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Howerton E, Dahlin K, Edholm CJ, Fox L, Reynolds M, Hollingsworth B, Lytle G, Walker M, Blackwood J, Lenhart S. The effect of governance structures on optimal control of two-patch epidemic models. J Math Biol 2023; 87:74. [PMID: 37861753 PMCID: PMC10589198 DOI: 10.1007/s00285-023-02001-8] [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: 09/09/2022] [Revised: 09/07/2023] [Accepted: 09/14/2023] [Indexed: 10/21/2023]
Abstract
Infectious diseases continue to pose a significant threat to the health of humans globally. While the spread of pathogens transcends geographical boundaries, the management of infectious diseases typically occurs within distinct spatial units, determined by geopolitical boundaries. The allocation of management resources within and across regions (the "governance structure") can affect epidemiological outcomes considerably, and policy-makers are often confronted with a choice between applying control measures uniformly or differentially across regions. Here, we investigate the extent to which uniform and non-uniform governance structures affect the costs of an infectious disease outbreak in two-patch systems using an optimal control framework. A uniform policy implements control measures with the same time varying rate functions across both patches, while these measures are allowed to differ between the patches in a non-uniform policy. We compare results from two systems of differential equations representing transmission of cholera and Ebola, respectively, to understand the interplay between transmission mode, governance structure and the optimal control of outbreaks. In our case studies, the governance structure has a meaningful impact on the allocation of resources and burden of cases, although the difference in total costs is minimal. Understanding how governance structure affects both the optimal control functions and epidemiological outcomes is crucial for the effective management of infectious diseases going forward.
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Affiliation(s)
- Emily Howerton
- Department of Biology and Center for Infectious Disease Dynamics, Pennsylvania State University, University Park, PA, USA
| | - Kyle Dahlin
- Center for the Ecology of Infectious Diseases, Odum School of Ecology, University of Georgia, Athens, GA, USA.
| | | | - Lindsey Fox
- Mathematics Discipline, Eckerd College, Saint Petersburg, FL, USA
| | - Margaret Reynolds
- Department of Mathematical Sciences, United States Military Academy, West Point, NY, USA
| | | | - George Lytle
- Department of Biology, Chemistry, Mathematics, and Computer Science, University of Montevallo, Montevallo, AL, USA
| | - Melody Walker
- Department of Medicine, University of Florida, Gainesville, FL, USA
| | - Julie Blackwood
- Department of Mathematics and Statistics, Williams College, Williamstown, MA, USA
| | - Suzanne Lenhart
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA
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2
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Saucedo O, Tien JH. Host movement, transmission hot spots, and vector-borne disease dynamics on spatial networks. Infect Dis Model 2022; 7:742-760. [PMID: 36439402 PMCID: PMC9672958 DOI: 10.1016/j.idm.2022.10.006] [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: 04/12/2022] [Revised: 09/04/2022] [Accepted: 10/26/2022] [Indexed: 11/06/2022] Open
Abstract
We examine how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics. Specifically, we consider a Ross-Macdonald-type disease model on n spatial locations that are coupled by host movement on a strongly connected, weighted, directed graph. We derive a closed form approximation to the domain reproduction number using a Laurent series expansion, and use this approximation to compute sensitivities of the basic reproduction number to model parameters. To illustrate how these results can be used to help inform mitigation strategies, as a case study we apply these results to malaria dynamics in Namibia, using published cell phone data and estimates for local disease transmission. Our analytical results are particularly useful for understanding drivers of transmission when mobility sinks and transmission hot spots do not coincide.
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Affiliation(s)
- Omar Saucedo
- Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
| | - Joseph H. Tien
- Department of Mathematics, The Ohio State University, Columbus, OH, USA
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3
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Vargas Bernal E, Saucedo O, Tien JH. Relating Eulerian and Lagrangian spatial models for vector-host disease dynamics through a fundamental matrix. J Math Biol 2022; 84:57. [PMID: 35676373 DOI: 10.1007/s00285-022-01761-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2021] [Revised: 01/21/2022] [Accepted: 05/11/2022] [Indexed: 11/26/2022]
Abstract
We explore the relationship between Eulerian and Lagrangian approaches for modeling movement in vector-borne diseases for discrete space. In the Eulerian approach we account for the movement of hosts explicitly through movement rates captured by a graph Laplacian matrix L. In the Lagrangian approach we only account for the proportion of time that individuals spend in foreign patches through a mixing matrix P. We establish a relationship between an Eulerian model and a Lagrangian model for the hosts in terms of the matrices L and P. We say that the two modeling frameworks are consistent if for a given matrix P, the matrix L can be chosen so that the residence times of the matrix P and the matrix L match. We find a sufficient condition for consistency, and examine disease quantities such as the final outbreak size and basic reproduction number in both the consistent and inconsistent cases. In the special case of a two-patch model, we observe how similar values for the basic reproduction number and final outbreak size can occur even in the inconsistent case. However, there are scenarios where the final sizes in both approaches can significantly differ by means of the relationship we propose.
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Affiliation(s)
| | - Omar Saucedo
- Department of Mathematics, Virginia Tech., Blacksburg, VA, USA
| | - Joseph Hua Tien
- Department of Mathematics, The Ohio State University, Columbus, OH, USA
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4
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Michalska-Smith M, VanderWaal K, Craft ME. Asymmetric host movement reshapes local disease dynamics in metapopulations. Sci Rep 2022; 12:9365. [PMID: 35672422 PMCID: PMC9171740 DOI: 10.1038/s41598-022-12774-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2021] [Accepted: 05/11/2022] [Indexed: 11/13/2022] Open
Abstract
Understanding how the movement of individuals affects disease dynamics is critical to accurately predicting and responding to the spread of disease in an increasingly interconnected world. In particular, it is not yet known how movement between patches affects local disease dynamics (e.g., whether pathogen prevalence remains steady or oscillates through time). Considering a set of small, archetypal metapopulations, we find three surprisingly simple patterns emerge in local disease dynamics following the introduction of movement between patches: (1) movement between identical patches with cyclical pathogen prevalence dampens oscillations in the destination while increasing synchrony between patches; (2) when patches differ from one another in the absence of movement, adding movement allows dynamics to propagate between patches, alternatively stabilizing or destabilizing dynamics in the destination based on the dynamics at the origin; and (3) it is easier for movement to induce cyclical dynamics than to induce a steady-state. Considering these archetypal networks (and the patterns they exemplify) as building blocks of larger, more realistically complex metapopulations provides an avenue for novel insights into the role of host movement on disease dynamics. Moreover, this work demonstrates a framework for future predictive modelling of disease spread in real populations.
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Affiliation(s)
- Matthew Michalska-Smith
- Department of Veterinary Population Medicine, University of Minnesota, St. Paul, MN, USA. .,Department of Plant Pathology, University of Minnesota, St. Paul, MN, USA.
| | - Kimberly VanderWaal
- Department of Veterinary Population Medicine, University of Minnesota, St. Paul, MN, USA
| | - Meggan E Craft
- Department of Veterinary Population Medicine, University of Minnesota, St. Paul, MN, USA.,Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, MN, USA
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5
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Ortiz DI, Piche-Ovares M, Romero-Vega LM, Wagman J, Troyo A. The Impact of Deforestation, Urbanization, and Changing Land Use Patterns on the Ecology of Mosquito and Tick-Borne Diseases in Central America. INSECTS 2021; 13:20. [PMID: 35055864 PMCID: PMC8781098 DOI: 10.3390/insects13010020] [Citation(s) in RCA: 19] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 11/16/2021] [Revised: 12/13/2021] [Accepted: 12/16/2021] [Indexed: 11/29/2022]
Abstract
Central America is a unique geographical region that connects North and South America, enclosed by the Caribbean Sea to the East, and the Pacific Ocean to the West. This region, encompassing Belize, Costa Rica, Guatemala, El Salvador, Honduras, Panama, and Nicaragua, is highly vulnerable to the emergence or resurgence of mosquito-borne and tick-borne diseases due to a combination of key ecological and socioeconomic determinants acting together, often in a synergistic fashion. Of particular interest are the effects of land use changes, such as deforestation-driven urbanization and forest degradation, on the incidence and prevalence of these diseases, which are not well understood. In recent years, parts of Central America have experienced social and economic improvements; however, the region still faces major challenges in developing effective strategies and significant investments in public health infrastructure to prevent and control these diseases. In this article, we review the current knowledge and potential impacts of deforestation, urbanization, and other land use changes on mosquito-borne and tick-borne disease transmission in Central America and how these anthropogenic drivers could affect the risk for disease emergence and resurgence in the region. These issues are addressed in the context of other interconnected environmental and social challenges.
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Affiliation(s)
- Diana I. Ortiz
- Biology Program, Westminster College, New Wilmington, PA 16172, USA
| | - Marta Piche-Ovares
- Laboratorio de Virología, Centro de Investigación en Enfermedades Tropicales (CIET), Universidad de Costa Rica, San José 11501, Costa Rica;
- Departamento de Virología, Escuela de Medicina Veterinaria, Universidad Nacional, Heredia 40104, Costa Rica
| | - Luis M. Romero-Vega
- Departamento de Patología, Escuela de Medicina Veterinaria, Universidad Nacional, Heredia 40104, Costa Rica;
- Laboratorio de Investigación en Vectores (LIVe), Centro de Investigación en Enfermedades Tropicales (CIET), Universidad de Costa Rica, San José 11501, Costa Rica;
| | - Joseph Wagman
- Malaria and Neglected Tropical Diseases Program, Center for Malaria Control and Elimination, PATH, Washington, DC 20001, USA;
| | - Adriana Troyo
- Laboratorio de Investigación en Vectores (LIVe), Centro de Investigación en Enfermedades Tropicales (CIET), Universidad de Costa Rica, San José 11501, Costa Rica;
- Departamento de Parasitología, Facultad de Microbiología, Universidad de Costa Rica, San José 11501, Costa Rica
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6
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Effect of daily human movement on some characteristics of dengue dynamics. Math Biosci 2021; 332:108531. [PMID: 33460675 DOI: 10.1016/j.mbs.2020.108531] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2020] [Revised: 12/09/2020] [Accepted: 12/10/2020] [Indexed: 11/21/2022]
Abstract
Human movement is a key factor in infectious diseases spread such as dengue. Here, we explore a mathematical modeling approach based on a system of ordinary differential equations to study the effect of human movement on characteristics of dengue dynamics such as the existence of endemic equilibria, and the start, duration, and amplitude of the outbreak. The model considers that every day is divided into two periods: high-activity and low-activity. Periodic human movement between patches occurs in discrete times. Based on numerical simulations, we show unexpected scenarios such as the disease extinction in regions where the local basic reproductive number is greater than 1. In the same way, we obtain scenarios where outbreaks appear despite the fact that the local basic reproductive numbers in these regions are less than 1 and the outbreak size depends on the length of high-activity and low-activity periods.
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7
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Núñez-López M, Alarcón Ramos L, Velasco-Hernández JX. Migration rate estimation in an epidemic network. APPLIED MATHEMATICAL MODELLING 2021; 89:1949-1964. [PMID: 32952269 PMCID: PMC7486824 DOI: 10.1016/j.apm.2020.08.025] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Revised: 07/31/2020] [Accepted: 08/06/2020] [Indexed: 05/07/2023]
Abstract
Most of the recent epidemic outbreaks in the world have as a trigger, a strong migratory component as has been evident in the recent Covid-19 pandemic. In this work we address the problem of migration of human populations and its effect on pathogen reinfections in the case of Dengue, using a Markov-chain susceptible-infected-susceptible (SIS) metapopulation model over a network. Our model postulates a general contact rate that represents a local measure of several factors: the population size of infected hosts that arrive at a given location as a function of total population size, the current incidence at neighboring locations, and the connectivity of the network where the disease spreads. This parameter can be interpreted as an indicator of outbreak risk at a given location. This parameter is tied to the fraction of individuals that move across boundaries (migration). To illustrate our model capabilities, we estimate from epidemic Dengue data in Mexico the dynamics of migration at a regional scale incorporating climate variability represented by an index based on precipitation data.
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Affiliation(s)
- M Núñez-López
- Department of Mathematics, ITAM Río Hondo 1, Ciudad de México 01080, México
| | - L Alarcón Ramos
- Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana, Cuajimalpa, Av. Vasco de Quiroga 4871, Cuajimalpa de Morelos, 05300, México
| | - J X Velasco-Hernández
- Instituto de Matemáticas, Universidad Nacional Autónoma de México, Boulevard Juriquilla No. 3001, Juriquilla, 76230, México
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8
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Mukhtar AYA, Munyakazi JB, Ouifki R. Assessing the role of human mobility on malaria transmission. Math Biosci 2019; 320:108304. [PMID: 31883985 DOI: 10.1016/j.mbs.2019.108304] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2019] [Revised: 12/12/2019] [Accepted: 12/12/2019] [Indexed: 11/26/2022]
Abstract
South Sudan accounts for a large proportion of all annual malaria cases in Africa. In recent years, the country has witnessed an unprecedented number of people on the move, refugees, internally displaced people, people who have returned to their counties or areas of origin, stateless people and other populations of concern, posing challenges to malaria control. Thus, one can claim that human mobility is one of the contributing factors to the resurgence of malaria. The aim of this paper is to assess the impact of human mobility on the burden of malaria disease in South Sudan. For this, we formulate an SIR-type model that describes the transmission dynamics of malaria disease between multiple patches. The proposed model is a system of stochastic differential equations consisting of ordinary differential equations perturbed by a stochastic Wiener process. For the deterministic part of the model, we calculate the basic reproduction number. Concerning the whole stochastic model, we use the maximum likelihood approach to fit the model to weekly malaria data of 2011 from Central Equatoria State, Western Bahr El Ghazal State and Warrap State. Using the parameters estimated on the fitted model, we simulate the future observation of the disease pattern. The disease was found to persist in the low transmission patches when there is human inflow in these patches and although the intervention coverage reaches 75%.
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Affiliation(s)
- Abdulaziz Y A Mukhtar
- Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa; DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-Mass), South Africa.
| | - Justin B Munyakazi
- Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa
| | - Rachid Ouifki
- Department of Mathematics and Applied Mathematics, Faculty of Natural & Agricultural Sciences, University of Pretoria, Private Bag X20, Hatfield 0028, South Africa
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9
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Gao D, van den Driessche P, Cosner C. Habitat fragmentation promotes malaria persistence. J Math Biol 2019; 79:2255-2280. [PMID: 31520106 DOI: 10.1007/s00285-019-01428-2] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2018] [Revised: 08/28/2019] [Indexed: 12/16/2022]
Abstract
Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.
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Affiliation(s)
- Daozhou Gao
- Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China
| | - P van den Driessche
- Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada
| | - Chris Cosner
- Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA.
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10
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Palmer C, Landguth E, Stone E, Johnson T. The dynamics of vector-borne relapsing diseases. Math Biosci 2018; 297:32-42. [PMID: 29339054 DOI: 10.1016/j.mbs.2018.01.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2016] [Revised: 04/04/2017] [Accepted: 01/08/2018] [Indexed: 11/20/2022]
Abstract
In this paper, we describe the dynamics of a vector-borne relapsing disease, such as tick-borne relapsing fever, using the methods of compartmental models. After some motivation and model description we provide a proof of a conjectured general form of the reproductive ratio R0, which is the average number of new infections produced by a single infected individual. A disease free equilibrium undergoes a bifurcation at R0=1 and we show that for an arbitrary number of relapses it is a transcritical bifurcation with a single branch of endemic equilibria that is locally asymptotically stable for R0 sufficiently close to 1. Furthermore, we show there is no backwards bifurcation. We then show that these results can be extended to variants of the model with an example that allows for variation in the number of relapses before recovery. Finally, we discuss implications of our results and directions for future research.
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Affiliation(s)
- Cody Palmer
- Department of Mathematical Sciences, University of Montana, USA.
| | - Erin Landguth
- Division of Biological Sciences, University of Montana, USA
| | - Emily Stone
- Department of Mathematical Sciences, University of Montana, USA
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11
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Ruktanonchai NW, Smith DL, De Leenheer P. Parasite sources and sinks in a patched Ross-Macdonald malaria model with human and mosquito movement: Implications for control. Math Biosci 2016; 279:90-101. [PMID: 27436636 DOI: 10.1016/j.mbs.2016.06.012] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/24/2015] [Revised: 05/05/2016] [Accepted: 06/30/2016] [Indexed: 11/19/2022]
Abstract
We consider the dynamics of a mosquito-transmitted pathogen in a multi-patch Ross-Macdonald malaria model with mobile human hosts, mobile vectors, and a heterogeneous environment. We show the existence of a globally stable steady state, and a threshold that determines whether a pathogen is either absent from all patches, or endemic and present at some level in all patches. Each patch is characterized by a local basic reproduction number, whose value predicts whether the disease is cleared or not when the patch is isolated: patches are known as "demographic sinks" if they have a local basic reproduction number less than one, and hence would clear the disease if isolated; patches with a basic reproduction number above one would sustain endemic infection in isolation, and become "demographic sources" of parasites when connected to other patches. Sources are also considered focal areas of transmission for the larger landscape, as they export excess parasites to other areas and can sustain parasite populations. We show how to determine the various basic reproduction numbers from steady state estimates in the patched network and knowledge of additional model parameters, hereby identifying parasite sources in the process. This is useful in the context of control of the infection on natural landscapes, because a commonly suggested strategy is to target focal areas, in order to make their corresponding basic reproduction numbers less than one, effectively turning them into sinks. We show that this is indeed a successful control strategy-albeit a conservative and possibly expensive one-in case either the human host, or the vector does not move. However, we also show that when both humans and vectors move, this strategy may fail, depending on the specific movement patterns exhibited by hosts and vectors.
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Affiliation(s)
| | - David L Smith
- Institute for Health Metrics and Evaluation, Department of Global Health, University of Washington, Seattle WA.
| | - Patrick De Leenheer
- Department of Mathematics and Department of Integrative Biology, Oregon State University, Corvallis, OR, United States.
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12
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From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges. Math Biosci 2015; 270:143-55. [PMID: 26474512 DOI: 10.1016/j.mbs.2015.10.002] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Abstract
Since their earliest days, humans have been struggling with infectious diseases. Caused by viruses, bacteria, protozoa, or even higher organisms like worms, these diseases depend critically on numerous intricate interactions between parasites and hosts, and while we have learned much about these interactions, many details are still obscure. It is evident that the combined host-parasite dynamics constitutes a complex system that involves components and processes at multiple scales of time, space, and biological organization. At one end of this hierarchy we know of individual molecules that play crucial roles for the survival of a parasite or for the response and survival of its host. At the other end, one realizes that the spread of infectious diseases by far exceeds specific locales and, due to today's easy travel of hosts carrying a multitude of organisms, can quickly reach global proportions. The community of mathematical modelers has been addressing specific aspects of infectious diseases for a long time. Most of these efforts have focused on one or two select scales of a multi-level disease and used quite different computational approaches. This restriction to a molecular, physiological, or epidemiological level was prudent, as it has produced solid pillars of a foundation from which it might eventually be possible to launch comprehensive, multi-scale modeling efforts that make full use of the recent advances in biology and, in particular, the various high-throughput methodologies accompanying the emerging -omics revolution. This special issue contains contributions from biologists and modelers, most of whom presented and discussed their work at the workshop From within Host Dynamics to the Epidemiology of Infectious Disease, which was held at the Mathematical Biosciences Institute at Ohio State University in April 2014. These contributions highlight some of the forays into a deeper understanding of the dynamics between parasites and their hosts, and the consequences of this dynamics for the spread and treatment of infectious diseases.
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13
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Gutierrez JB, Galinski MR, Cantrell S, Voit EO. WITHDRAWN: From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges. Math Biosci 2015:S0025-5564(15)00085-1. [PMID: 25890102 DOI: 10.1016/j.mbs.2015.04.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Abstract
This article has been withdrawn at the request of the author(s) and/or editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at http://www.elsevier.com/locate/withdrawalpolicy.
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Affiliation(s)
- Juan B Gutierrez
- Department of Mathematics, Institute of Bioinformatics, University of Georgia, Athens, GA 30602, United States .
| | - Mary R Galinski
- Emory University School of Medicine, Division of Infectious Diseases, Emory Vaccine Center, Yerkes National Primate Research Center, Emory University, 954 Gatewood Road, Atlanta, GA 30329, United States .
| | - Stephen Cantrell
- Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States .
| | - Eberhard O Voit
- Department of Biomedical Engineering, Georgia Institute of Technology, 313 Ferst Drive, Suite 4103, Atlanta, GA 30332-0535, United States .
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