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Baysazan E, Berker AN, Mandal H, Kaygusuz H. COVID-19 modeling based on real geographic and population data. Turk J Med Sci 2023; 53:333-339. [PMID: 36945958 PMCID: PMC10387910 DOI: 10.55730/1300-0144.5589] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Accepted: 12/31/2022] [Indexed: 02/26/2023] Open
Abstract
BACKGROUND : Intercity travel is one of the most important parameters for combating a pandemic. The ongoing COVID-19 pandemic has resulted in different computational studies involving intercity connections. In this study, the effects of intercity connections during an epidemic such as COVID-19 are evaluated using a new network model. METHODS This model considers the actual geographic neighborhood and population density data. This new model is applied to actual Turkish data by means of provincial connections and populations. A Monte Carlo algorithm with a hybrid lattice model is applied to a lattice with 8802 data points. RESULTS Around Monte Carlo step 70, the number of active cases in Türkiye reaches up to 8.0% of the total population, which is followed by a second wave at around Monte Carlo step 100. The number of active cases vanishes around Monte Carlo step 160. Starting with İstanbul, the epidemic quickly expands between steps 60 and 100. Simulation results fit the actual mortality data in Türkiye. DISCUSSION This model is quantitatively very efficient in modeling real-world COVID-19 epidemic data based on populations and geographical intercity connections, by means of estimating the number of deaths, disease spread, and epidemic termination.
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Affiliation(s)
- Emir Baysazan
- TEBIP High Performers Program, Council of Higher Education, İstanbul University, İstanbul, Turkey
| | - Ahmet Nihat Berker
- Faculty of Engineering and Natural Sciences, Kadir Has University, İstanbul, Turkey; TÜBİTAK Research Institute for Fundamental Sciences, Kocaeli, Turkey; Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
| | - Hasan Mandal
- The Scientific and Technological Research Council of Türkiye (TÜBİTAK), Ankara, Turkey
| | - Hakan Kaygusuz
- Department of Basic Sciences, Faculty of Engineering and Architecture, Altınbaş University, İstanbul, Turkey; SUNUM Nanotechnology Research Center, Sabancı University, İstanbul, Turkey
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2
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Triambak S, Mahapatra DP, Mallick N, Sahoo R. A new logistic growth model applied to COVID-19 fatality data. Epidemics 2021; 37:100515. [PMID: 34763160 PMCID: PMC8556694 DOI: 10.1016/j.epidem.2021.100515] [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: 08/10/2020] [Revised: 08/02/2021] [Accepted: 10/21/2021] [Indexed: 12/23/2022] Open
Abstract
BACKGROUND Recent work showed that the temporal growth of the novel coronavirus disease (COVID-19) follows a sub-exponential power-law scaling whenever effective control interventions are in place. Taking this into consideration, we present a new phenomenological logistic model that is well-suited for such power-law epidemic growth. METHODS We empirically develop the logistic growth model using simple scaling arguments, known boundary conditions and a comparison with available data from four countries, Belgium, China, Denmark and Germany, where (arguably) effective containment measures were put in place during the first wave of the pandemic. A non-linear least-squares minimization algorithm is used to map the parameter space and make optimal predictions. RESULTS Unlike other logistic growth models, our presented model is shown to consistently make accurate predictions of peak heights, peak locations and cumulative saturation values for incomplete epidemic growth curves. We further show that the power-law growth model also works reasonably well when containment and lock down strategies are not as stringent as they were during the first wave of infections in 2020. On the basis of this agreement, the model was used to forecast COVID-19 fatalities for the third wave in South Africa, which was in progress during the time of this work. CONCLUSION We anticipate that our presented model will be useful for a similar forecasting of COVID-19 induced infections/deaths in other regions as well as other cases of infectious disease outbreaks, particularly when power-law scaling is observed.
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Affiliation(s)
- S Triambak
- Department of Physics and Astronomy, University of the Western Cape, P/B X17, Bellville 7535, South Africa.
| | - D P Mahapatra
- Department of Physics, Utkal University, Vani Vihar, Bhubaneshwar 751004, India.
| | - N Mallick
- Department of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India
| | - R Sahoo
- Department of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India
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Triambak S, Mahapatra DP. A random walk Monte Carlo simulation study of COVID-19-like infection spread. PHYSICA A 2021; 574:126014. [PMID: 33875903 PMCID: PMC8047309 DOI: 10.1016/j.physa.2021.126014] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2020] [Revised: 03/05/2021] [Indexed: 05/30/2023]
Abstract
Recent analysis of early COVID-19 data from China showed that the number of confirmed cases followed a subexponential power-law increase, with a growth exponent of around 2.2 (Maier and Brockmann, 2020). The power-law behavior was attributed to a combination of effective containment and mitigation measures employed as well as behavioral changes by the population. In this work, we report a random walk Monte Carlo simulation study of proximity-based infection spread. Control interventions such as lockdown measures and mobility restrictions are incorporated in the simulations through a single parameter, the size of each step in the random walk process. The step size l is taken to be a multiple of 〈 r 〉 , which is the average separation between individuals. Three temporal growth regimes (quadratic, intermediate power-law and exponential) are shown to emerge naturally from our simulations. For l = 〈 r 〉 , we get intermediate power-law growth exponents that are in general agreement with available data from China. On the other hand, we obtain a quadratic growth for smaller step sizes l ≲ 〈 r 〉 ∕ 2 , while for large l the growth is found to be exponential. We further performed a comparative case study of early fatality data (under varying levels of lockdown conditions) from three other countries, India, Brazil and South Africa. We show that reasonable agreement with these data can be obtained by incorporating small-world-like connections in our simulations.
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Affiliation(s)
- S Triambak
- Department of Physics and Astronomy, University of the Western Cape, P/B X17, Bellville 7535, South Africa
| | - D P Mahapatra
- Department of Physics, Utkal University, Vani Vihar, Bhubaneshwar 751004, India
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Hamada M, Takasu F. Equilibrium properties of the spatial SIS model as a point pattern dynamics - How is infection distributed over space? J Theor Biol 2019; 468:12-26. [PMID: 30738052 DOI: 10.1016/j.jtbi.2019.02.005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2018] [Revised: 02/01/2019] [Accepted: 02/06/2019] [Indexed: 10/27/2022]
Abstract
We revisit the classical epidemiological SIS model as a stochastic point pattern dynamics with special focus on its spatial distribution at equilibrium. In this model, each point on a continuous space is either susceptible S or infectious I, and infection occurs with an infection kernel as a function of distance from I to S. This stochastic process has been mathematically described by the hierarchical dynamics of the probabilities that a point, a pair made by two points, and a triplet made by three points, etc., is in a specific configuration of status. Using a simple closure thereby triplet probabilities that appear in the dynamics are approximated, we show that the average singlet probabilities and the pair probabilities that describe spatial distributions of Ss and Is at equilibrium can be explicitly derived using the infection kernel; Is are spatially clustered in the same order of the infection kernel. The results highlight the advantage of point pattern approach to model spatial population dynamics in general ecology where local interactions among individuals likely depend on distance between them.
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Affiliation(s)
- Miki Hamada
- Graduate School of Humanities and Sciences, Nara Women's University, Kita-Uoya Nishimachi, Nara 630-8506, Japan.
| | - Fugo Takasu
- Department of Environmental Science, Nara Women's University, Kita-Uoya Nishimachi, Nara 630-8506, Japan.
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5
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Dynamics analysis of SIR epidemic model with correlation coefficients and clustering coefficient in networks. J Theor Biol 2018; 449:1-13. [PMID: 29649430 DOI: 10.1016/j.jtbi.2018.04.007] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2017] [Revised: 04/02/2018] [Accepted: 04/04/2018] [Indexed: 11/22/2022]
Abstract
In this paper, the correlation coefficients between nodes in states are used as dynamic variables, and we construct SIR epidemic dynamic models with correlation coefficients by using the pair approximation method in static networks and dynamic networks, respectively. Considering the clustering coefficient of the network, we analytically investigate the existence and the local asymptotic stability of each equilibrium of these models and derive threshold values for the prevalence of diseases. Additionally, we obtain two equivalent epidemic thresholds in dynamic networks, which are compared with the results of the mean field equations.
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6
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Hiebeler DE, Audibert A, Strubell E, Michaud IJ. An epidemiological model of internet worms with hierarchical dispersal and spatial clustering of hosts. J Theor Biol 2017; 418:8-15. [DOI: 10.1016/j.jtbi.2017.01.035] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2016] [Revised: 12/20/2016] [Accepted: 01/19/2017] [Indexed: 10/20/2022]
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Leclerc M, Doré T, Gilligan CA, Lucas P, Filipe JAN. Estimating the delay between host infection and disease (incubation period) and assessing its significance to the epidemiology of plant diseases. PLoS One 2014; 9:e86568. [PMID: 24466153 PMCID: PMC3899291 DOI: 10.1371/journal.pone.0086568] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2013] [Accepted: 12/11/2013] [Indexed: 11/18/2022] Open
Abstract
Knowledge of the incubation period of infectious diseases (time between host infection and expression of disease symptoms) is crucial to our epidemiological understanding and the design of appropriate prevention and control policies. Plant diseases cause substantial damage to agricultural and arboricultural systems, but there is still very little information about how the incubation period varies within host populations. In this paper, we focus on the incubation period of soilborne plant pathogens, which are difficult to detect as they spread and infect the hosts underground and above-ground symptoms occur considerably later. We conducted experiments on Rhizoctonia solani in sugar beet, as an example patho-system, and used modelling approaches to estimate the incubation period distribution and demonstrate the impact of differing estimations on our epidemiological understanding of plant diseases. We present measurements of the incubation period obtained in field conditions, fit alternative probability models to the data, and show that the incubation period distribution changes with host age. By simulating spatially-explicit epidemiological models with different incubation-period distributions, we study the conditions for a significant time lag between epidemics of cryptic infection and the associated epidemics of symptomatic disease. We examine the sensitivity of this lag to differing distributional assumptions about the incubation period (i.e. exponential versus Gamma). We demonstrate that accurate information about the incubation period distribution of a pathosystem can be critical in assessing the true scale of pathogen invasion behind early disease symptoms in the field; likewise, it can be central to model-based prediction of epidemic risk and evaluation of disease management strategies. Our results highlight that reliance on observation of disease symptoms can cause significant delay in detection of soil-borne pathogen epidemics and mislead practitioners and epidemiologists about the timing, extent, and viability of disease control measures for limiting economic loss.
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Affiliation(s)
- Melen Leclerc
- UMR 1349 Institute for Genetics Environment and Plant Protection, Institut National de la Recherche Agronomique – Agrocampus Ouest – Université Rennes 1, Le Rheu, France
- UR 546 Biostatistics and Spatial Processes Unit, Institut National de la Recherche Agronomique, Avignon, France
- UAR 1240 Unité Impacts Ecologiques des Innovations en Production Végétale, Institut National de la Recherche Agronomique, Thiverval-Grignon, France
| | - Thierry Doré
- UMR 211 Agronomie, AgroParisTech, Thiverval-Grignon, France
- UMR 211 Agronomie, Institut National de la Recherche Agronomique, Thiverval-Grignon, France
| | - Christopher A. Gilligan
- Epidemiology and Modelling Group, Department of Plant Sciences, University of Cambridge, Cambridge, United Kingdom
| | - Philippe Lucas
- UMR 1349 Institute for Genetics Environment and Plant Protection, Institut National de la Recherche Agronomique – Agrocampus Ouest – Université Rennes 1, Le Rheu, France
| | - João A. N. Filipe
- Epidemiology and Modelling Group, Department of Plant Sciences, University of Cambridge, Cambridge, United Kingdom
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Del Genio CI, House T. Endemic infections are always possible on regular networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:040801. [PMID: 24229103 DOI: 10.1103/physreve.88.040801] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2013] [Indexed: 06/02/2023]
Abstract
We study the dependence of the largest component in regular networks on the clustering coefficient, showing that its size changes smoothly without undergoing a phase transition. We explain this behavior via an analytical approach based on the network structure, and provide an exact equation describing the numerical results. Our work indicates that intrinsic structural properties always allow the spread of epidemics on regular networks.
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Affiliation(s)
- Charo I Del Genio
- Warwick Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom and Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom and Warwick Infectious Disease Epidemiology Research (WIDER) Centre, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom and Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, Dresden D-01187, Germany
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Markham DC, Simpson MJ, Baker RE. Simplified method for including spatial correlations in mean-field approximations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062702. [PMID: 23848710 DOI: 10.1103/physreve.87.062702] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2013] [Indexed: 06/02/2023]
Abstract
Biological systems involving proliferation, migration, and death are observed across all scales. For example, they govern cellular processes such as wound healing, as well as the population dynamics of groups of organisms. In this paper, we provide a simplified method for correcting mean-field approximations of volume-excluding birth-death-movement processes on a regular lattice. An initially uniform distribution of agents on the lattice may give rise to spatial heterogeneity, depending on the relative rates of proliferation, migration, and death. Many frameworks chosen to model these systems neglect spatial correlations, which can lead to inaccurate predictions of their behavior. For example, the logistic model is frequently chosen, which is the mean-field approximation in this case. This mean-field description can be corrected by including a system of ordinary differential equations for pairwise correlations between lattice site occupancies at various lattice distances. In this work we discuss difficulties with this method and provide a simplification in the form of a partial differential equation description for the evolution of pairwise spatial correlations over time. We test our simplified model against the more complex corrected mean-field model, finding excellent agreement. We show how our model successfully predicts system behavior in regions where the mean-field approximation shows large discrepancies. Additionally, we investigate regions of parameter space where migration is reduced relative to proliferation, which has not been examined in detail before and find our method is successful at correcting the deviations observed in the mean-field model in these parameter regimes.
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Affiliation(s)
- Deborah C Markham
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, United Kingdom.
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KOSHIBA SHINKO, SENO HIROMI. A MATHEMATICAL MODEL FOR SPATIALLY EXPANDING INFECTED AREA OF EPIDEMICS TRANSMITTED THROUGH HETEROGENEOUSLY DISTRIBUTED SUSCEPTIBLE UNITS. J BIOL SYST 2011. [DOI: 10.1142/s0218339005001471] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Little is known about the effect of environmental heterogeneity on the spatial expansion of epidemics. In this work, to focus on the question of how the extent of epidemic damage depends on the spatial distribution of susceptible units, we develop a mathematical model with a simple stochastic process, and analyze it. We assume that the unit of infection is immobile, as town, plant, etc. and classify the units into three classes: susceptible, infective and recovered. We consider the range expanded by infected units, the infected rangeR, assuming a certain generalized relation between R and the total number of infected units k, making use of an index, a sort of fractal dimension, to characterize the spatial distribution of infected units. From the results of our modeling analysis, we show that the expected velocity of spatial expansion of infected range is significantly affected by the fractal nature of spatial distribution of immobile susceptible units, and is temporally variable. When the infection finally terminates at a moment, the infected range at the moment is closely related to the nature of spatial distribution of immobile susceptible units, which is explicitly demonstrated in our analysis.
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Affiliation(s)
- SHINKO KOSHIBA
- Department of Information and Computer Sciences, Faculty of Science, Nara Women's University, Nara 630-8506, Japan
| | - HIROMI SENO
- Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, Higashi-hiroshima 739-8526, Japan
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12
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Raghib M, Hill NA, Dieckmann U. A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics. J Math Biol 2010; 62:605-53. [DOI: 10.1007/s00285-010-0345-9] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2009] [Revised: 03/29/2010] [Indexed: 11/30/2022]
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13
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Critical parameters for modelling the spread of foot-and-mouth disease in wildlife. Epidemiol Infect 2009; 138:125-38. [DOI: 10.1017/s0950268809002829] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022] Open
Abstract
SUMMARYA series of simulation experiments was conducted to determine how estimates of the latent and infectious periods, number of neighbours (contacts) and population size impact on the predicted magnitude and distribution of foot-and-mouth disease (FMD) outbreaks in white-tailed deer in southern Texas. Outbreaks were simulated using a previously developed and applied susceptible–latent–infected–recovered geographic automata model. There were substantial differences in the estimated predicted number of deer and locations infected, based on the model parameters used (3779–119 879 deer infected and 227–6526 locations affected). There were also substantial differences in the spatial risk of infection based on the model parameters used. The predicted spread of FMD was found to be most sensitive to the assumed latent period and the assumed number of contacts. How these parameters are estimated is likely to be critical in studies on the impact of FMD spread in situations in which wildlife reservoirs might potentially exist.
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15
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Long-range correlations improve understanding of the influence of network structure on contact dynamics. Theor Popul Biol 2008; 73:383-94. [DOI: 10.1016/j.tpb.2007.12.006] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2006] [Revised: 10/31/2007] [Accepted: 12/11/2007] [Indexed: 11/22/2022]
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Davidson RS, Marion G, Hutchings MR. Effects of host social hierarchy on disease persistence. J Theor Biol 2008; 253:424-33. [PMID: 18485373 DOI: 10.1016/j.jtbi.2008.03.021] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2007] [Revised: 03/20/2008] [Accepted: 03/21/2008] [Indexed: 11/18/2022]
Abstract
The effects of social hierarchy on population dynamics and epidemiology are examined through a model which contains a number of fundamental features of hierarchical systems, but is simple enough to allow analytical insight. In order to allow for differences in birth rates, contact rates and movement rates among different sets of individuals the population is first divided into subgroups representing levels in the hierarchy. Movement, representing dominance challenges, is allowed between any two levels, giving a completely connected network. The model includes hierarchical effects by introducing a set of dominance parameters which affect birth rates in each social level and movement rates between social levels, dependent upon their rank. Although natural hierarchies vary greatly in form, the skewing of contact patterns, introduced here through non-uniform dominance parameters, has marked effects on the spread of disease. A simple homogeneous mixing differential equation model of a disease with SI dynamics in a population subject to simple birth and death process is presented and it is shown that the hierarchical model tends to this as certain parameter regions are approached. Outside of these parameter regions correlations within the system give rise to deviations from the simple theory. A Gaussian moment closure scheme is developed which extends the homogeneous model in order to take account of correlations arising from the hierarchical structure, and it is shown that the results are in reasonable agreement with simulations across a range of parameters. This approach helps to elucidate the origin of hierarchical effects and shows that it may be straightforward to relate the correlations in the model to measurable quantities which could be used to determine the importance of hierarchical corrections. Overall, hierarchical effects decrease the levels of disease present in a given population compared to a homogeneous unstructured model, but show higher levels of disease than structured models with no hierarchy. The separation between these three models is greatest when the rate of dominance challenges is low, reducing mixing, and when the disease prevalence is low. This suggests that these effects will often need to be considered in models being used to examine the impact of control strategies where the low disease prevalence behaviour of a model is critical.
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Affiliation(s)
- Ross S Davidson
- Animal Nutrition and Health Department, SAC, West Mains Road, Edinburgh EH9 3JG, UK.
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Highfield L, Ward MP, Laffan SW. Representation of animal distributions in space: how geostatistical estimates impact simulation modeling of foot-and-mouth disease spread. Vet Res 2008; 39:17. [PMID: 18258171 DOI: 10.1051/vetres:2007055] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2007] [Accepted: 10/22/2007] [Indexed: 11/14/2022] Open
Abstract
Modeling potential disease spread in wildlife populations is important for predicting, responding to and recovering from a foreign animal disease incursion. To make spatial epidemic predictions, the target animal species of interest must first be represented in space. We conducted a series of simulation experiments to determine how estimates of the spatial distribution of white-tailed deer impact the predicted magnitude and distribution of foot-and-mouth disease (FMD) outbreaks. Outbreaks were simulated using a susceptible-infected-recovered geographic automata model. The study region was a 9-county area (24 000 km(2)) of southern Texas. Methods used for creating deer distributions included dasymetric mapping, kriging and remotely sensed image analysis. The magnitudes and distributions of the predicted outbreaks were evaluated by comparing the median number of deer infected and median area affected (km(2)), respectively. The methods were further evaluated for similar predictive power by comparing the model predicted outputs with unweighted pair group method with arithmetic mean (UPGMA) clustering. There were significant differences in the estimated number of deer in the study region, based on the geostatistical estimation procedure used (range: 385 939-768 493). There were also substantial differences in the predicted magnitude of the FMD outbreaks (range: 1 563-8 896) and land area affected (range: 56-447 km(2)) for the different estimated animal distributions. UPGMA clustering indicated there were two main groups of distributions, and one outlier. We recommend that one distribution from each of these two groups be used to model the range of possible outbreaks. Methods included in cluster 1 (such as county-level disaggregation) could be used in conjunction with any of the methods in cluster 2, which included kriging, NDVI split by ecoregion, or disaggregation at the regional level, to represent the variability in the model predicted outbreak distributions. How animal populations are represented needs to be considered in all spatial disease spread models.
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Affiliation(s)
- Linda Highfield
- Department of Veterinary Integrative Biosciences, Texas A&M University College of Veterinary Medicine & Biomedical Sciences, College Station, TX 77845-4458, USA.
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Swain D, Hutchings M, Marion G. Using a spatially explicit model to understand the impact of search rate and search distance on spatial heterogeneity within an herbivore grazing system. Ecol Modell 2007. [DOI: 10.1016/j.ecolmodel.2006.11.031] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Walker DM, Pérez-Barbería F, Marion G. Stochastic modelling of ecological processes using hybrid Gibbs samplers. Ecol Modell 2006. [DOI: 10.1016/j.ecolmodel.2006.04.008] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Abstract
An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints. Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.
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Affiliation(s)
- David Hiebeler
- Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA.
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22
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A Mathematical Model for Invasion Range of Population Dispersion Through a Patchy Environment. Biol Invasions 2005. [DOI: 10.1007/s10530-005-5211-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Doran RJ, Laffan SW. Simulating the spatial dynamics of foot and mouth disease outbreaks in feral pigs and livestock in Queensland, Australia, using a susceptible-infected-recovered cellular automata model. Prev Vet Med 2005; 70:133-52. [PMID: 15967247 DOI: 10.1016/j.prevetmed.2005.03.002] [Citation(s) in RCA: 79] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2004] [Revised: 01/18/2005] [Accepted: 03/10/2005] [Indexed: 11/25/2022]
Abstract
We describe an approach to modelling the spatio-temporal spread of foot and mouth disease through feral animal and unfenced livestock populations. We used a susceptible-infected-recovered model, implemented in a cellular automata framework, to assess the spread of FMD across two regions of Queensland, Australia. Following a sensitivity analysis on the infectious states, scenario analyses were conducted using feral pigs only as the susceptible population, and then with the addition of livestock, and initiated in the wet season and in the dry season. The results indicate that, depending on the season the outbreak is initiated, and without the implementation of control measures, an outbreak of Foot and Mouth Disease around Winton could continue unchecked, while an outbreak around Cape York may die out naturally. The approach explicitly incorporates the spatial relationships between the populations through which the disease spreads and provides a framework by which the spread of disease outbreaks can be explored through varying the model parameters. It highlights the emergence and importance of spatio-temporal patterns, something that previous modelling of FMD in feral animal and unfenced livestock populations has lacked.
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Affiliation(s)
- Ruth J Doran
- School of Resources, Environment and Society, Australian National University, Canberra 0200, Australia
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Marion G, Swain DL, Hutchings MR. Understanding foraging behaviour in spatially heterogeneous environments. J Theor Biol 2005; 232:127-42. [PMID: 15498600 DOI: 10.1016/j.jtbi.2004.08.005] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2003] [Revised: 06/29/2004] [Accepted: 08/04/2004] [Indexed: 10/26/2022]
Abstract
The role of stochasticity and spatial heterogeneity in foraging systems is investigated. We formulate a spatially explicit model which describes the behaviour of grazing animals in response to local information using simple stochastic rules. In particular the model reflects the biology in that decisions to move to a new location are based on visual assessment of the sward height in a surrounding neighbourhood, whilst the decision to graze the current location is based on the residual sward height and olfactory assessment of local faecal contamination. It is assumed that animals do not interact directly, but do so through modification of, and response to a common environment. Spatial heterogeneity is shown to have significant effects including reducing the equilibrium intake rate and increasing the optimal stocking density, and must therefore be taken into account by resource managers. We demonstrate the relationship between the stochastic spatial model and its non-spatial deterministic counterpart, and in the process derive a moment-closure approximation to the full process, which can be regarded as an intermediate, or pseudo-spatial model. The role of spatial heterogeneity is emphasized, and better understood by comparing the results obtained from each approach. The relative efficiency of random and directed searching behaviour in spatially heterogeneous environments is explored for both clean and contaminated pastures, and the impact of faecal avoidance behaviour assessed.
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Affiliation(s)
- Glenn Marion
- Biomathematics & Statistics Scotland, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK.
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25
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Jeger MJ. Analysis of disease progress as a basis for evaluating disease management practices. ANNUAL REVIEW OF PHYTOPATHOLOGY 2004; 42:61-82. [PMID: 15283660 DOI: 10.1146/annurev.phyto.42.040803.140427] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The relationship between epidemiology and disease management is long-standing but sometimes tenuous. It may seem self-evident that improved understanding of epidemic processes will lead to more effective control practices but this remains a testable proposition rather than demonstrated reality. A wide range of models differing in mathematical sophistication and computational complexity has been proposed as a means of achieving a greater understanding of epidemiology and carrying this through to improved management. The potential exists to align these modeling approaches to evaluation of control practices and prediction of the consequent epidemic outcomes, but these have yet to make a major impact on practical disease management. For the immediate future simpler pragmatic approaches for analysis of disease progress, using nonlinear growth functions and/or integrated measures such as area under disease progress curves, will play a key role in informing tactical and strategic decisions on control treatments. These approaches have proved useful in describing control effectiveness and, in some cases, optimizing or changing control practices.
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Affiliation(s)
- M J Jeger
- Department of Agricultural Sciences, Imperial College London, Wye Campus, Ashford, Kent TN25 5AH, United Kingdom.
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26
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Abstract
The nature of pathogen transport mechanisms strongly determines the spatial pattern of disease and, through this, the dynamics and persistence of epidemics in plant populations. Up to recently, the range of possible mechanisms or interactions assumed by epidemic models has been limited: either independent of the location of individuals (mean-field models) or restricted to local contacts (between nearest neighbours or decaying exponentially with distance). Real dispersal processes are likely to lie between these two extremes, and many are well described by long-tailed contact kernels such as power laws. We investigate the effect of different spatial dispersal mechanisms on the spatio-temporal spread of disease epidemics by simulating a stochastic Susceptible-infective model motivated by previous data analyses. Both long-term stationary behaviour (in the presence of a control or recovery process) and transient behaviour (which varies widely within and between epidemics) are examined. We demonstrate the relationship between epidemic size and disease pattern (characterized by spatial autocorrelation), and its dependence on dispersal and infectivity parameters. Special attention is given to boundary effects, which can decrease disease levels significantly relative to standard, periodic geometries in cases of long-distance dispersal. We propose and test a definition of transient duration which captures the dependence of transients on dispersal mechanisms. We outline an analytical approach that represents the behaviour of the spatially-explicit model, and use it to prove that the epidemic size is predicted exactly by the mean-field model (in the limit of an infinite system) when dispersal is sufficiently long ranged (i.e. when the power-law exponent a</=2).
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Affiliation(s)
- J A N Filipe
- Department of Plant Sciences, The University of Cambridge, Downing Street, Cambridge CB2 3EA, UK.
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27
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Bolker BM. Combining endogenous and exogenous spatial variability in analytical population models. Theor Popul Biol 2003; 64:255-70. [PMID: 14522167 DOI: 10.1016/s0040-5809(03)00090-x] [Citation(s) in RCA: 78] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Abstract
Analytically tractable models of dynamics in continuous space rarely incorporate both endogenous and exogenous spatial heterogeneity. We use spatial moment equations in combination with simulation models to analyze the combined effects of endogenous and exogenous variability on population viability in a simple single-population model where landscape heterogeneity and local population density both affect mortality rate. The equations partition the effects of heterogeneity into an effect of local crowding and an effect of habitat association caused by differential mortality. Exogenous heterogeneity in mortality rate increases population viability through habitat association and decreases it through increased crowding; the net effect of exogenous heterogeneity is generally to improve population viability. This result is contrary to some (but not all) conclusions in the literature, which usually focus on the effects of fragmentation rather than the benefits of refuges to short-dispersing individuals.
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Affiliation(s)
- Benjamin M Bolker
- Zoology Department, University of Florida, Box 118525, 318 Carr Hall, Gainesville, FL 32611-8525, USA.
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28
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Abstract
To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here we evaluate the effectiveness of the pair-edge approximation on a spatially heterogeneous lattice in which some sites are unoccupiable, or "dead". This model has several possible interpretations, including a spatial SIS epidemic model, in which some sites are occupied by immobile host-species individuals while others are empty. We find that, as in the homogeneous model, the pair-edge approximation is significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of "dead" sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveals limitations of pair approximation in the presence of spatial heterogeneity.
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Affiliation(s)
- Nikkala A Thomson
- Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA.
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29
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Filipe JAN, Maule MM. Analytical methods for predicting the behaviour of population models with general spatial interactions. Math Biosci 2003; 183:15-35. [PMID: 12604133 DOI: 10.1016/s0025-5564(02)00224-9] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.
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Affiliation(s)
- J A N Filipe
- Biomathematics and Statistics Scotland, The King's Buildings (JCMB), EH9 3JZ, Edinburgh, UK.
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30
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Marion G, Mao X, Renshaw E, Liu J. Spatial heterogeneity and the stability of reaction states in autocatalysis. PHYSICAL REVIEW E 2002; 66:051915. [PMID: 12513531 DOI: 10.1103/physreve.66.051915] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2002] [Revised: 07/05/2002] [Indexed: 11/07/2022]
Abstract
The impact of stochasticity and spatial heterogeneity on the quadratic autocatalytic system is studied. In a nonspatial setting the reactive state of the system is found to be unstable in small volumes where internal fluctuations drive the system to the unreactive state. This phenomena is of potential importance to the stability of reactions in biological cells. A simple spatial model is constructed by linking N nonspatial models via migration of reactants controlled by a mixing rate lambda. Simulation of this stochastic process demonstrates the importance of such mixing in controlling the impact of internal fluctuations on the stability of the autocatalytic reaction. For high mixing rate the mean reactant levels in equilibrium correspond to the well-mixed deterministic system, although a significant degree of spatial heterogeneity remains. For intermediate mixing rates, mean reactant levels vary continuously with lambda, where the interaction of internal fluctuations with limited spatial mixing modifies the reactive states of the deterministic system. However, there is a threshold below which mixing is unable to control internal fluctuations which drive the system into the unreactive state. Thus a critical minimum level of communication between the cells is required to stabilize the reaction across the entire system. Approximate analytic results, obtained using moment-closure techniques, support these findings and demonstrate the relationship between the spatial stochastic and nonspatial deterministic models.
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Affiliation(s)
- Glenn Marion
- Biomathematics & Statistics Scotland, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom.
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31
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Pascual M, Roy M, Franc A. Simple temporal models for ecological systems with complex spatial patterns. Ecol Lett 2002. [DOI: 10.1046/j.1461-0248.2002.00334.x] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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33
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Abstract
Following the first model of wildlife tuberculosis (in European badgers) there has been a spate of papers modelling wildlife TB. These have looked at population parameters and disease dynamics in the badger and possum. Recent papers in particular have looked at various methods of controlling the wildlife vector to reduce the incidence of TB in cattle. The author examines the role of modelling to show what insights it has given us, which issues have not been addressed, and where the shortfalls lie. Particular attention will be paid to a comparison between models of badgers and possums, and between simple and more complex models, and possible areas of future research will be revealed.
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Affiliation(s)
- G C Smith
- Central Science Laboratory, York, UK.
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34
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Abstract
Pair approximation has frequently proved effective for deriving qualitative information about lattice-based stochastic spatial models for population, epidemic and evolutionary dynamics. Pair approximation is a moment closure method in which the mean-field description is supplemented by approximate equations for the frequencies of neighbor-site pairs of each possible type. A limitation of pair approximation relative to moment closure for continuous space models is that all modes of interaction between individuals (e.g., dispersal of offspring, competition, or disease transmission) are assumed to operate over a single spatial scale determined by the size of the interaction neighborhood. In this paper I present a multiscale pair approximation which allows different sized neighborhoods for each type of interaction. To illustrate and test the approximation I consider a spatial single-species logistic model in which offspring are dispersed across a birth neighborhood and established individuals have a death rate depending on the population density in a competition neighborhood, with one of these neighborhoods nested inside the other. Analysis of the steady-state equations yields several qualitative predictions that are confirmed by simulations of the model, and numerical solutions of the dynamic equations provide a close approximation to the transient behavior of the stochastic model on a large lattice. The multiscale pair approximation thus provides a useful intermediate between the standard pair approximation for a single interaction neighborhood, and a complete set of moment equations for more spatially detailed models.
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Affiliation(s)
- S P Ellner
- Department of Ecology and Evolutionary Biology, Cornell University, E145 Corson Hall, Ithaca, NY 14853-2701, USA.
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Madden LV, Hughes G, Irwin ME. Coupling Disease-Progress-Curve and Time-of-Infection Functions for Predicting Yield Loss of Crops. PHYTOPATHOLOGY 2000; 90:788-800. [PMID: 18944499 DOI: 10.1094/phyto.2000.90.8.788] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
ABSTRACT A general approach was developed to predict the yield loss of crops in relation to infection by systemic diseases. The approach was based on two premises: (i) disease incidence in a population of plants over time can be described by a nonlinear disease progress model, such as the logistic or monomolecular; and (ii) yield of a plant is a function of time of infection (t) that can be represented by the (negative) exponential or similar model (zeta(t)). Yield loss of a population of plants on a proportional scale (L) can be written as the product of the proportion of the plant population newly infected during a very short time interval (X'(t)dt) and zeta(t), integrated over the time duration of the epidemic. L in the model can be expressed in relation to directly interpretable parameters: maximum per-plant yield loss (alpha, typically occurring at t = 0); the decline in per-plant loss as time of infection is delayed (gamma; units of time(-1)); and the parameters that characterize disease progress over time, namely, initial disease incidence (X(0)), rate of disease increase (r; units of time(-1)), and maximum (or asymptotic) value of disease incidence (K). Based on the model formulation, L ranges from alphaX(0) to alphaK and increases with increasing X(0), r, K, alpha, and gamma(-1). The exact effects of these parameters on L were determined with numerical solutions of the model. The model was expanded to predict L when there was spatial heterogeneity in disease incidence among sites within a field and when maximum per-plant yield loss occurred at a time other than the beginning of the epidemic (t > 0). However, the latter two situations had a major impact on L only at high values of r. The modeling approach was demonstrated by analyzing data on soybean yield loss in relation to infection by Soybean mosaic virus, a member of the genus Potyvirus. Based on model solutions, strategies to reduce or minimize yield losses from a given disease can be evaluated.
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36
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Madden LV, Hughes G. An effective sample size for predicting plant disease incidence in a spatial hierarchy. PHYTOPATHOLOGY 1999; 89:770-781. [PMID: 18944705 DOI: 10.1094/phyto.1999.89.9.770] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
ABSTRACT For aggregated or heterogeneous disease incidence, one can predict the proportion of sampling units diseased at a higher scale (e.g., plants) based on the proportion of diseased individuals and heterogeneity of diseased individuals at a lower scale (e.g., leaves) using a function derived from the beta-binomial distribution. Here, a simple approximation for the beta-binomial-based function is derived. This approximation has a functional form based on the binomial distribution, but with the number of individuals per sampling unit (n) replaced by a parameter (v) that has similar interpretation as, but is not the same as, the effective sample size (n(deff) ) often used in survey sampling. The value of v is inversely related to the degree of heterogeneity of disease and generally is intermediate between n(deff) and n in magnitude. The choice of v was determined iteratively by finding a parameter value that allowed the zero term (probability that a sampling unit is disease free) of the binomial distribution to equal the zero term of the beta-binomial. The approximation function was successfully tested on observations of Eutypa dieback of grapes collected over several years and with simulated data. Unlike the beta-binomial-based function, the approximation can be rearranged to predict incidence at the lower scale from observed incidence data at the higher scale, making group sampling for heterogeneous data a more practical proposition.
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