear effect. We have investigated whether fear can play an important role or not to spread and control the infectious diseases like COVID-19, SARS etc. in a bounded region. The basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. Stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity greater than a certain threshold value.We have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. Numerical simulation suggests that the fear factor is an important role which is observed through Hopf-bifurcation.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49441766","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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