In this paper we study the early stages of atherosclerosis via a mathematical model of partial differential equations of reaction-diffusion type. The model includes several key species and identifies endothelial hyperpermeability, believed to be a precursor on the onset of atherosclerosis. We reduce the system to a monotone system and provide a biological interpretation for the stability analysis according to endothelial functionality. We investigate as well the existence of solutions of traveling waves type along with numerical simulations. The obtained results are in good agreement with current biological knowledge. Likewise, they confirm and generalize results of mathematical models previously performed in literature. Then, we study the non monotone reduced model and prove the existence of perturbed solutions and perturbed waves, particularly in the bistable case. Finally, we consider the complete model proposed initially, perform numerical simulations and provide more specific results. We study the consistency between the reduced and complete models for a certain range of parameters. We elaborate bifurcation diagrams showing the evolution of inflammation upon endothelial permeability and LDL accumulation. We show that the regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to certain extent, lead to plaque regression.
{"title":"Mathematical Modeling of Inflammatory Processes of Atherosclerosis","authors":"G. Abi Younes, N. El Khatib","doi":"10.1051/mmnp/2022004","DOIUrl":"https://doi.org/10.1051/mmnp/2022004","url":null,"abstract":"In this paper we study the early stages of atherosclerosis via a mathematical model of partial differential equations of reaction-diffusion type. The model includes several key species and identifies endothelial hyperpermeability, believed to be a precursor on the onset of atherosclerosis. We reduce the system to a monotone system and provide a biological interpretation for the stability analysis according to endothelial functionality. We investigate as well the existence of solutions of traveling waves type along with numerical simulations. The obtained results are in good agreement with current biological knowledge. Likewise, they confirm and generalize results of mathematical models previously performed in literature. Then, we study the non monotone reduced model and prove the existence of perturbed solutions and perturbed waves, particularly in the bistable case. Finally, we consider the complete model proposed initially, perform numerical simulations and provide more specific results. We study the consistency between the reduced and complete models for a certain range of parameters. We elaborate bifurcation diagrams showing the evolution of inflammation upon endothelial permeability and LDL accumulation. We show that the regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to certain extent, lead to plaque regression.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41706537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Toxoplasma gondii (T. gondii) is a parasitic pathogen that causes serious brain diseases in fetuses and patients with immunodeficiency, particularly AIDS patients. In the field of immunology, a large number of studies have shown that effector CD8 + T cells respond to T. gondii infection in the brain tissue through controlling the proliferation of intracellular parasites and killing infected brain cells. These protective mechanisms do not occur without T cell movement and searching for infected cells, as a fundamental feature of the immune system. Following infection with a pathogen in a tissue, in their search for infected cells, CD8 + T cells can perform different stochastic searches, including Levy and Brownian random walks. Statistical analysis of CD8 + T cells in response to infected brain cells could be described by a Levy random walk., In this work, by considering a Levy distribution for the displacements, we propose a space fractional-order diffusion equation for the T cell density in the infected brain tissue. Furthermore, we derive a mathematical model representing CD8 + T cell response to infected brain cells. By solving the model equations numerically, we perform a comparison between Levy and Brownian search strategies.
{"title":"A fractional diffusion model of CD8+T cells response to parasitic infection in the brain","authors":"A. Farhadi, E. Hanert","doi":"10.1051/mmnp/2022003","DOIUrl":"https://doi.org/10.1051/mmnp/2022003","url":null,"abstract":"Toxoplasma gondii (T. gondii) is a parasitic pathogen that causes serious brain diseases in fetuses and patients with immunodeficiency, particularly AIDS patients. In the field of immunology, a large number of studies have shown that effector CD8 + T cells respond to T. gondii infection in the brain tissue through controlling the proliferation of intracellular parasites and killing infected brain cells. These protective mechanisms do not occur without T cell movement and searching for infected cells, as a fundamental feature of the immune system. Following infection with a pathogen in a tissue, in their search for infected cells, CD8 + T cells can perform different stochastic searches, including Levy and Brownian random walks. Statistical analysis of CD8 + T cells in response to infected brain cells could be described by a Levy random walk., In this work, by considering a Levy distribution for the displacements, we propose a space fractional-order diffusion equation for the T cell density in the infected brain tissue. Furthermore, we derive a mathematical model representing CD8 + T cell response to infected brain cells. By solving the model equations numerically, we perform a comparison between Levy and Brownian search strategies.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47795529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with a stochastic population model with Allee effect and jumps. First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last, some numerical results are given. The results show that if attack rate $a$ is in the intermediate range or very large, the population will go extinct. Under the premise that attack rate $a$ is less than growth rate $r$, if the noise intensity or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean. The results in this paper generalize and improve the previous related results.
{"title":"Dynamics of a stochastic population model with Allee effect and jumps","authors":"Rong Liu, Guirong Liu","doi":"10.1051/mmnp/2022002","DOIUrl":"https://doi.org/10.1051/mmnp/2022002","url":null,"abstract":"This paper is concerned with a stochastic population model with Allee effect and jumps.\u0000\u0000First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and\u0000\u0000persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last,\u0000\u0000some numerical results are given. The results show that if attack rate $a$ is in the intermediate range or very large,\u0000\u0000the population will go extinct. Under the premise that attack rate $a$ is less than growth rate $r$, if the noise intensity\u0000\u0000or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean.\u0000\u0000The results in this paper generalize and improve the previous related results.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49214800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.1101/2021.11.01.21265729
Y. Ko, V. M. Mendoza, Y. Seo, J. Lee, Y. Kim, D. Kwon, E. Jung
Background: Early vaccination efforts and non-pharmaceutical interventions were insufficient to prevent a surge of coronavirus disease 2019 (COVID-19) cases triggered by the Delta variant. This study aims to understand how vaccination and variants contribute to the spread of COVID-19 so that appropriate measures are implemented. Methods: A compartment model that includes age, vaccination, and infection with the Delta or non-Delta variants was developed. We estimated the transmission rates using maximum likelihood estimation and phase-dependent reduction effect of non-pharmaceutical interventions (NPIs) according to government policies from 26 February to 8 October 2021. We extended our model simulation until 31 December considering the initiation of eased NPIs. Furthermore, we also performed simulations to examine the effect of NPIs, arrival timing of Delta variant, and speed of vaccine administration. Results: The estimated transmission rate matrices show distinct pattern, with the transmission rates of younger age groups (0~39 years) much larger than non-Delta. Social distancing (SD) level 2 and SD4 in Korea were associated with transmission reduction factors of 0.64 to 0.69 and 0.70 to 0.78, respectively. The easing of NPIs to a level comparable to SD2 should be initiated not earlier than 16 October to keep the number of severe cases below the capacity of Korean healthcare system. Simulation results also showed that a surge prompted by the spread of the Delta variant can be prevented if the number of people vaccinated daily was larger. Conclusions: Simulations showed that the timing of easing and intensity of NPIs, vaccination speed, and screening measures are key factors in preventing another epidemic wave.
{"title":"Quantifying the effects of non-pharmaceutical and pharmaceutical interventions against COVID-19 epidemic in the Republic of Korea: Mathematical model-based approach considering age groups and the Delta variant","authors":"Y. Ko, V. M. Mendoza, Y. Seo, J. Lee, Y. Kim, D. Kwon, E. Jung","doi":"10.1101/2021.11.01.21265729","DOIUrl":"https://doi.org/10.1101/2021.11.01.21265729","url":null,"abstract":"Background: Early vaccination efforts and non-pharmaceutical interventions were insufficient to prevent a surge of coronavirus disease 2019 (COVID-19) cases triggered by the Delta variant. This study aims to understand how vaccination and variants contribute to the spread of COVID-19 so that appropriate measures are implemented. Methods: A compartment model that includes age, vaccination, and infection with the Delta or non-Delta variants was developed. We estimated the transmission rates using maximum likelihood estimation and phase-dependent reduction effect of non-pharmaceutical interventions (NPIs) according to government policies from 26 February to 8 October 2021. We extended our model simulation until 31 December considering the initiation of eased NPIs. Furthermore, we also performed simulations to examine the effect of NPIs, arrival timing of Delta variant, and speed of vaccine administration. Results: The estimated transmission rate matrices show distinct pattern, with the transmission rates of younger age groups (0~39 years) much larger than non-Delta. Social distancing (SD) level 2 and SD4 in Korea were associated with transmission reduction factors of 0.64 to 0.69 and 0.70 to 0.78, respectively. The easing of NPIs to a level comparable to SD2 should be initiated not earlier than 16 October to keep the number of severe cases below the capacity of Korean healthcare system. Simulation results also showed that a surge prompted by the spread of the Delta variant can be prevented if the number of people vaccinated daily was larger. Conclusions: Simulations showed that the timing of easing and intensity of NPIs, vaccination speed, and screening measures are key factors in preventing another epidemic wave.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"62335697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
As cyber attacks have become more frequent, cyber insurance premiums have increased, resulting in the need for better modeling of cyber risk. Jevtic and Lanchier[20] proposed a dynamic structural model of aggregate loss distribution for cyber risk of small-and-medium-sized enterprises under the assumption of a tree-based local-area-network topology that consists of the combination of a Poisson process, homogeneous random trees, bond percolation processes, and cost topology. Their model assumes that the contagion spreads through the edges of the network with the same fixed probability in both directions, thus overlooking a dynamic cyber security environment implemented in most networks, and their results give an exact expression for the mean of the aggregate loss but only a rough upper bound for the variance. In this paper, we consider a bidirectional version of their percolation model in which the contagion spreads through the edges of the network with a certain probability moving toward the lower level assets of the network but with another probability moving toward the higher level assets of the network. Also, our different mathematical approach leads to exact expressions for both the mean and the variance of the aggregate loss, and therefore an exact expression for the insurance premiums.
{"title":"Exact insurance premiums for cyber risk of small and medium-sized enterprises","authors":"Stefano Chiaradonna, N. Lanchier","doi":"10.1051/mmnp/2022041","DOIUrl":"https://doi.org/10.1051/mmnp/2022041","url":null,"abstract":"As cyber attacks have become more frequent, cyber insurance premiums have increased, resulting in the need for better modeling of cyber risk. Jevtic and Lanchier[20] proposed a dynamic structural model of aggregate loss distribution for cyber risk of small-and-medium-sized enterprises under the assumption of a tree-based local-area-network topology that consists of the combination of a Poisson process, homogeneous random trees, bond percolation processes, and cost topology. Their model assumes that the contagion spreads through the edges of the network with the same fixed probability in both directions, thus overlooking a dynamic cyber security environment implemented in most networks, and their results give an exact expression for the mean of the aggregate loss but only a rough upper bound for the variance. In this paper, we consider a bidirectional version of their percolation model in which the contagion spreads through the edges of the network with a certain probability moving toward the lower level assets of the network but with another probability moving toward the higher level assets of the network. Also, our different mathematical approach leads to exact expressions for both the mean and the variance of the aggregate loss, and therefore an exact expression for the insurance premiums.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44046379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-14DOI: 10.21203/rs.3.rs-1113138/v1
L. Fern'andez, C. Pola, Judith Sáinz-Pardo:
We mathematically justify metronomic chemotherapy as the best strategy to apply most cytotoxic drugs in oncology for both curative and palliative approaches, assuming the classical pharmacokinetic model together with the Emax pharmacodynamic and the Norton-Simon hypothesis.From the mathematical point of view, we will consider two mixed-integer nonlinear optimization problems, where the unknowns are the number of the doses and the quantity of each one, adjusting the administration times a posteriori.Mathematics Subject Classification: 93C15, 92C50, 90C30
{"title":"A mathematical justification for metronomic chemotherapy in oncology","authors":"L. Fern'andez, C. Pola, Judith Sáinz-Pardo:","doi":"10.21203/rs.3.rs-1113138/v1","DOIUrl":"https://doi.org/10.21203/rs.3.rs-1113138/v1","url":null,"abstract":"\u0000 We mathematically justify metronomic chemotherapy as the best strategy to apply most cytotoxic drugs in oncology for both curative and palliative approaches, assuming the classical pharmacokinetic model together with the Emax pharmacodynamic and the Norton-Simon hypothesis.From the mathematical point of view, we will consider two mixed-integer nonlinear optimization problems, where the unknowns are the number of the doses and the quantity of each one, adjusting the administration times a posteriori.Mathematics Subject Classification: 93C15, 92C50, 90C30","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44855525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $Omega_{0}.$
{"title":"Mathematical analysis of an age structured epidemic model with a quarantine class","authors":"B. Ainseba, T. Touaoula, Z. Sari","doi":"10.1051/mmnp/2021049","DOIUrl":"https://doi.org/10.1051/mmnp/2021049","url":null,"abstract":"In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $Omega_{0}.$","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49326548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A hyperbolic model for diffusion, nonlinear transport (or advection) and production of a scalar quantity, is considered. The model is based on a constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by means of a relaxation time relation. The production or source term is assumed to be of logistic type. This paper studies the existence and spectral stability properties of spatially periodic traveling wave solutions to this system. It is shown that a family of subcharacteristic periodic waves emerges from a local Hopf bifurcation around a critical value of the wave speed. These waves have bounded fundamental period and small-amplitude. In addition, it is shown that these waves are spectrally unstable as solutions to the hyperbolic system. For that purpose, it is proved that the Floquet spectrum of the linearized operator around a wave can be approximated by a linear operator whose point spectrum intersects the unstable half plane of complex numbers with positive real part.
{"title":"Spectral instability of small-amplitude periodic waves for hyperbolic non-Fickian diffusion advection models with logistic source","authors":"E. Alvarez, Ricardo Murillo, R. Plaza","doi":"10.1051/mmnp/2022020","DOIUrl":"https://doi.org/10.1051/mmnp/2022020","url":null,"abstract":"A hyperbolic model for diffusion, nonlinear transport (or advection) and production of a scalar quantity, is considered. The model is based on a constitutive law of Cattaneo-Maxwell type expressing non-Fickian diffusion by means of a relaxation time relation. The production or source term is assumed to be of logistic type. This paper studies the existence and spectral stability properties of spatially periodic traveling wave solutions to this system. It is shown that a family of subcharacteristic periodic waves emerges from a local Hopf bifurcation around a critical value of the wave speed. These waves have bounded fundamental period and small-amplitude. In addition, it is shown that these waves are spectrally unstable as solutions to the hyperbolic system. For that purpose, it is proved that the Floquet spectrum of the linearized operator around a wave can be approximated by a linear operator whose point spectrum intersects the unstable half plane of complex numbers with positive real part.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57787110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Subdural ECoG data recorded from the matrix of electrodes during syllable pronunciation are analyzed by the method of circular-linear regression. Phase waves in 1D electrode arrays and in the whole 2D set of electrodes are detected, and their spatial organization and temporal evolution are studied. Phase portraits of wave vectors indicate the presence of sources, sinks, and saddle points. The analysis of temporal evolution of phase portraits shows that they changed more at the beginning of syllable pronunciation. Furthermore, wave sources were more stable in their localization during the pronunciation. Overall, in spite of large variability of phase portraits, they represent some characterization of the dynamics of electric potential in the cerebral cortex.
{"title":"Analysis of phase waves in the ECoG data","authors":"Alexandre Aksenov, A. Beuter","doi":"10.1051/mmnp/2021045","DOIUrl":"https://doi.org/10.1051/mmnp/2021045","url":null,"abstract":"Subdural ECoG data recorded from the matrix of electrodes during syllable pronunciation are analyzed by the method of circular-linear regression. Phase waves in 1D electrode arrays and in the whole 2D set of electrodes are detected, and their spatial organization and temporal evolution are studied. Phase portraits of wave vectors indicate the presence of sources, sinks, and saddle points. The analysis of temporal evolution of phase portraits shows that they changed more at the beginning of syllable pronunciation. Furthermore, wave sources were more stable in their localization during the pronunciation. Overall, in spite of large variability of phase portraits, they represent some characterization of the dynamics of electric potential in the cerebral cortex.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48952369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In our model of the COVID-19 epidemic, infected individuals can be of four types, according whether they are asymptomatic ($A$) or symptomatic ($I$), and use a contact tracing mobile phone app ($Y$) or not ($N$). We denote by $f$ the fraction of $A$'s, by $y$ the fraction of $Y$'s and by $R_0$ the average number of secondary infections from a random infected individual. We investigate the effect of non-digital interventions and of digital interventions, depending on the willingness to quarantine, parameterized by four cooperating probabilities. For a given `effective' $R_0$ obtained with non-digital interventions, we use non-negative matrix theory and stopping line techniques to characterize mathematically the minimal fraction $y_0$ of app users needed to curb the epidemic. We show that under a wide range of scenarios, the threshold $y_0$ as a function of $R_0$ rises steeply from 0 at $R_0=1$ to prohibitively large values (of the order of $60-70%$ up) whenever $R_0$ is above 1.3. Our results show that moderate rates of adoption of a contact tracing app can reduce $R_0$ but are by no means sufficient to reduce it below 1 unless it is already very close to 1 thanks to non-digital interventions.
{"title":"A mathematical assessment of the efficiency of quarantining and contact tracing in curbing the COVID-19 epidemic","authors":"A. Lambert","doi":"10.1051/mmnp/2021042","DOIUrl":"https://doi.org/10.1051/mmnp/2021042","url":null,"abstract":"In our model of the COVID-19 epidemic, infected individuals can be of four types, according whether they are asymptomatic ($A$) or symptomatic ($I$), and use a contact tracing mobile phone app ($Y$) or not ($N$). We denote by $f$ the fraction of $A$'s, by $y$ the fraction of $Y$'s and by $R_0$ the average number of secondary infections from a random infected individual.\u0000We investigate the effect of non-digital interventions and of digital interventions, depending on the willingness to quarantine, parameterized by four cooperating probabilities.\u0000For a given `effective' $R_0$ obtained with non-digital interventions, we use non-negative matrix theory and stopping line techniques to characterize mathematically the minimal fraction $y_0$ of app users needed to curb the epidemic. We show that under a wide range of scenarios, the threshold $y_0$ as a function of $R_0$ rises steeply from 0 at $R_0=1$ to prohibitively large values (of the order of $60-70%$ up) whenever $R_0$ is above 1.3. Our results show that moderate rates of adoption of a contact tracing app can reduce $R_0$ but are by no means sufficient to reduce it below 1 unless it is already very close to 1 thanks to non-digital interventions.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46407584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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