Homogeneous Mixing Model Research Articles

Light absorption by brown carbon over the South-East Atlantic Ocean

Abstract. Biomass burning emissions often contain brown carbon (BrC), which represents a large family of light-absorbing organics that are chemically complex, thus making it difficult to estimate their absorption of incoming solar radiation, resulting in large uncertainties in the estimation of the global direct radiative effect of aerosols. Here we investigate the contribution of BrC to the total light absorption of biomass burning aerosols over the South-East Atlantic Ocean with different optical models, utilizing a suite of airborne measurements from the ORACLES 2018 campaign. An effective refractive index of black carbon (BC), meBC=1.95+ikeBC, that characterizes the absorptivity of all absorbing components at 660 nm wavelength was introduced to facilitate the attribution of absorption at shorter wavelengths, i.e. 470 nm. Most values of the imaginary part of the effective refractive index, keBC, were larger than those commonly used for BC from biomass burning emissions, suggesting contributions from absorbers besides BC at 660 nm. The TEM-EDX single-particle analysis further suggests that these long-wavelength absorbers might include iron oxides, as iron is found to be present only when large values of keBC are derived. Using this effective BC refractive index, we find that the contribution of BrC to the total absorption at 470 nm (RBrC,470) ranges from ∼8 %–22 %, with the organic aerosol mass absorption coefficient (MACOA,470) at this wavelength ranging from 0.30±0.27 to 0.68±0.08 m2 g−1. The core–shell model yielded much higher estimates of MACOA,470 and RBrC,470 than homogeneous mixing models, underscoring the importance of model treatment. Absorption attribution using the Bruggeman mixing Mie model suggests a minor BrC contribution of 4 % at 530 nm, while its removal would triple the BrC contribution to the total absorption at 470 nm obtained using the AAE (absorption Ångström exponent) attribution method. Thus, it is recommended that the application of any optical properties-based attribution method use absorption coefficients at the longest possible wavelength to minimize the influence of BrC and to account for potential contributions from other absorbing materials.

African swine fever detection and transmission estimates using homogeneous versus heterogeneous model formulation in stochastic simulations within pig premises.

African swine fever (ASF) is one of the most important foreign animal diseases to the U.S. swine industry. Stakeholders in the swine production sector are on high alert as they witness the devastation of ongoing outbreaks in some of its most important trade partner countries. Efforts to improve preparedness for ASF outbreak management are proceeding in earnest and mathematical modeling is an integral part of these efforts. This study aimed to assess the impact on within-herd transmission dynamics of ASF when the models used to simulate transmission assume there is homogeneous mixing of animals within a barn. Barn-level heterogeneity was explicitly captured using a stochastic, individual pig-based, heterogeneous transmission model that considers three types of infection transmission, (1) within-pen via nose-to-nose contact; (2) between-pen via nose-to-nose contact with pigs in adjacent pens; and (3) both between- and within-pen via distance-independent mechanisms (e.g., via fomites). Predictions were compared between the heterogeneous and the homogeneous Gillespie models. Results showed that the predicted mean number of infectious pigs at specific time points differed greatly between the homogeneous and heterogeneous models for scenarios with low levels of between-pen contacts via distance-independent pathways and the differences between the two model predictions were more pronounced for the slow contact rate scenario. The heterogeneous transmission model results also showed that it may take significantly longer to detect ASF, particularly in large barns when transmission predominantly occurs via nose-to-nose contact between pigs in adjacent pens. The findings emphasize the need for completing preliminary explorations when working with homogeneous mixing models to ascertain their suitability to predict disease outcomes.

Simulation and Evaluation of Precaution Strategies for COVID-19 Pandemic: A Case Study

The world now is living in Corona Virus Disease (COVID-19) pandemic era, which forced almost all businesses and companies to lockdown, which as a result led to economic fluctuation and social issues. Some precautions strategies were introduced and applied by governments to allow businesses to reopen to the public. This paper aims to model an agent-based crowd simulation using unity game engine with C# programming language in order to explore the effectiveness of the current precautions to limit the spread of COVID-19. Two important models are adopted in this work, which are the SIR disease spreading model and the homogeneous mixing behavior model. Utilizing these models, several parameters are chosen to control the viral spread among the agents in five different realistic scenarios. After several simulations, it was determined that the number of agents in the scene and wearing masks have the highest impact on limiting the viral spread.

Immunization using a heterogeneous geo-spatial population model: A qualitative perspective on COVID-19 vaccination strategies

Epidemic modeling has been a key tool for understanding the impact of global viral outbreaks for over two decades. Recent developments of the COVID-19 pandemic have accelerated research using compartmental models, like SI, SIR, SEIR, with their appropriate modifications. However, there is a large body of recent research consolidated on homogeneous population mixing models, which are known to offer reduced tractability, and render conclusions hard to quantify. As such, based on our recent work, introducing the heterogeneous geo-spatial mobility population model (GPM), we adapt a modified SIR-V (susceptible-infected-recovered-vaccinated) epidemic model which embodies the idea of patient relapse from R back to S, vaccination of R and S patients (reducing their infectiousness), thus altering the infectiousness of V patients (from λn to λr). Simulation results spanning over a period of t = 2000 days (6 years, the period « 2020-2025) compare the impact of an epidemic outbreak with variable vaccination strategies, starting after 1 year (as is the case of COVID-19). The infected proportion in the remaining 5-year period is analyzed using vaccination rates from rv = 0 (no vaccination) to rv = 1. While rv < 0.4 is less effective during the earlier stages, all strategies with rv > 0.4 show a similar downward convergence reducing the number of infected by more than half, compared to no vaccination. Given the complexity of epidemic processes, we conclude that higher vaccination rates yield similar results, but a minimal rv = 0.4 (40% of population over five years) should be targeted.

Methods for Rapid Mobility Estimation to Support Outbreak Response.

When pressed for time, outbreak investigators often use homogeneous mixing models to model infectious diseases in data-poor regions. But recent outbreaks such as the 2014 Ebola outbreak in West Africa have shown the limitations of this approach in an era of increasing urbanization and connectivity. Both outbreak detection and predictive modeling depend on realistic estimates of human and disease mobility, but these data are difficult to acquire in a timely manner. This is especially true when dealing with an emerging outbreak in an under-resourced nation. Weighted travel networks with realistic estimates for population flows are often proprietary, expensive, or nonexistent. Here we propose a method for rapidly generating a mobility model from open-source data. As an example, we use road and river network data, along with population estimates, to construct a realistic model of human movement between health zones in the Democratic Republic of the Congo (DRC). Using these mobility data, we then fit an epidemic model to real-world surveillance data from the recent Ebola outbreak in the Nord Kivu region of the DRC to illustrate a potential use of the generated mobility estimation. In addition to providing a way for rapid risk estimation, this approach brings together novel techniques to merge diverse GIS datasets that can then be used to address issues that pertain to public health and global health security.

A dynamic power-law sexual network model of gonorrhoea outbreaks.

Human networks of sexual contacts are dynamic by nature, with partnerships forming and breaking continuously over time. Sexual behaviours are also highly heterogeneous, so that the number of partners reported by individuals over a given period of time is typically distributed as a power-law. Both the dynamism and heterogeneity of sexual partnerships are likely to have an effect in the patterns of spread of sexually transmitted diseases. To represent these two fundamental properties of sexual networks, we developed a stochastic process of dynamic partnership formation and dissolution, which results in power-law numbers of partners over time. Model parameters can be set to produce realistic conditions in terms of the exponent of the power-law distribution, of the number of individuals without relationships and of the average duration of relationships. Using an outbreak of antibiotic resistant gonorrhoea amongst men have sex with men as a case study, we show that our realistic dynamic network exhibits different properties compared to the frequently used static networks or homogeneous mixing models. We also consider an approximation to our dynamic network model in terms of a much simpler branching process. We estimate the parameters of the generation time distribution and offspring distribution which can be used for example in the context of outbreak reconstruction based on genomic data. Finally, we investigate the impact of a range of interventions against gonorrhoea, including increased condom use, more frequent screening and immunisation, concluding that the latter shows great promise to reduce the burden of gonorrhoea, even if the vaccine was only partially effective or applied to only a random subset of the population.

Measurability of the epidemic reproduction number in data-driven contact networks

The basic reproduction number is one of the conceptual cornerstones of mathematical epidemiology. Its classical definition as the number of secondary cases generated by a typical infected individual in a fully susceptible population finds a clear analytical expression in homogeneous and stratified mixing models. Along with the generation time (the interval between primary and secondary cases), the reproduction number allows for the characterization of the dynamics of an epidemic. A clear-cut theoretical picture, however, is hardly found in real data. Here, we infer from highly detailed sociodemographic data two multiplex contact networks representative of a subset of the Italian and Dutch populations. We then simulate an infection transmission process on these networks accounting for the natural history of influenza and calibrated on empirical epidemiological data. We explicitly measure the reproduction number and generation time, recording all individual-level transmission events. We find that the classical concept of the basic reproduction number is untenable in realistic populations, and it does not provide any conceptual understanding of the epidemic evolution. This departure from the classical theoretical picture is not due to behavioral changes and other exogenous epidemiological determinants. Rather, it can be simply explained by the (clustered) contact structure of the population. Finally, we provide evidence that methodologies aimed at estimating the instantaneous reproduction number can operationally be used to characterize the correct epidemic dynamics from incidence data.

Modeling the Heterogeneity of Dengue Transmission in a City.

Dengue fever is one of the most important vector-borne diseases in the world, and modeling its transmission dynamics allows for determining the key influence factors and helps to perform interventions. The heterogeneity of mosquito bites of humans during the spread of dengue virus is an important factor that should be considered when modeling the dynamics. However, traditional models generally assumed homogeneous mixing between humans and vectors, which is inconsistent with reality. In this study, we proposed a compartmental model with negative binomial distribution transmission terms to model this heterogeneity at the population level. By including the aquatic stage of mosquitoes and incorporating the impacts of the environment and climate factors, an extended model was used to simulate the 2014 dengue outbreak in Guangzhou, China, and to simulate the spread of dengue in different scenarios. The results showed that a high level of heterogeneity can result in a small peak size in an outbreak. As the level of heterogeneity decreases, the transmission dynamics approximate the dynamics predicted by the corresponding homogeneous mixing model. The simulation results from different scenarios showed that performing interventions early and decreasing the carrying capacity for mosquitoes are necessary for preventing and controlling dengue epidemics. This study contributes to a better understanding of the impact of heterogeneity during the spread of dengue virus.

Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model.

Mathematical models have been used to understand the transmission dynamics of infectious diseases and to assess the impact of intervention strategies. Traditional mathematical models usually assume a homogeneous mixing in the population, which is rarely the case in reality. Here, we construct a new transmission function by using as the probability density function a negative binomial distribution, and we develop a compartmental model using it to model the heterogeneity of contact rates in the population. We explore the transmission dynamics of the developed model using numerical simulations with different parameter settings, which characterize different levels of heterogeneity. The results show that when the reproductive number, , is larger than one, a low level of heterogeneity results in dynamics similar to those predicted by the homogeneous mixing model. As the level of heterogeneity increases, the dynamics become more different. As a test case, we calibrated the model with the case incidence data for severe acute respiratory syndrome (SARS) in Beijing in 2003, and the estimated parameters demonstrated the effectiveness of the control measures taken during that period.

Incubation periods under various anti-retroviral therapies in homogeneous mixing and age-structured dynamical models: A theoretical approach

With the launch of second line anti-retroviral therapy for HIV infected individuals, there has been an increased expectation on surviving period of people with HIV. We consider previously well-known models in HIV epidemiology where the parameter for incubation period is used as one of the important components to explain the dynamics of the variables. Such models are extended here to explain the dynamics with respect to a given therapy that prolongs life of an HIV infected individual. A deconvolution method is demonstrated for estimation of parameters in the situations when no-therapy and multiple therapies are given to the infected population. The models and deconvolution method are extended in order to study the impact of therapy in age-structured populations. A generalization for a situation when n-types of therapies are available is given. Models are demonstrated using hypothetical data and sensitivity of the parameters are also computed.

On the early epidemic dynamics for pairwise models

The relationship between the basic reproduction number R0 and the exponential growth rate, specific to pair approximation models, is derived for the SIS, SIR and SEIR deterministic models without demography. These models are extended by including a random rewiring of susceptible individuals from infectious (and exposed) neighbours. The derived relationship between the exponential growth rate and R0 appears as formally consistent with those derived from homogeneous mixing models, enabling us to measure the transmission potential using the early growth rate of cases. On the other hand, the algebraic expression of R0 for the SEIR pairwise model shows that its value is affected by the average duration of the latent period, in contrast to what happens for the homogeneous mixing SEIR model. Numerical simulations on complex contact networks are performed to check the analytical assumptions and predictions.

Epidemic growth rate and household reproduction number in communities of households, schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.

Investigation of an inhomogeneous turbulent mixing model for conditional moment closure applied to autoignition

The present paper examines the case of autoignition of high pressure methane jets in a shock tube over a range of pre-heated air temperatures in engine-relevant conditions. The two objectives of the present paper are: (i) to examine the effect of the inhomogeneous mixing model on the autoignition predictions relative to the results obtained using homogeneous mixing models and (ii) to see if the magnitude of the change can explain the discrepancy between the predictions of ignition delay previously obtained with homogeneous mixing models and the experimental data. The governing equation of the scalar dissipation rate is solved for transient conditions and two different formulations of the same model are tested and compared: one using the linear model for the conditional velocity and one including the gradient diffusion model. The predicted ignition kernel location and time delay over a range of pre-combustion air temperatures are compared with results obtained using two homogeneous turbulent mixing models and available experimental data. The profiles of conditional velocity and the conditional scalar dissipation rate are examined. Issues related to the conditional velocity model are discussed. It is found that the differences in the predictions are due to the mixing model only. The inhomogeneous model using the gradient conditional velocity model produces much larger ignition delays compared to the other models, whereas the inhomogeneous form including the linear model does not produce any significant differences. The effect of the turbulent inhomogeneous model is larger at high air temperatures and decreases with decreasing air temperatures. In comparison with the measured ignition delays, the inhomogeneous-Gradient model brings a small improvement at high air temperatures over the results from the turbulent homogeneous models. At low air temperatures, other parameters need to be investigated in order to bring the predicted ignition delays and locations within the experimental data scatter.

Calculation of Disease Dynamics in a Population of Households

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.

A simulation analysis to characterize the dynamics of vaccinating behaviour on contact networks

BackgroundHuman behavior influences infectious disease transmission, and numerous "prevalence-behavior" models have analyzed this interplay. These previous analyses assumed homogeneously mixing populations without spatial or social structure. However, spatial and social heterogeneity are known to significantly impact transmission dynamics and are particularly relevant for certain diseases. Previous work has demonstrated that social contact structure can change the individual incentive to vaccinate, thus enabling eradication of a disease under a voluntary vaccination policy when the corresponding homogeneous mixing model predicts that eradication is impossible due to free rider effects. Here, we extend this work and characterize the range of possible behavior-prevalence dynamics on a network.MethodsWe simulate transmission of a vaccine-prevetable infection through a random, static contact network. Individuals choose whether or not to vaccinate on any given day according to perceived risks of vaccination and infection.ResultsWe find three possible outcomes for behavior-prevalence dynamics on this type of network: small final number vaccinated and final epidemic size (due to rapid control through voluntary ring vaccination); large final number vaccinated and significant final epidemic size (due to imperfect voluntary ring vaccination), and little or no vaccination and large final epidemic size (corresponding to little or no voluntary ring vaccination). We also show that the social contact structure enables eradication under a broad range of assumptions, except when vaccine risk is sufficiently high, the disease risk is sufficiently low, or individuals vaccinate too late for the vaccine to be effective.ConclusionFor populations where infection can spread only through social contact network, relatively small differences in parameter values relating to perceived risk or vaccination behavior at the individual level can translate into large differences in population-level outcomes such as final size and final number vaccinated. The qualitative outcome of rational, self interested behaviour under a voluntary vaccination policy can vary substantially depending on interactions between social contact structure, perceived vaccine and disease risks, and the way that individual vaccination decision-making is modelled.

The Effect of Time Distribution Shape on a Complex Epidemic Model

In elaborating a model of the progress of an epidemic, it is necessary to make assumptions about the distributions of latency times and infectious times. In many models, the often implicit assumption is that these times are independent and exponentially distributed. We explore the effects of altering the distribution of latency and infectious times in a complex epidemic model with regional divisions connected by a travel intensity matrix. We show a delay in spread with more realistic latency times. More realistic infectiousness times lead to faster epidemics. The effects are similar but accentuated when compared to a purely homogeneous mixing model.

Mathematical modelling of tuberculosis epidemics

The strengths and limitations of using homogeneous mixing and heterogeneous mixing epidemic models are explored in the context of the transmission dynamics of tuberculosis. The focus is on three types of models: a standard incidence homogeneous mixing model, a non-homogeneous mixing model that incorporates 'household' contacts, and an age-structured model. The models are parameterized using demographic and epidemiological data and the patterns generated from these models are compared. Furthermore, the effects of population growth, stochasticity, clustering of contacts, and age structure on disease dynamics are explored. This framework is used to asses the possible causes for the observed historical decline of tuberculosis notifications.

When individual behaviour matters: homogeneous and network models in epidemiology.

Heterogeneity in host contact patterns profoundly shapes population-level disease dynamics. Many epidemiological models make simplifying assumptions about the patterns of disease-causing interactions among hosts. In particular, homogeneous-mixing models assume that all hosts have identical rates of disease-causing contacts. In recent years, several network-based approaches have been developed to explicitly model heterogeneity in host contact patterns. Here, we use a network perspective to quantify the extent to which real populations depart from the homogeneous-mixing assumption, in terms of both the underlying network structure and the resulting epidemiological dynamics. We find that human contact patterns are indeed more heterogeneous than assumed by homogeneous-mixing models, but are not as variable as some have speculated. We then evaluate a variety of methodologies for incorporating contact heterogeneity, including network-based models and several modifications to the simple SIR compartmental model. We conclude that the homogeneous-mixing compartmental model is appropriate when host populations are nearly homogeneous, and can be modified effectively for a few classes of non-homogeneous networks. In general, however, network models are more intuitive and accurate for predicting disease spread through heterogeneous host populations.

The effect of population structure on the emergence of drug resistance during influenza pandemics.

The spread of H5N1 avian influenza and the recent high numbers of confirmed human cases have raised international concern about the possibility of a new pandemic. Therefore, antiviral drugs are now being stockpiled to be used as a first line of defence. The large-scale use of antivirals will however exert a strong selection pressure on the virus, and may lead to the emergence of drug-resistant strains. A few mathematical models have been developed to assess the emergence of drug resistance during influenza pandemics. These models, however, neglected the spatial structure of large populations and the stochasticity of epidemic and demographic processes. To assess the impact of population structure and stochasticity, we modify and extend a previous model of influenza epidemics into a metapopulation model which takes into account the division of large populations into smaller units, and develop deterministic and stochastic versions of the model. We find that the dynamics in a fragmented population is less explosive, and, as a result, prophylaxis will prevent more infections and lead to fewer resistant cases in both the deterministic and stochastic model. While in the deterministic model the final level of resistance during treatment is not affected by fragmentation, in the stochastic model it is. Our results enable us to qualitatively extrapolate the prediction of deterministic, homogeneous-mixing models to more realistic scenarios.