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Vanalli C, Mari L, Casagrandi R, Boag B, Gatto M, Cattadori IM. Modeling the contribution of antibody attack rates to single and dual helminth infections in a natural system. Math Biosci 2023; 360:109010. [PMID: 37088125 DOI: 10.1016/j.mbs.2023.109010] [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: 11/18/2022] [Revised: 04/12/2023] [Accepted: 04/14/2023] [Indexed: 04/25/2023]
Abstract
Within-host models of infection can provide important insights into the processes that affect parasite spread and persistence in host populations. However, modeling can be limited by the availability of empirical data, a problem commonly encountered in natural systems. Here, we used six years of immune-infection observations of two gastrointestinal helminths (Trichostrongylus retortaeformis and Graphidium strigosum) from a population of European rabbits (Oryctolagus cuniculus) to develop an age-dependent, mathematical model that explicitly included species-specific and cross-reacting antibody (IgA and IgG) responses to each helminth in hosts with single or dual infections. Different models of single infection were formally compared to test alternative mechanisms of parasite regulation. The two models that best described single infections of each helminth species were then coupled through antibody cross-immunity to examine how the presence of one species could alter the host immune response to, and the within-host dynamics of, the other species. For both single infections, model selection suggested that either IgA or IgG responses could equally explain the observed parasite intensities by host age. However, the antibody attack rate and affinity level changed between the two helminths, it was stronger against T. retortaeformis than against G. strigosum and caused contrasting age-intensity profiles. When the two helminths coinfect the same host, we found variation of the species-specific antibody response to both species together with an asymmetric cross-immune response driven by IgG. Lower attack rate and affinity of antibodies in dual than single infections contributed to the significant increase of both helminth intensities. By combining mathematical modeling with immuno-infection data, our work provides a tractable model framework for disentangling some of the complexities generated by host-parasite and parasite-parasite interactions in natural systems.
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Affiliation(s)
- Chiara Vanalli
- Center for Infectious Disease Dynamics and Department of Biology, The Pennsylvania State University, University Park, 16802 PA, USA.
| | - Lorenzo Mari
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, Italy
| | - Renato Casagrandi
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, Italy
| | - Brian Boag
- The James Hutton Institute, DD2 5DA Invergowrie, UK
| | - Marino Gatto
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, Italy
| | - Isabella M Cattadori
- Center for Infectious Disease Dynamics and Department of Biology, The Pennsylvania State University, University Park, 16802 PA, USA
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2
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Tuncer N, Le TT. Structural and practical identifiability analysis of outbreak models. Math Biosci 2018; 299:1-18. [PMID: 29477671 DOI: 10.1016/j.mbs.2018.02.004] [Citation(s) in RCA: 34] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2017] [Revised: 02/06/2018] [Accepted: 02/17/2018] [Indexed: 02/02/2023]
Abstract
Estimating the reproduction number of an emerging infectious disease from an epidemiological data is becoming more essential in evaluating the current status of an outbreak. However, these studies are lacking the fundamental prerequisite in parameter estimation problem, namely the structural identifiability of the epidemic model, which determines the possibility of uniquely determining the model parameters from the epidemic data. In this paper, we perform both structural and practical identifiability analysis to classical epidemic models such as SIR (Susceptible-Infected-Recovered), SEIR (Susceptible-Exposed-Infected-Recovered) and an epidemic model with the treatment class (SITR). We performed structural identifiability analysis on these epidemic models using a differential algebra approach to investigate the well-posedness of the parameter estimation problem. Parameters of these models are estimated from different data types, namely prevalence, cumulative incidences and treated individuals. Furthermore, we carried out practical identifiability analysis on these models using Monte Carlo simulations and Fisher's Information Matrix. Our study shows that the SIR model is both structurally and practically identifiable from the prevalence data. It is also structurally identifiable to cumulative incidence observations, but due to high correlations of the parameters, it is practically unidentifiable from the cumulative incidence data. Furthermore, we found that none of these simple epidemic models are practically identifiable from the cumulative incidence data which is the standard type of epidemiological data provided by CDC or WHO. Our analysis with simple SIR model suggest that the health agencies, if possible, should report prevalence rather than incidence data.
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Affiliation(s)
- Necibe Tuncer
- Department of Mathematical Sciences, Florida Atlantic University, Science Building, Room 234, 777 Glades Road, Boca Raton, FL 33431, USA.
| | - Trang T Le
- Department of Mathematics, University of Tulsa, 800 S Tucker Drive, Tulsa, OK 74104, USA.
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Ho LST, Xu J, Crawford FW, Minin VN, Suchard MA. Birth/birth-death processes and their computable transition probabilities with biological applications. J Math Biol 2018; 76:911-944. [PMID: 28741177 PMCID: PMC5783825 DOI: 10.1007/s00285-017-1160-3] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2016] [Revised: 04/04/2017] [Indexed: 01/20/2023]
Abstract
Birth-death processes track the size of a univariate population, but many biological systems involve interaction between populations, necessitating models for two or more populations simultaneously. A lack of efficient methods for evaluating finite-time transition probabilities of bivariate processes, however, has restricted statistical inference in these models. Researchers rely on computationally expensive methods such as matrix exponentiation or Monte Carlo approximation, restricting likelihood-based inference to small systems, or indirect methods such as approximate Bayesian computation. In this paper, we introduce the birth/birth-death process, a tractable bivariate extension of the birth-death process, where rates are allowed to be nonlinear. We develop an efficient algorithm to calculate its transition probabilities using a continued fraction representation of their Laplace transforms. Next, we identify several exemplary models arising in molecular epidemiology, macro-parasite evolution, and infectious disease modeling that fall within this class, and demonstrate advantages of our proposed method over existing approaches to inference in these models. Notably, the ubiquitous stochastic susceptible-infectious-removed (SIR) model falls within this class, and we emphasize that computable transition probabilities newly enable direct inference of parameters in the SIR model. We also propose a very fast method for approximating the transition probabilities under the SIR model via a novel branching process simplification, and compare it to the continued fraction representation method with application to the 17th century plague in Eyam. Although the two methods produce similar maximum a posteriori estimates, the branching process approximation fails to capture the correlation structure in the joint posterior distribution.
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Affiliation(s)
- Lam Si Tung Ho
- Department of Biostatistics, University of California, Los Angeles, Los Angeles, CA, USA.
| | - Jason Xu
- Department of Biomathematics, University of California, Los Angeles, Los Angeles, CA, USA
| | | | - Vladimir N Minin
- Departments of Statistics and Biology, University of Washington, Seattle, WA, USA
| | - Marc A Suchard
- Departments of Biomathematics, Biostatistics and Human Genetics, University of California, Los Angeles, Los Angeles, WA, USA
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4
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King AA, Domenech de Cellès M, Magpantay FMG, Rohani P. Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola. Proc Biol Sci 2016; 282:20150347. [PMID: 25833863 PMCID: PMC4426634 DOI: 10.1098/rspb.2015.0347] [Citation(s) in RCA: 121] [Impact Index Per Article: 15.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Abstract
As an emergent infectious disease outbreak unfolds, public health response is
reliant on information on key epidemiological quantities, such as transmission
potential and serial interval. Increasingly, transmission models fit to
incidence data are used to estimate these parameters and guide policy. Some
widely used modelling practices lead to potentially large errors in parameter
estimates and, consequently, errors in model-based forecasts. Even more
worryingly, in such situations, confidence in parameter estimates and forecasts
can itself be far overestimated, leading to the potential for large errors that
mask their own presence. Fortunately, straightforward and computationally
inexpensive alternatives exist that avoid these problems. Here, we first use a
simulation study to demonstrate potential pitfalls of the standard practice of
fitting deterministic models to cumulative incidence data. Next, we demonstrate
an alternative based on stochastic models fit to raw data from an early phase of
2014 West Africa Ebola virus disease outbreak. We show not only that bias is
thereby reduced, but that uncertainty in estimates and forecasts is better
quantified and that, critically, lack of model fit is more readily diagnosed. We
conclude with a short list of principles to guide the modelling response to
future infectious disease outbreaks.
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Affiliation(s)
- Aaron A King
- Department of Ecology & Evolutionary Biology, University of Michigan, Ann Arbor, MI 48109, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
| | | | - Felicia M G Magpantay
- Department of Ecology & Evolutionary Biology, University of Michigan, Ann Arbor, MI 48109, USA
| | - Pejman Rohani
- Department of Ecology & Evolutionary Biology, University of Michigan, Ann Arbor, MI 48109, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
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5
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Drovandi CC, Pettitt AN, Lee A. Bayesian Indirect Inference Using a Parametric Auxiliary Model. Stat Sci 2015. [DOI: 10.1214/14-sts498] [Citation(s) in RCA: 48] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Kennedy DA, Dukic V, Dwyer G. Pathogen growth in insect hosts: inferring the importance of different mechanisms using stochastic models and response-time data. Am Nat 2014; 184:407-23. [PMID: 25141148 PMCID: PMC10495239 DOI: 10.1086/677308] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
Pathogen population dynamics within individual hosts can alter disease epidemics and pathogen evolution, but our understanding of the mechanisms driving within-host dynamics is weak. Mathematical models have provided useful insights, but existing models have only rarely been subjected to rigorous tests, and their reliability is therefore open to question. Most models assume that initial pathogen population sizes are so large that stochastic effects due to small population sizes, so-called demographic stochasticity, are negligible, but whether this assumption is reasonable is unknown. Most models also assume that the dynamic effects of a host's immune system strongly affect pathogen incubation times or "response times," but whether such effects are important in real host-pathogen interactions is likewise unknown. Here we use data for a baculovirus of the gypsy moth to test models of within-host pathogen growth. By using Bayesian statistical techniques and formal model-selection procedures, we are able to show that the response time of the gypsy moth virus is strongly affected by both demographic stochasticity and a dynamic response of the host immune system. Our results imply that not all response-time variability can be explained by host and pathogen variability, and that immune system responses to infection may have important effects on population-level disease dynamics.
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Affiliation(s)
- David A. Kennedy
- Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637
- Center for Infectious Disease Dynamics, Pennsylvania State University, University Park, Pennsylvania 16802
- Fogarty International Center, National Institutes of Health, Bethesda, Maryland 20892
| | - Vanja Dukic
- Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80305
| | - Greg Dwyer
- Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637
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Drovandi CC, Pettitt AN. Bayesian experimental design for models with intractable likelihoods. Biometrics 2013; 69:937-48. [PMID: 24131221 DOI: 10.1111/biom.12081] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2013] [Revised: 06/01/2013] [Accepted: 06/01/2013] [Indexed: 11/28/2022]
Abstract
In this paper we present a methodology for designing experiments for efficiently estimating the parameters of models with computationally intractable likelihoods. The approach combines a commonly used methodology for robust experimental design, based on Markov chain Monte Carlo sampling, with approximate Bayesian computation (ABC) to ensure that no likelihood evaluations are required. The utility function considered for precise parameter estimation is based upon the precision of the ABC posterior distribution, which we form efficiently via the ABC rejection algorithm based on pre-computed model simulations. Our focus is on stochastic models and, in particular, we investigate the methodology for Markov process models of epidemics and macroparasite population evolution. The macroparasite example involves a multivariate process and we assess the loss of information from not observing all variables.
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Affiliation(s)
- Christopher C Drovandi
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
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Ross J. On parameter estimation in population models III: Time-inhomogeneous processes and observation error. Theor Popul Biol 2012; 82:1-17. [DOI: 10.1016/j.tpb.2012.03.001] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2011] [Revised: 11/10/2011] [Accepted: 03/05/2012] [Indexed: 10/28/2022]
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Basáñez MG, McCarthy JS, French MD, Yang GJ, Walker M, Gambhir M, Prichard RK, Churcher TS. A research agenda for helminth diseases of humans: modelling for control and elimination. PLoS Negl Trop Dis 2012; 6:e1548. [PMID: 22545162 PMCID: PMC3335861 DOI: 10.1371/journal.pntd.0001548] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022] Open
Abstract
Mathematical modelling of helminth infections has the potential to inform policy and guide research for the control and elimination of human helminthiases. However, this potential, unlike in other parasitic and infectious diseases, has yet to be realised. To place contemporary efforts in a historical context, a summary of the development of mathematical models for helminthiases is presented. These efforts are discussed according to the role that models can play in furthering our understanding of parasite population biology and transmission dynamics, and the effect on such dynamics of control interventions, as well as in enabling estimation of directly unobservable parameters, exploration of transmission breakpoints, and investigation of evolutionary outcomes of control. The Disease Reference Group on Helminth Infections (DRG4), established in 2009 by the Special Programme for Research and Training in Tropical Diseases (TDR), was given the mandate to review helminthiases research and identify research priorities and gaps. A research and development agenda for helminthiasis modelling is proposed based on identified gaps that need to be addressed for models to become useful decision tools that can support research and control operations effectively. This agenda includes the use of models to estimate the impact of large-scale interventions on infection incidence; the design of sampling protocols for the monitoring and evaluation of integrated control programmes; the modelling of co-infections; the investigation of the dynamical relationship between infection and morbidity indicators; the improvement of analytical methods for the quantification of anthelmintic efficacy and resistance; the determination of programme endpoints; the linking of dynamical helminth models with helminth geostatistical mapping; and the investigation of the impact of climate change on human helminthiases. It is concluded that modelling should be embedded in helminth research, and in the planning, evaluation, and surveillance of interventions from the outset. Modellers should be essential members of interdisciplinary teams, propitiating a continuous dialogue with end users and stakeholders to reflect public health needs in the terrain, discuss the scope and limitations of models, and update biological assumptions and model outputs regularly. It is highlighted that to reach these goals, a collaborative framework must be developed for the collation, annotation, and sharing of databases from large-scale anthelmintic control programmes, and that helminth modellers should join efforts to tackle key questions in helminth epidemiology and control through the sharing of such databases, and by using diverse, yet complementary, modelling approaches.
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Affiliation(s)
- María-Gloria Basáñez
- Department of Infectious Disease Epidemiology, School of Public Health, Faculty of Medicine (St Mary's campus), Imperial College London, London, UK.
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Hartig F, Calabrese JM, Reineking B, Wiegand T, Huth A. Statistical inference for stochastic simulation models - theory and application. Ecol Lett 2011; 14:816-27. [DOI: 10.1111/j.1461-0248.2011.01640.x] [Citation(s) in RCA: 269] [Impact Index Per Article: 20.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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11
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Drovandi CC, Pettitt AN, Faddy MJ. Approximate Bayesian computation using indirect inference. J R Stat Soc Ser C Appl Stat 2011. [DOI: 10.1111/j.1467-9876.2010.00747.x] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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12
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Drovandi CC, Pettitt AN. Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation. Biometrics 2010; 67:225-33. [DOI: 10.1111/j.1541-0420.2010.01410.x] [Citation(s) in RCA: 127] [Impact Index Per Article: 9.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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13
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Applying predator-prey theory to modelling immune-mediated, within-host interspecific parasite interactions. Parasitology 2010; 137:1027-38. [PMID: 20152061 DOI: 10.1017/s0031182009991788] [Citation(s) in RCA: 50] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Abstract
Predator-prey models are often applied to the interactions between host immunity and parasite growth. A key component of these models is the immune system's functional response, the relationship between immune activity and parasite load. Typically, models assume a simple, linear functional response. However, based on the mechanistic interactions between parasites and immunity we argue that alternative forms are more likely, resulting in very different predictions, ranging from parasite exclusion to chronic infection. By extending this framework to consider multiple infections we show that combinations of parasites eliciting different functional responses greatly affect community stability. Indeed, some parasites may stabilize other species that would be unstable if infecting alone. Therefore hosts' immune systems may have adapted to tolerate certain parasites, rather than clear them and risk erratic parasite dynamics. We urge for more detailed empirical information relating immune activity to parasite load to enable better predictions of the dynamic consequences of immune-mediated interspecific interactions within parasite communities.
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Krishnarajah I, Marion G, Gibson G. Novel bivariate moment-closure approximations. Math Biosci 2007; 208:621-43. [PMID: 17300816 DOI: 10.1016/j.mbs.2006.12.002] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2005] [Revised: 11/27/2006] [Accepted: 12/04/2006] [Indexed: 10/23/2022]
Abstract
Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.
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