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Li A, Zou X. R 0 May Not Tell Us Everything: Transient Disease Dynamics of Some SIR Models Over Patchy Environments. Bull Math Biol 2024; 86:41. [PMID: 38491224 DOI: 10.1007/s11538-024-01271-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2023] [Accepted: 02/16/2024] [Indexed: 03/18/2024]
Abstract
This paper examines the short-term or transient dynamics of SIR infectious disease models in patch environments. We employ reactivity of an equilibrium and amplification rates, concepts from ecology, to analyze how dispersals/travels between patches, spatial heterogeneity, and other disease-related parameters impact short-term dynamics. Our findings reveal that in certain scenarios, due to the impact of spatial heterogeneity and the dispersals, the short-term disease dynamics over a patch environment may disagree with the long-term disease dynamics that is typically reflected by the basic reproduction number. Such an inconsistence can mislead the public, public healthy agencies and governments when making public health policy and decisions, and hence, these findings are of practical importance.
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Affiliation(s)
- Ao Li
- Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada
| | - Xingfu Zou
- Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada.
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Trevisin C, Lemaitre JC, Mari L, Pasetto D, Gatto M, Rinaldo A. Epidemicity of cholera spread and the fate of infection control measures. J R Soc Interface 2022; 19:20210844. [PMID: 35259956 PMCID: PMC8905172 DOI: 10.1098/rsif.2021.0844] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
Abstract
The fate of ongoing infectious disease outbreaks is predicted through reproduction numbers, defining the long-term establishment of the infection, and epidemicity indices, tackling the reactivity of the infectious pool to new contagions. Prognostic metrics of unfolding outbreaks are of particular importance when designing adaptive emergency interventions facing real-time assimilation of epidemiological evidence. Our aim here is twofold. First, we propose a novel form of the epidemicity index for the characterization of cholera epidemics in spatial models of disease spread. Second, we examine in hindsight the survey of infections, treatments and containment measures carried out for the now extinct 2010–2019 Haiti cholera outbreak, to suggest that magnitude and timing of non-pharmaceutical and vaccination interventions imply epidemiological responses recapped by the evolution of epidemicity indices. Achieving negative epidemicity greatly accelerates fading of infections and thus proves a worthwhile target of containment measures. We also show that, in our model, effective reproduction numbers and epidemicity indices are explicitly related. Therefore, providing an upper bound to the effective reproduction number (significantly lower than the unit threshold) warrants negative epidemicity and, in turn, a rapidly fading outbreak preventing coalescence of sparse local sub-threshold flare-ups.
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Affiliation(s)
- Cristiano Trevisin
- Laboratory of Ecohydrology ENAC/IIE/ECHO, École polytechinque fédérale de Lausanne (EPFL), Lausanne 1015, Switzerland
| | - Joseph C Lemaitre
- Laboratory of Ecohydrology ENAC/IIE/ECHO, École polytechinque fédérale de Lausanne (EPFL), Lausanne 1015, Switzerland
| | - Lorenzo Mari
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano 20133, Italy
| | - Damiano Pasetto
- Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca' Foscari Venezia, Venezia 30172, Italy
| | - Marino Gatto
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano 20133, Italy
| | - Andrea Rinaldo
- Laboratory of Ecohydrology ENAC/IIE/ECHO, École polytechinque fédérale de Lausanne (EPFL), Lausanne 1015, Switzerland.,Dipartimento ICEA, Università degli studi di Padova, Padova 35131, Italy
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Mari L, Casagrandi R, Bertuzzo E, Pasetto D, Miccoli S, Rinaldo A, Gatto M. The epidemicity index of recurrent SARS-CoV-2 infections. Nat Commun 2021; 12:2752. [PMID: 33980858 PMCID: PMC8115165 DOI: 10.1038/s41467-021-22878-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2020] [Accepted: 03/30/2021] [Indexed: 01/29/2023] Open
Abstract
Several indices can predict the long-term fate of emerging infectious diseases and the effect of their containment measures, including a variety of reproduction numbers (e.g. [Formula: see text]). Other indices evaluate the potential for transient increases of epidemics eventually doomed to disappearance, based on generalized reactivity analysis. They identify conditions for perturbations to a stable disease-free equilibrium ([Formula: see text]) to grow, possibly causing significant damage. Here, we introduce the epidemicity index e0, a threshold-type indicator: if e0 > 0, initial foci may cause infection peaks even if [Formula: see text]. Therefore, effective containment measures should achieve a negative epidemicity index. We use spatially explicit models to rank containment measures for projected evolutions of the ongoing pandemic in Italy. There, we show that, while the effective reproduction number was below one for a sizable timespan, epidemicity remained positive, allowing recurrent infection flare-ups well before the major epidemic rebounding observed in the fall.
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Affiliation(s)
- Lorenzo Mari
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy.
| | - Renato Casagrandi
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy
| | - Enrico Bertuzzo
- Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca' Foscari Venezia, Venice, Italy
| | - Damiano Pasetto
- Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca' Foscari Venezia, Venice, Italy
| | - Stefano Miccoli
- Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy
| | - Andrea Rinaldo
- Laboratory of Ecohydrology, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.
- Dipartimento ICEA, Università di Padova, Padua, Italy.
| | - Marino Gatto
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy.
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Pillonetto G, Bisiacco M, Palù G, Cobelli C. Tracking the time course of reproduction number and lockdown's effect on human behaviour during SARS-CoV-2 epidemic: nonparametric estimation. Sci Rep 2021; 11:9772. [PMID: 33963235 PMCID: PMC8105401 DOI: 10.1038/s41598-021-89014-9] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2020] [Accepted: 04/14/2021] [Indexed: 01/10/2023] Open
Abstract
Understanding the SARS-CoV-2 dynamics has been subject of intense research in the last months. In particular, accurate modeling of lockdown effects on human behaviour and epidemic evolution is a key issue in order e.g. to inform health-care decisions on emergency management. In this regard, the compartmental and spatial models so far proposed use parametric descriptions of the contact rate, often assuming a time-invariant effect of the lockdown. In this paper we show that these assumptions may lead to erroneous evaluations on the ongoing pandemic. Thus, we develop a new class of nonparametric compartmental models able to describe how the impact of the lockdown varies in time. Our estimation strategy does not require significant Bayes prior information and exploits regularization theory. Hospitalized data are mapped into an infinite-dimensional space, hence obtaining a function which takes into account also how social distancing measures and people's growing awareness of infection's risk evolves as time progresses. This also permits to reconstruct a continuous-time profile of SARS-CoV-2 reproduction number with a resolution never reached before in the literature. When applied to data collected in Lombardy, the most affected Italian region, our model illustrates how people behaviour changed during the restrictions and its importance to contain the epidemic. Results also indicate that, at the end of the lockdown, around [Formula: see text] of people in Lombardy and [Formula: see text] in Italy was affected by SARS-CoV-2, with the fatality rate being 1.14%. Then, we discuss how the situation evolved after the end of the lockdown showing that the reproduction number dangerously increased in the summer, due to holiday relax, reaching values larger than one on August 1, 2020. Finally, we also document how Italy faced the second wave of infection in the last part of 2020. Since several countries still observe a growing epidemic and others could be subject to other waves, the proposed reproduction number tracking methodology can be of great help to health care authorities to prevent SARS-CoV-2 diffusion or to assess the impact of lockdown restrictions on human behaviour to contain the spread.
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Affiliation(s)
- G Pillonetto
- Department of Information Engineering, University of Padova, Padova, Italy.
| | - M Bisiacco
- Department of Information Engineering, University of Padova, Padova, Italy
| | - G Palù
- Department of Molecular Medicine, Professor Emeritus, University of Padova, Padova, Italy
- Member of the Scientific Technical Committee, Italian Ministry of Health, Rome, Italy
| | - C Cobelli
- Member of Consiglio Superiore di Sanità, Italian Ministry of Health, Rome, Italy
- Dipartimento di Salute della Donna e del Bambino, Professor Emeritus, University of Padova, Padova, Italy
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Bisiacco M, Pillonetto G. COVID-19 epidemic control using short-term lockdowns for collective gain. ANNUAL REVIEWS IN CONTROL 2021; 52:573-586. [PMID: 34849089 PMCID: PMC8616743 DOI: 10.1016/j.arcontrol.2021.10.017] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2021] [Accepted: 10/29/2021] [Indexed: 05/17/2023]
Abstract
While many efforts are currently devoted to vaccines development and administration, social distancing measures, including severe restrictions such as lockdowns, remain fundamental tools to contain the spread of COVID-19. A crucial point for any government is to understand, on the basis of the epidemic curve, the right temporal instant to set up a lockdown and then to remove it. Different strategies are being adopted with distinct shades of intensity. USA and Europe tend to introduce restrictions of considerable temporal length. They vary in time: a severe lockdown may be reached and then gradually relaxed. An interesting alternative is the Australian model where short and sharp responses have repeatedly tackled the virus and allowed people a return to near normalcy. After a few positive cases are detected, a lockdown is immediately set. In this paper we show that the Australian model can be generalized and given a rigorous mathematical analysis, casting strategies of the type short-term pain for collective gain in the context of sliding-mode control, an important branch of nonlinear control theory. This allows us to gain important insights regarding how to implement short-term lockdowns, obtaining a better understanding of their merits and possible limitations. Effects of vaccines administration in improving the control law's effectiveness are also illustrated. Our model predicts the duration of the severe lockdown to be set to maintain e.g. the number of people in intensive care under a certain threshold. After tuning our strategy exploiting data collected in Italy, it turns out that COVID-19 epidemic could be e.g. controlled by alternating one or two weeks of complete lockdown with one or two months of freedom, respectively. Control strategies of this kind, where the lockdown's duration is well circumscribed, could be important also to alleviate coronavirus impact on economy.
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Affiliation(s)
- Mauro Bisiacco
- Department of Information Engineering, University of Padova, Padova, Italy
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Nkurunziza G, Omara T, Kiwanuka Nakiguli C, Mukasa P, Byamugisha D, Ntambi E. Physicochemical Quality of Water from Chuho Springs, Kisoro District, Uganda. FRENCH-UKRAINIAN JOURNAL OF CHEMISTRY 2021. [DOI: 10.17721/fujcv9i2p12-26] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In the current study, water from Chuho springs used as the main water source in Kisoro municipality, Uganda were assessed for their suitability as drinking water. The temperature, turbidity, conductivity, total dissolved solids, dissolved oxygen, biological oxygen demand, total hardness, total alkalinity, calcium, magnesium, phosphates, iron, copper, arsenic, chlorides and the fluoride content of the water samples were determined. Not all the parameters met World Health Organizations’ guidelines for drinking water. Temperature, dissolved oxygen and fluorides were outside the recommended limits of 15 ℃, 10-12 mg/L and 1.5 mg/L, respectively. Further studies should assess the microbiological and sanitary profile of the springs.
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Chen R. Transient inconsistency between population density and fisheries yields without bycatch species extinction. Ecol Evol 2020; 10:12372-12384. [PMID: 33209295 PMCID: PMC7663084 DOI: 10.1002/ece3.6868] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2020] [Revised: 09/10/2020] [Accepted: 09/11/2020] [Indexed: 11/06/2022] Open
Abstract
Recent studies have demonstrated the great advantages of marine reserves in solving bycatch problems by maintaining the persistence (i.e., avoid extinction) of endangered species without sacrificing the fisheries yields of target species. However, transient phenomena rather than equilibrium states of population dynamics still require further research. Here, with a simple and general model, the transient dynamics of the target fish species are investigated under management which minimizes extinction risk of the bycatch species. An interesting finding is that fisheries yields can strongly fluctuate even if population density both inside and outside marine reserve only slightly varies (or vice versa), leading to transient inconsistency between the population densities and fisheries yields. This finding suggests that population density dynamics of the target fish species cannot be used to predict the transient phenomena of fisheries yields (or vice versa) in fisheries management. However, the unpredictability can be receded as the sensitivity analyses show that a large marine reserve size and low escapement rate can shorten the transient duration.
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Affiliation(s)
- Renfei Chen
- School of Life ScienceShanxi Normal UniversityLinfenChina
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Carraro L, Bertuzzo E, Fronhofer EA, Furrer R, Gounand I, Rinaldo A, Altermatt F. Generation and application of river network analogues for use in ecology and evolution. Ecol Evol 2020; 10:7537-7550. [PMID: 32760547 PMCID: PMC7391543 DOI: 10.1002/ece3.6479] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Revised: 05/21/2020] [Accepted: 05/26/2020] [Indexed: 01/19/2023] Open
Abstract
Several key processes in freshwater ecology are governed by the connectivity inherent to dendritic river networks. These have extensively been analyzed from a geomorphological and hydrological viewpoint, yet structures classically used in ecological modeling have been poorly representative of the structure of real river basins, often failing to capture well-known scaling features of natural rivers. Pioneering work identified optimal channel networks (OCNs) as spanning trees reproducing all scaling features characteristic of natural stream networks worldwide. While OCNs have been used to create landscapes for studies on metapopulations, biodiversity, and epidemiology, their generation has not been generally accessible.Given the increasing interest in dendritic riverine networks by ecologists and evolutionary biologists, we here present a method to generate OCNs and, to facilitate its application, we provide the R-package OCNet. Owing to the stochastic process generating OCNs, multiple network replicas spanning the same surface can be built; this allows performing computational experiments whose results are irrespective of the particular shape of a single river network. The OCN construct also enables the generation of elevational gradients derived from the optimal network configuration, which can constitute three-dimensional landscapes for spatial studies in both terrestrial and freshwater realms. Moreover, the package provides functions that aggregate OCNs into an arbitrary number of nodes, calculate several descriptors of river networks, and draw relevant network features.We describe the main functionalities of the package and its integration with other R-packages commonly used in spatial ecology. Moreover, we exemplify the generation of OCNs and discuss an application to a metapopulation model for an invasive riverine species.In conclusion, OCNet provides a powerful tool to generate realistic river network analogues for various applications. It thereby allows the design of spatially realistic studies in increasingly impacted ecosystems and enhances our knowledge on spatial processes in freshwater ecology in general.
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Affiliation(s)
- Luca Carraro
- Department of Aquatic EcologySwiss Federal Institute of Aquatic Science and Technology (Eawag)DübendorfSwitzerland
- Department of Evolutionary Biology and Environmental StudiesUniversity of ZurichZürichSwitzerland
| | - Enrico Bertuzzo
- Department of Environmental Sciences, Informatics and StatisticsUniversity of Venice Ca' FoscariVeniceItaly
| | | | - Reinhard Furrer
- Department of Mathematics and Department of Computational ScienceUniversity of ZurichZürichSwitzerland
| | - Isabelle Gounand
- Department of Aquatic EcologySwiss Federal Institute of Aquatic Science and Technology (Eawag)DübendorfSwitzerland
- Department of Evolutionary Biology and Environmental StudiesUniversity of ZurichZürichSwitzerland
- CNRSUPECCNRSIRDINRA, Institut d’écologie et des sciences de l'environnement, IEESSorbonne UniversitéParisFrance
| | - Andrea Rinaldo
- Laboratory of EcohydrologySwiss Federal Institute of Technology in Lausanne (EPFL)LausanneSwitzerland
- Department of Civil, Environmental and Architectural EngineeringUniversity of PaduaPadovaItaly
| | - Florian Altermatt
- Department of Aquatic EcologySwiss Federal Institute of Aquatic Science and Technology (Eawag)DübendorfSwitzerland
- Department of Evolutionary Biology and Environmental StudiesUniversity of ZurichZürichSwitzerland
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