Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment

Vaccination and antiviral treatment are two important prevention and control measures for the spread of influenza. However, the benefit of antiviral use can be compromised if drug-resistant strains arise. In this paper, we develop a mathematical model to explore the impact of vaccination and antiviral treatment on the transmission dynamics of influenza. The model includes both drug-sensitive and resistant strains. Analytical results of the model show that the quantities ℛ _SC and ℛ _RC , which represent the control reproduction numbers of the sensitive and resistant strains, respectively, provide threshold conditions that determine the competitive outcomes of the two strains. These threshold conditions can be used to gain important insights into the effect of vaccination and treatment on the prevention and control of influenza. Numerical simulations are also conducted to confirm and extend the analytic results. The findings imply that higher levels of treatment may lead to an increase of epidemic size, and the extent to which this occurs depends on other factors such as the rates of vaccination and resistance development. This suggests that antiviral treatment should be implemented appropriately.     

Impacts of Incubation Delay on the Dynamics of an Eco-Epidemiological System—A Theoretical Study

Parasite and predator play significant role in trophic interaction, productivity and stability of an ecosystem. In this paper, we have studied a host-parasite-predator interaction that incorporates incubation delay. How the qualitative and quantitative behaviors of the system alter with the incubation delay have been discussed both from mathematical and biological point of views. It is observed that for a lower infection rate, the system is stable for all delays; but for a higher infection rate, there exists a threshold value of the delay above which the system is unstable and below which the system is stable leading to the persistence of all the species. Also, the instability arising from the incubation delay may be controlled if somehow the growth rate of predator population is increased. Numerical studies have also been performed to illustrate different analytical findings. Research is supported by UGC, India; F No. 32-173/2006(SR).     

Conditions for Global Dynamic Stability of a Class of Resource-Bounded Model Ecosystems

This paper studies a class of dynamical systems that model multi-species ecosystems. These systems are ‘resource bounded’ in the sense that species compete to utilize an underlying limiting resource or substrate. This boundedness means that the relevant state space can be reduced to a simplex, with coordinates representing the proportions of substrate utilized by the various species. If the vector field is inward pointing on the boundary of the simplex, the state space is forward invariant under the system flow, a requirement that can be interpreted as the presence of non-zero exogenous recruitment. We consider conditions under which these model systems have a unique interior equilibrium that is globally asymptotically stable. The systems we consider generalize classical multi-species Lotka–Volterra systems, the behaviour of which is characterized by properties of the community (or interaction) matrix. However, the more general systems considered here are not characterized by a single matrix, but rather a family of matrices. We develop a set of ‘explicit conditions’ on the basis of a notion of ‘uniform diagonal dominance’ for such a family of matrices, that allows us to extract a set of sufficient conditions for global asymptotic stability based on properties of a single, derived matrix. Examples of these explicit conditions are discussed.     

Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics

Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein–Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.     

Bistability Analysis of an Apoptosis Model in the Presence of Nitric Oxide

Bistability in apoptosis, or programmed cell death, is crucial for the healthy functioning of multicellular organisms. The aim in this study is to show the presence of bistability in a mitochondria-dependent apoptosis model under nitric oxide effects using chemical reaction network theory. The model equations are a set of coupled ordinary differential equations arising from the assumed mass action kinetics. Whether these equations have a capacity for bistability (cell survival and apoptosis) is determined using a modular approach in which the model is decomposed into modules. Each module contains only a subset of the whole model and is analyzed separately. It is seen that bistability in a module is preserved throughout the whole model after adding the remaining reactions in the pathway on these modules. It is also found that inhibitor effect of some proteins and the appearance of a reacting protein in a later stage as a product is a desired feature but not sufficient for bistability (in the absence of cooperativity effects). On the whole model, two apoptotic and two cell survival states are obtained depending on the initial cell conditions. The results suggest that the antiapoptotic effects of nitric oxide species are responsible for the bistable character of the apoptotic pathway when cooperativity is not assumed in the apoptosome formation.     

Modeling the Dynamics of Viral Evolution Considering Competition Within Individual Hosts and at Population Level: The Effects of Treatment

We consider two viral strains competing against each other within individual hosts (at cellular level) and at population level (for infecting hosts) by studying two cases. In the first case, the strains do not mutate into each other. In this case, we found that each individual in the population can be infected by only one strain and that co-existence in the population is possible only when the strain that has the greater basic intracellular reproduction number, R _0c , has the smaller population number R _0p . Treatment against the one strain shifts the population equilibrium toward the other strain in a complicated way (see Appendix B). In the second case, we assume that the strain that has the greater intracellular number R _0c can mutate into the other strain. In this case, individual hosts can be simultaneously infected by both strains (co-existence within the host). Treatment shifts the prevalence of the two strains within the hosts, depending on the mortality induced by the treatment, which is, in turn, dependent upon the doses given to each individual. The relative proportions of the strains at the population level, under treatment, depend both on the relative proportions within the hosts (which is determined by the dosage of treatment) and on the number of individuals treated per unit time, that is, the rate of treatment. Implications for cases of real diseases are briefly discussed.     

Stability and Bifurcations in an Epidemic Model with Varying Immunity Period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.     

Phenologically-Structured Predator-Prey Dynamics with Temperature Dependence

Studies document the fact that temperature changes strongly affect interactions in many consumer-resource systems through altered, or shifted, phenologies. The mistiming of events, such as migration or emergence times, or the contraction or expansion of development times can upset the normal synchronization and lead to increased or decreased predation events. In this paper, we formulate a continuous time, phenologically-structured model of predator-prey interactions that is driven by temperature variations. It is particularly applicable to arthropod interactions because their development rates are so strongly temperature related. The model takes the form of a system of partial differential-integral equations for the species’ population densities in development-time variables. In special cases, the model is analytically tractable and we find a closed-form solution. By calculating density variations under different temperature regimes, the model gives a quantitative method for assessing the effects of global temperature change on consumer-resource interactions.     

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

Dynamics of Circulating γδ T Cell Activity in an Immunocompetent Mouse Model of High-Grade Glioma

Human γδ T cells are potent effectors against glioma cell lines in vitro and in human/mouse xenograft models of glioblastoma, however, this effect has not been investigated in an immunocompetent mouse model. In this report, we established GL261 intracranial gliomas in syngeneic WT C57BL/6 mice and measured circulating γδ T cell count, phenotype, Vγ/Vδ repertoire, tumor histopathology, NKG2D ligands expression, and T cell invasion at day 10–12 post-injection and at end stage. Circulating γδ T cells transiently increased and upregulated Annexin V expression at post-tumor day 10–12 followed by a dramatic decline in γδ T cell count at end stage. T cell receptor repertoire showed no changes in Vγ1, Vγ4, Vγ7 or Vδ1 subsets from controls at post-tumor day 10–12 or at end stage except for an end-stage increase in the Vδ4 population. Approximately 12% of γδ T cells produced IFN-γ. IL-17 and IL-4 producing γδ T cells were not detected. Tumor progression was the same in TCRδ^-/- C57BL/6 mice as that observed in WT mice, suggesting that γδ T cells exerted neither a regulatory nor a sustainable cytotoxic effect on the tumor. WT mice that received an intracranial injection of γδ T cells 15m following tumor placement showed evidence of local tumor growth inhibition but this was insufficient to confer a survival advantage over untreated controls. Taken together, our findings suggest that an early nonspecific proliferation of γδ T cells followed by their depletion occurs in mice implanted with syngeneic GL261 gliomas. The mechanism by which γδ T cell expansion occurs remains a subject for further investigation of the mechanisms responsible for this immune response in the setting of high-grade glioma.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.     

Interaction of Tumor with Its Micro-environment: A Mathematical Model

This paper is concerned with early development of transformed epithelial cells (TECs) in the presence of fibroblasts in the tumor micro-environment. These two types of cells interact by means of cytokines such as transforming growth factor (TGF-β) and epidermal growth factor (EGF) secreted, respectively, by the TECs and the fibroblasts. As this interaction proceeds, TGF-β induces fibroblasts to differentiate into myofibroblasts which secrete EGF at a larger rate than fibroblasts. We monitor the entire process in silico, in a setup which mimics experiments in a Tumor Chamber Invasion Assay, where a semi-permeable membrane coated by extracellular matrix (ECM) is placed between two chambers, one containing TECs and another containing fibroblasts. We develop a mathematical model, based on a system of PDEs, that includes the interaction between TECs, fibroblasts, myofibroblasts, TGF-β, and EGF, and we show how model parameters affect tumor progression. The model is used to generate several hypotheses on how to slow tumor growth and invasion. In an Appendix, it is proved that the mathematical model has a unique global in-time solution.     

Vegetation Dynamics on an Abandoned Vacuum-Mined Peatland: 5 Years of Monitoring

We studied from 1998 to 2002 the fine-scale vegetation dynamics of a poorly regenerated vacuum-mined bog located in southern Quebec. We selected mined sites that have been abandoned for 14 years and monitored the vascular and non-vascular plants, and some hydrological characteristics. We focussed our study on the monitoring of cotton-grass (Eriophorum vaginatum L.) tussocks. Major changes in the plant cover were observed during the five-year period, such as a decrease (26–31%) in the number of cotton-grass tussocks and an increase in the ericaceous shrub cover. The water table level (lower than 40 cm below the soil surface) and frost heaving appear to be the main factors explaining the failure of cotton-grass and of other typical bog plant species to colonize abandoned mined surfaces. The ericaceous shrub cover, although increasing, was still sparse even after two decades of abandonment, and it may take several additional decades before a complete shrub cover establishes itself. While the natural revegetation process of this vacuum-mined bog is still dynamic after two decades of abandonment, there is no evidence to suggest that vegetation assemblages will eventually resemble those of undisturbed peatlands.     

Visualizing Dynamics of Cell Signaling In Vivo with a Phase Separation-Based Kinase Reporter

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A Mathematical Model of Schistosoma mansoni in Biomphalaria glabrata with Control Strategies

We describe and analyze a mathematical model for schistosomiasis in which infected snails are distinguished from susceptible through increased mortality and no reproduction. We based the model on the same derivation as Anderson and May (J. Anim. Ecol. 47:219–247, 1978), Feng and Milner (A New Mathematical Model of Schistosomiasis, Mathematical Models in Medical and Health Science, Nashville, TN, 1997. Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, pp. 117–128, 1998), and May and Anderson (J. Anim. Ecol. 47:249–267, 1978), but used logistic growth both in human and snail hosts. We introduce a parameter r, the effective coverage of medical treatment/prevention to control the infection. We determine a reproductive number for the disease directly related to its persistence and extinction. Finally, we obtain a critical value for r that indicates the minimum treatment effort needed in order to clear out the disease from the population.