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  • 1.
    Andersson, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Exploring voltage-dependent ion channels in silico by hysteretic conductance2010In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 226, no 1, p. 16-27Article in journal (Refereed)
    Abstract [en]

    Kinetic models of voltage-dependent ion channels are normally inferred from time records of macroscopic current relaxation or microscopic single channel data. A complementary explorative approach is outlined. Hysteretic conductance refers to conductance delays in response to voltage changes, delays at either macroscopic or microscopic levels of observation. It enables complementary assessments of model assumptions and gating schemes of voltage-dependent channels, e.g. independent versus cooperative gating, and multiple gating modes. Under the Hodgkin-Huxley condition of independent gating, and under ideal measurement conditions, hysteretic conductance makes it also possible to estimate voltage-dependent rate functions. The argument is mainly theoretical, based on experimental observations, and illustrated by simulations of Markov kinetic models.

  • 2. Ball, Frank
    et al.
    Pellis, Lorenzo
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Reproduction numbers for epidemic models with households and other social structures II: Comparisons and implications for vaccination2016In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 274, p. 108-139Article in journal (Refereed)
    Abstract [en]

    In this paper we consider epidemic models of directly transmissible SIR (susceptible -> infective -> recovered) and SEIR (with an additional latent class) infections in fully-susceptible populations with a social structure, consisting either of households or of households and workplaces. We review most reproduction numbers defined in the literature for these models, including the basic reproduction number R-0 introduced in the companion paper of this, for which we provide a simpler, more elegant derivation. Extending previous work, we provide a complete overview of the inequalities among these reproduction numbers and resolve some open questions. Special focus is put on the exponential-growth-associated reproduction number R-r, which is loosely defined as the estimate of R-0 based on the observed exponential growth of an emerging epidemic obtained when the social structure is ignored. We show that for the vast majority of the models considered in the literature R-r >= R-0 when R-0 >= 1 and R-r <= R-0 when R-0 <= 1. We show that, in contrast to models without social structure, vaccination of a fraction 1 - 1/R-0 of the population, chosen uniformly at random, with a perfect vaccine is usually insufficient to prevent large epidemics. In addition, we provide significantly sharper bounds than the existing ones for bracketing the critical vaccination coverage between two analytically tractable quantities, which we illustrate by means of extensive numerical examples.

  • 3.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Stochastic epidemic models: A survey2010In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 225, no 1, p. 24-35Article in journal (Refereed)
    Abstract [en]

    This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.

  • 4.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ball, Frank
    An epidemic model with infector and exposure dependent severity2009In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 218, no 2, p. 105-120Article in journal (Refereed)
    Abstract [en]

    A stochastic epidemic model allowing for both mildly and severely infectious individuals is defined, where an individual can become severely infectious directly upon infection or if additionally exposed to infection. It is shown that, assuming a large community, the initial phase of the epidemic may be approximated by a suitable branching process and that the main part of an epidemic that becomes established admits a law of large numbers and a central limit theorem, leading to a normal approximation for the final outcome of such an epidemic. Effects of vaccination prior to an outbreak are studied and the critical vaccination coverage, above which only small outbreaks can occur, is derived. The results are illustrated by simulations that demonstrate that the branching process and normal approximations work well for finite communities, and by numerical examples showing that the final outcome may be close to discontinuous in certain model parameters and that the fraction mildly infected may actually increase as an effect of vaccination.

  • 5.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindenstrand, David
    Stockholm University, Faculty of Science, Department of Mathematics.
    Epidemic modelling: aspects where stochasticity matters2009In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 22, no 2, p. 109-116Article in journal (Refereed)
    Abstract [en]

    Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R0 is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.

  • 6.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindenstrand, David
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inhomogeneous epidemics on weighted networks2012In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 240, no 2, p. 124-131Article in journal (Refereed)
    Abstract [en]

    A social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights, e.g., reflecting the number of sex-contacts between the individuals. An epidemic model is defined on the network such that individuals are heterogeneous in terms of how susceptible and infectious they are. The basic reproduction number R-0 is derived and studied for various examples, but also the size and probability of a major outbreak. The qualitative conclusion is that R-0 gets larger as the community becomes more heterogeneous but that different heterogeneities (degree distribution, weight, susceptibility and infectivity) can sometimes have the cumulative effect of homogenizing the community, thus making R-0 smaller. The effect on the probability and final size of an outbreak is more complicated.

  • 7.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ouédraogo, Désiré
    SEIRS epidemics with disease fatalities in growing populations2018In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 296, p. 45-59Article, review/survey (Refereed)
    Abstract [en]

    An SEIRS epidemic with disease fatalities is introduced in a growing population (modelled as a super-critical linear birth and death process). The study of the initial phase of the epidemic is stochastic, while the analysis of the major outbreaks is deterministic. Depending on the values of the parameters, the following scenarios are possible. i) The disease dies out quickly, only infecting few; ii) the epidemic takes off, the number of infected individuals grows exponentially, but the fraction of infected individuals remains negligible; iii) the epidemic takes off, the number of infected grows initially quicker than the population, the disease fatalities diminish the growth rate of the population, but it remains super critical, and the fraction of infected go to an endemic equilibrium; iv) the epidemic takes off, the number of infected individuals grows initially quicker than the population, the diseases fatalities turn the exponential growth of the population to an exponential decay.

  • 8.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Traoré, Ali
    Stockholm University, Faculty of Science, Department of Mathematics. Université Ouaga, Ouagadougou, Burkina Faso.
    A stochastic vector-borne epidemic model: Quasi-stationarity and extinction2017In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 289, p. 89-95Article in journal (Refereed)
    Abstract [en]

    We consider a stochastic model describing the spread of a vector borne disease in a community where individuals (hosts and vectors) die and new individuals (hosts and vectors) are born. The time to extinction of the disease, T-Q, starting in quasi-stationary (conditional on non extinction) is studied. Properties of the limiting distribution are used to obtain an approximate expression for E(T-Q), the mean-parameter in the exponential distribution of the time to extinction, for a finite population. It is then investigated numerically and by means of simulations how E(T-Q) and its approximations depend on the different model parameters.

  • 9. Cai, Liming
    et al.
    Li, Xuezhi
    Tuncer, Necibe
    Martcheva, Maia
    Lashari, Abid Ali
    Stockholm University, Faculty of Science, Department of Mathematics.
    Optimal control of a malaria model with asymptomatic class and superinfection2017In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 288, p. 94-108Article in journal (Refereed)
    Abstract [en]

    In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R0,R0, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R0<1,R0<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.

  • 10.
    Deijfen, Maria
    Stockholm University, Faculty of Science, Department of Mathematics.
    Epidemics and vaccination on weighted graphs2011In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 232, no 1, p. 57-65Article in journal (Refereed)
    Abstract [en]

    A Reed-Frost epidemic with inhomogeneous infection probabilities on a graph with prescribed degree distribution is studied. Each edge (u, v) in the graph is equipped with two weights W((u,v)) and W((v,u)) that represent the (subjective) strength of the connection and determine the probability that u infects v in case u is infected and vice versa. Expressions for the epidemic threshold are derived for i.i.d. weights and for weights that are functions of the degrees. For i.i.d. weights, a variation of the so called acquaintance vaccination strategy is analyzed where vertices are chosen randomly and neighbors of these vertices with large edge weights are vaccinated. This strategy is shown to outperform the strategy where the neighbors are chosen randomly in the sense that the basic reproduction number is smaller for a given vaccination coverage.

  • 11.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Olsson, Fredrik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    A new general analytical approach for modeling patterns of genetic differentiation and effective size of subdivided populations over time2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 258, p. 113-133Article in journal (Refereed)
    Abstract [en]

    The main purpose of this paper is to develop a theoretical framework for assessing effective population size and genetic divergence in situations with structured populations that consist of various numbers of more or less interconnected subpopulations. We introduce a general infinite allele model for a diploid, monoecious and subdivided population, with subpopulation sizes varying overtime, including local subpopulation extinction and recolonization, bottlenecks, cyclic census size changes or exponential growth. Exact matrix analytic formulas are derived for recursions of predicted (expected) gene identities and gene diversities, identity by descent and coalescence probabilities, and standardized variances of allele frequency change. This enables us to compute and put into a general framework a number of different types of genetically effective population sizes (N-e) including variance, inbreeding, nucleotide diversity, and eigenvalue effective size. General expressions for predictions (g(ST)) of the coefficient of gene differentiation G(ST) are also derived. We suggest that in order to adequately describe important properties of a subdivided population with respect to allele frequency change and maintenance of genetic variation over time, single values of g(ST) and N-e are not enough. Rather, the temporal dynamic patterns of these properties are important to consider. We introduce several schemes for weighting subpopulations that enable effective size and expected genetic divergence to be calculated and described as functions of time, globally for the whole population and locally for any group of subpopulations. The traditional concept of effective size is generalized to situations where genetic drift is confounded by external sources, such as immigration and mutation. Finally, we introduce a general methodology for state space reduction, which greatly decreases the computational complexity of the matrix analytic formulas.

  • 12.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tyvand, Peder A.
    Miloh, Touvia
    Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation2016In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 272, p. 100-112Article in journal (Refereed)
    Abstract [en]

    The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a. Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.

  • 13.
    Lindenstrand, David
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Svensson, Åke
    Stockholm University, Faculty of Science, Department of Mathematics.
    Estimation of the Malthusian parameter in an stochastic epidemic model using martingale methods2013In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 246, no 2, p. 272-279Article in journal (Refereed)
    Abstract [en]

    Data, on the number of infected, gathered from a large epidemic outbreak can be used to estimate parameters related to the strength and speed of the spread. The Malthusian parameter, which determines the initial growth rate of the epidemic is often of crucial interest. Using a simple epidemic SEIR model with known generation time distribution, we define and analyze an estimate, based on martingale methods. We derive asymptotic properties of the estimate and compare them to the results from simulations of the epidemic. The estimate uses all the information contained in the epidemic curve, in contrast to estimates which only use data from the start of the outbreak. 

  • 14.
    Lindholm, Mathias
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the time to extinction for a two-type version of Bartlett's epidemic model2008In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 212, no 1, p. 99-108Article in journal (Refereed)
    Abstract [en]

    We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible infectious recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.

  • 15.
    Lopes, Fabio Marcellus
    Stockholm University, Faculty of Science, Department of Mathematics.
    Epidemics on a weighted network with tunable degree-degree correlation2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 253, p. 40-49Article in journal (Refereed)
    Abstract [en]

    We propose a weighted version of the standard configuration model which allows for a tunable degree-degree correlation. A social network is modeled by a weighted graph generated by this model, where the edge weights indicate the intensity or type of contact between the individuals. An inhomogeneous Reed-Frost epidemic model is then defined on the network, where the inhomogeneity refers to different disease transmission probabilities related to the edge weights. By tuning the model we study the impact of different correlation patterns on the network and epidemics therein. Our results suggest that the basic reproduction number R-0 of the epidemic increases (decreases) when the degree-degree correlation coefficient rho increases (decreases). Furthermore, we show that such effect can be amplified or mitigated depending on the relation between degree and weight distributions as well as the choice of the disease transmission probabilities. In addition, for a more general model allowing additional heterogeneity in the disease transmission probabilities we show that rho can have the opposite effect on R-0.

  • 16.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Equilibrium distributions and simulation methods for age structured populations2015In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 268, p. 45-51Article in journal (Refereed)
    Abstract [en]

    A simulation method is presented for the demographic and genetic variation of age structured haploid populations. First, we use matrix analytic methods to derive an equilibrium distribution for the age class sizes conditioned on the total population size. Knowledge of this distribution eliminates the need of a burn-in time in simulations. Next, we derive the distribution of the alleles at a polymorphic locus in various age classes given the allele frequencies in the total population and the age size composition. For the time dynamics, we start by simulating the dynamics for the total population. In order to generate the inheritance of the alleles, we derive their distribution conditionally on the simulated population sizes. This method enables a fast simulation procedure of multiple loci in linkage equilibrium.

  • 17.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Simulation methods for age structured populationsIn: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134Article in journal (Refereed)
  • 18. Pellis, Lorenzo
    et al.
    Ball, Frank
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R-02012In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 235, no 1, p. 85-97Article in journal (Refereed)
    Abstract [en]

    The basic reproduction number R-0 is one of the most important quantities in epidemiology. However, for epidemic models with explicit social structure involving small mixing units such as households, its definition is not straightforward and a wealth of other threshold parameters has appeared in the literature. In this paper, we use branching processes to define R-0, we apply this definition to models with households or other more complex social structures and we provide methods for calculating it.

  • 19.
    Svensson, Åke
    Stockholm University, Faculty of Science, Department of Mathematics.
    The influence of assumptions on generation time distributions in epidemic models2015In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 270, no Part A, p. 81-89Article in journal (Refereed)
    Abstract [en]

    A simple class of stochastic models for epidemic spread in finite, but large, populations is studied. The purpose is to investigate how assumptions about the times between primary and secondary infections influences the outcome of the epidemic. Of particular interest is how assumptions of individual variability in infectiousness relates to variability of the epidemic curve. The main concern is the final size of the epidemic and the time scale at which it evolves. The theoretical results are illustrated by simulations.

  • 20. Tomba, Gianpaolo Scalia
    et al.
    Svensson, Åke
    Stockholm University, Faculty of Science, Department of Mathematics.
    Asikainen, Tommi
    Giesecke, Johan
    Some model based considerations on observing generation times for communicable diseases2010In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 223, no 1, p. 24-31Article in journal (Refereed)
    Abstract [en]

    The generation time of an infectious disease is usually defined as the time from the moment one person becomes infected until that person infects another person. The concept is similar to ""generation gap"" in demography, with new infections replacing births in a population. Originally applied to diseases such as measles where at least the first generations are clearly discernible, the concept has recently been extended to other diseases, such as influenza, where time order of infections is usually much less apparent. By formulating the relevant statistical questions within a simple yet basic mathematical model for infection spread, it is possible to derive theoretical properties of observations in various situations e.g. in ""isolation"", in households, or during large outbreaks. In each case, it is shown that the sampling distribution of observations depends on a number of factors, usually not considered in the literature and that must be taken into account in order to achieve unbiased inference about the generation time distribution. Some implications of these findings for statistical inference methods in epidemic spread models are discussed.

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