Modeling the transmission dynamics and control of hepatitis B virus in China

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40-50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. We propose a mathematical model to understand the transmission dynamics and prevalence of HBV infection in China. Based on the data reported by the Ministry of Health of China, the model provides an approximate estimate of the basic reproduction number R₀=2.406. This indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization program and control measures. Although China made a great progress in increasing coverage among infants with hepatitis B vaccine, it has a long and hard battle to fight in order to significantly reduce the incidence and eventually eradicate the virus.     

A general multi-strain model with environmental transmission: Invasion conditions for the disease-free and endemic states

Although many infectious diseases of humans and wildlife are transmitted via an environmental reservoir, the theory of environmental transmission remains poorly elaborated. Here we introduce an SIR-type multi-strain disease transmission model with perfect cross immunity where environmental transmission is broadly defined by three axioms. We establish the conditions under which a multi-strain endemic state is invaded by another strain which is both directly and environmentally transmitted. We discuss explicit forms for environmental transmission terms and apply our newly derived invasion conditions to a two-strain system. Then, we consider the case of two strains with matching basic reproduction numbers (i.e., R₀), one directly transmitted only and the other both directly and environmentally transmitted, invading each other's endemic state. We find that the strain which is only directly transmitted can invade the endemic state of the strain with mixed transmission. However, the endemic state of the first strain is neutrally stable to invasion by the second strain. Thus, our results suggest that environmental transmission makes the endemic state less resistant to invasion.     

Modeling the effect of information quality on risk behavior change and the transmission of infectious diseases

How the quality of information about the prevalence of an infectious disease affects individuals’ incentives to adopt self-protective actions to reduce the risk of infection is studied using an economic/game-theoretic model of epidemics. In the model, agents make inferences regarding the current prevalence of a disease by observing the health status of a subset of the population. Therefore, the higher the number of agents whose infection status can be observed, the better one’s information about the current prevalence is. In particular, it is assumed that an agent’s estimate of the current prevalence depends on observations of the current health status of other agents and on the agent’s estimate of past prevalence, and that the agent places more weight on the current observations in forming an estimate of the prevailing prevalence when the number of observations increases. It is shown that the likelihood of eradicating an infectious disease through behavioral changes depends critically on the amount of information that individuals have access to, which also determines whether prevalence will be relatively stable or will exhibit cyclical patterns over time. Increasing the amount of information that individuals possess may lower the likelihood of eradication.     

A growth and division model for retinoblastoma

A stochastic growth and division model for studying a two hit cancer is developed and applied to retinoblastoma. Retinoblastoma occurs if both genes coding for a tumor suppressor protein on homologous chromosomes become defective. Germinal cases occur when a patient or carrier, born with one defective gene, suffers a second insult to any progeny retinal cell. Somatic cases are far less likely as two hits to the same cell during development are required. Details of the disease, germinal or somatic, unilateral or bilateral, in combination with case data allow for the estimation of the two parameters of the model: mutation rate, estimated at p=7x10(-7) per chromosome per cell division, and carrier frequency, estimated at f=40 per million. The model indicates that carriers of the disease arise from similar mutations to germ cells; in particular, heridary transmission can occur for only a generation or two before dying out. The results show that a stochastic simulation of a multi-hit cancer is feasible and may predict tumor growth dynamics. A simulation run will have to consist of a few million cells in order to observe even a small number of mutations. And several dozens such runs will have to be simulated.     

Public avoidance and epidemics: insights from an economic model

In this paper, we present a mathematical model of infectious disease transmission in which people can engage in public avoidance behavior to minimize the likelihood of acquiring an infection. The framework employs the economist's theory of utility maximization to model people's decision regarding their level of public avoidance. We derive the reproductive number of a disease which determines whether an endemic equilibrium exists or not. We show that when the contact function exhibits saturation, an endemic equilibrium must be unique. Otherwise, multiple endemic equilibria that differ in disease prevalence can coexist, and which one the population gets to depends on initial conditions. Even when a unique endemic equilibrium exists, people's preferences and the initial conditions may determine whether the disease will eventually die out or become endemic. Public health policies that increase the recovery rate or encourage self-quarantine by infected people can be beneficial to the community by lowering disease prevalence. However, it is also possible for these policies to worsen the situation and cause prevalence to rise since these measures give people less incentive to engage in public avoidance behavior. We also show that implementing policies that result in a higher level of public avoidance behavior in equilibrium does not necessarily lower prevalence and can result in more infections.     

A two-age-classes dengue transmission model

In this paper, we discuss a two-age-classes dengue transmission model with vaccination. The reason to divide the human population into two age classes is for practical purpose, as vaccination is usually concentrated in one age class. We assume that a constant rate of individuals in the child-class is vaccinated. We analyze a threshold number which is equivalent to the basic reproduction number. If there is an undeliberate vaccination to infectious children, which worsens their condition as the time span of being infectious increases, then paradoxically, vaccination can be counter productive. The paradox, stating that vaccination makes the basic reproduction number even bigger, can occur if the worsening effect is greater than a certain threshold, a function of the human demographic and epidemiological parameters, which is independent of the level of vaccination. However, if the worsening effect is to increase virulence so that one will develop symptoms, then the vaccination is always productive. In both situations, screening should take place before vaccination. In general, the presence of class division has obscured the known rule of thumb for vaccination.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

Modeling amantadine treatment of influenza A virus in vitro

We analyzed the dynamics of an influenza A/Albany/1/98 (H3N2) viral infection, using a set of mathematical models highlighting the differences between in vivo and in vitro infection. For example, we found that including virion loss due to cell entry was critical for the in vitro model but not for the in vivo model. Experiments were performed on influenza virus-infected MDCK cells in vitro inside a hollow-fiber (HF) system, which was used to continuously deliver the drug amantadine. The HF system captures the dynamics of an influenza infection, and is a controlled environment for producing experimental data which lend themselves well to mathematical modeling. The parameter estimates obtained from fitting our mathematical models to the HF experimental data are consistent with those obtained earlier for a primary infection in a human model. We found that influenza A/Albany/1/98 (H3N2) virions under normal experimental conditions at rapidly lose infectivity with a half-life of , and that the lifespan of productively infected MDCK cells is . Finally, using our models we estimated that the maximum efficacy of amantadine in blocking viral infection is ∼74%, and showed that this low maximum efficacy is likely due to the rapid development of drug resistance.     

A simple model of retina-LGN transmission

To gain a deeper understanding of the transmission of visual signals from retina through the lateral geniculate nucleus (LGN), we have used a simple leaky integrate and-fire model to simulate a relay cell in the LGN. The simplicity of the model was motivated by two questions: (1) Can an LGN model that is driven by a retinal spike train recorded as synaptic (‘S’) potentials, but does not include a diverse array of ion channels, nor feedback inputs from the cortex, brainstem, and thalamic reticular nucleus, accurately simulate the LGN discharge on a spike-for-spike basis? (2) Are any special synaptic mechanisms, beyond simple summation of currents, necessary to model experimental recordings? We recorded cat relay cell responses to spatially homogeneous small or large spots, with luminance that was rapidly modulated in a pseudo-random fashion. Model parameters for each cell were optimized with a Simplex algorithm using a short segment of the recording. The model was then tested on a much longer, distinct data set consisting of responses to numerous repetitions of the noisy stimulus. For LGN cells that spiked in response to a sufficiently large fraction of retinal inputs, we found that this simplified model accurately predicted the firing times of LGN discharges. This suggests that modulations of the efficacy of the retino-geniculate synapse by pre-synaptic facilitation or depression are not necessary in order to account for the LGN responses generated by our stimuli, and that post-synaptic summation is sufficient.     

Analysis of a dengue disease transmission model

A model for the transmission of dengue fever in a constant human population and variable vector population is discussed. A complete global analysis is given, which uses the results of the theory of competitive systems and stability of periodic orbits, to establish the global stability of the endemic equilibrium. The control measures of the vector population are discussed in terms of the threshold condition, which governs the existence and stability of the endemic equilibrium.     

Development of a mathematical model for mechanical transmission of trypanosomes and other pathogens of cattle transmitted by tabanids

Mechanical transmission of pathogens by biting insects is a non-specific phenomenon in which pathogens are transmitted from the blood of an infected host to another host during interrupted feeding of the insects. A large range of pathogens can be mechanically transmitted, e.g. hemoparasites, bacteria and viruses. Some pathogens are almost exclusively mechanically transmitted, while others are also cyclically transmitted. For agents transmitted both cyclically and mechanically (mixed transmission), such as certain African pathogenic trypanosomes, the relative impact of mechanical versus cyclical transmission is essentially unknown. We have developed a mathematical model of pathogen transmission by a defined insect population to evaluate the importance of mechanical transmission. Based on a series of experiments aimed at demonstrating mechanical transmission of African trypanosomes by tabanids, the main parameters of the model were either quantified (host parasitaemia, mean individual insect burden, initial prevalence of infection) or estimated (unknown parameters). This model allows us to simulate the evolution of pathogen prevalence under various predictive circumstances, including control measures and could be used to assess the risk of mechanical transmission under field conditions. If adjustments of parameters are provided, this model could be generalized to other pathogenic agents present in the blood of their hosts (Bovine Leukemia virus, Anaplasma, etc.) or other biting insects such as biting muscids (stomoxyines) and hippoboscids.     

Social networks of disease spread in the lower Illinois valley: a simulation approach

This study illustrates the use of disease modeling and simulation techniques to the study of the spread of disease within and between social networks. A Reed-Frost type model of disease spread is used to construct a simulation of the spread of tuberculosis within three prehistoric populations of the Lower Illinois River Valley during Middle Woodland, Late Woodland, and Mississippian times. A high and low population size was modeled for each time period. Late Woodland model 2 (low population estimate) is the only model that experienced pathogen extinction with host survival. The rest of the models experienced rapid and severe host population decline. The results of the simulation suggest that a social network size of between 180 and 440 persons is required under the conditions of this model for host-pathogen coexistence (i.e., endemicity) to occur. The severe population decline experienced by these populations suggests that tuberculosis as modeled here could not have existed in these populations. Future refinements of modeling and simulation techniques can provide additional insights into how disease spreads among social contacts.     

An individual-based model of chaparral vegetation response to frequent wildfires

The Santa Monica Mountains are home to many species of chaparral shrubs that provide vegetative cover and whose deep roots contribute to the stability of the steep slopes. Recently, native chaparral have been threatened by an unprecedented drought and frequent wildfires in Southern California. Besides the damage from the wildfires themselves, there is the potential for subsequent structural losses due to erosion and landslides. In this paper, we develop a mathematical model that predicts the impact of drought and frequent wildfires on chaparral plant community structure. We begin by classifying chaparral into two life history types based on their response to wildfires. Nonsprouters are completely killed by a fire, but their seeds germinate in response to fire cues. Facultative sprouters survive by resprouting but also rely on seed germination for post-fire recovery. The individual-based model presented here simulates the growth, seed dispersal, and resprouting behavior of individual shrubs across two life history types as they compete for space and resources in a rectangular domain. The model also incorporates varying annual rainfall and fire frequency as well as the competition between plants for scarce resources. The parameters were fit using seedling and resprout survivorship data as well as point quarter sampling data from 1986 to 2014 at a biological preserve within the natural landscape of the Malibu campus of Pepperdine University. The simulations from our model reproduce the change in plant community structure at our study site which includes the local extinction of the nonsprouter Ceanothus megacarpus due to shortened fire return intervals. Our simulations predict that a combination of extreme drought and frequent wildfires will drastically reduce the overall density of chaparral, increasing the likelihood of invasion by highly flammable exotic grasses. The simulations further predict that the majority of surviving shrubs will be facultative sprouting species such as Malosma laurina.     

Modeling and numerical simulation of biotin carboxylase kinetics: implications for half-sites reactivity

Biotin carboxylase catalyzes the ATP-dependent carboxylation of biotin and is one component of the multienzyme complex acetyl-CoA carboxylase that catalyzes the first committed step in fatty acid synthesis in all organisms. In Escherichia coli, biotin carboxylase exists as a homodimer where each subunit contains a complete active site. In a previous study (Janiyani, K., Bordelon, T., Waldrop, G.L., Cronan Jr., J.E., 2001. J. Biol. Chem. 276, 29864-29870), hybrid dimers were constructed where one subunit was wild-type and the other contained an active site mutation that reduced activity at least 100-fold. The activity of the hybrid dimers was only slightly greater than the activity of the mutant homodimers and far less than the expected 50% activity for completely independent active sites. Thus, there is communication between the two subunits of biotin carboxylase. The dominant negative effect of the mutations on the wild-type active site was interpreted as alternating catalytic cycles of the active sites in the homodimer. In order to test the hypothesis of oscillating catalytic cycles, mathematical modeling and numerical simulations of the kinetics of wild-type, hybrid dimers, and mutant homodimers of biotin carboxylase were performed. Numerical simulations of biotin carboxylase kinetics were the most similar to the experimental data when an oscillating active site model was used. In contrast, alternative models where the active sites were independent did not agree with the experimental data. Thus, the numerical simulations of the proposed kinetic model support the hypothesis that the two active sites of biotin carboxylase alternate their catalytic cycles.     

A fully continuous individual-based model of tumor cell evolution

The aim of this work is to develop and study a fully continuous individual-based model (IBM) for cancer tumor invasion into a spatial environment of surrounding tissue. The IBM improves previous spatially discrete models, because it is continuous in all variables (including spatial variables), and thus not constrained to lattice frameworks. The IBM includes four types of individual elements: tumor cells, extracellular macromolecules (MM), a matrix degradative enzyme (MDE), and oxygen. The algorithm underlying the IBM is based on the dynamic interaction of these four elements in the spatial environment, with special consideration of mutation phenotypes. A set of stochastic differential equations is formulated to describe the evolution of the IBM in an equivalent way. The IBM is scaled up to a system of partial differential equations (PDE) representing the limiting behavior of the IBM as the number of cells and molecules approaches infinity. Both models (IBM and PDE) are numerically simulated with two kinds of initial conditions: homogeneous MM distribution and heterogeneous MM distribution. With both kinds of initial MM distributions spatial fingering patterns appear in the tumor growth. The output of both simulations is quite similar.     

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

A precise definition of the basic reproduction number, , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if , then the disease free equilibrium is locally asymptotically stable; whereas if , then it is unstable. Thus, is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for near one. This criterion, together with the definition of , is illustrated by treatment, multigroup, staged progression, multistrain and vector–host models and can be applied to more complex models. The results are significant for disease control.     

A dynamical model of human immune response to influenza A virus infection

We present a simplified dynamical model of immune response to uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Innate immunity is represented by interferon-induced resistance to infection of respiratory epithelial cells and by removal of infected cells by effector cells (cytotoxic T-cells and natural killer cells). Adaptive immunity is represented by virus-specific antibodies. Similar in spirit to the recent model of Bocharov and Romanyukha [1994. Mathematical model of antiviral immune response. III. Influenza A virus infection. J. Theor. Biol. 167, 323-360], the model is constructed as a system of 10 ordinary differential equations with 27 parameters characterizing the rates of various processes contributing to the course of disease. The parameters are derived from published experimental data or estimated so as to reproduce available data about the time course of IAV infection in a na?ve host. We explore the effect of initial viral load on the severity and duration of the disease, construct a phase diagram that sheds insight into the dynamics of the disease, and perform sensitivity analysis on the model parameters to explore which ones influence the most the onset, duration and severity of infection. To account for the variability and speed of adaptation of the adaptive response to a particular virus strain, we introduce a variable that quantifies the antigenic compatibility between the virus and the antibodies currently produced by the organism. We find that for small initial viral load the disease progresses through an asymptomatic course, for intermediate value it takes a typical course with constant duration and severity of infection but variable onset, and for large initial viral load the disease becomes severe. This behavior is robust to a wide range of parameter values. The absence of antibody response leads to recurrence of disease and appearance of a chronic state with nontrivial constant viral load.     

A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation

The acute inflammatory response, triggered by a variety of biological or physical stresses on an organism, is a delicate system of checks and balances that, although aimed at promoting healing and restoring homeostasis, can result in undesired and occasionally lethal physiological responses. In this work, we derive a reduced conceptual model for the acute inflammatory response to infection, built up from consideration of direct interactions of fundamental effectors. We harness this model to explore the importance of dynamic anti-inflammation in promoting resolution of infection and homeostasis. Further, we offer a clinical correlation between model predictions and potential therapeutic interventions based on modulation of immunity by anti-inflammatory agents.     

A neural network model of reliably optimized spike transmission

We studied the detailed structure of a neuronal network model in which the spontaneous spike activity is correctly optimized to match the experimental data and discuss the reliability of the optimized spike transmission. Two stochastic properties of the spontaneous activity were calculated: the spike-count rate and synchrony size. The synchrony size, expected to be an important factor for optimization of spike transmission in the network, represents a percentage of observed coactive neurons within a time bin, whose probability approximately follows a power-law. We systematically investigated how these stochastic properties could matched to those calculated from the experimental data in terms of the log-normally distributed synaptic weights between excitatory and inhibitory neurons and synaptic background activity induced by the input current noise in the network model. To ensure reliably optimized spike transmission, the synchrony size as well as spike-count rate were simultaneously optimized. This required changeably balanced log-normal distributions of synaptic weights between excitatory and inhibitory neurons and appropriately amplified synaptic background activity. Our results suggested that the inhibitory neurons with a hub-like structure driven by intensive feedback from excitatory neurons were a key factor in the simultaneous optimization of the spike-count rate and synchrony size, regardless of different spiking types between excitatory and inhibitory neurons.     

A biochemically semi-detailed model of auxin-mediated vein formation in plant leaves

We present here a model intended to capture the biochemistry of vein formation in plant leaves. The model consists of three modules. Two of these modules, those describing auxin signaling and transport in plant cells, are biochemically detailed. We couple these modules to a simple model for PIN (auxin efflux carrier) protein localization based on an extracellular auxin sensor. We study the single-cell responses of this combined model in order to verify proper functioning of the modeled biochemical network. We then assemble a multicellular model from the single-cell building blocks. We find that the model can, under some conditions, generate files of polarized cells, but not true veins.