•Measles epidemic model is constructed to assess the controlling measles transmission and preventing by vaccination.•Global stability analysis is based on constructing a Lyapunov ...function.•Sensitivity analysis is helpful to design control strategies.•Schedule vaccination is the cause of occurring backward bifurcation which is impact to the strategy of disease control by vaccination.
A deterministic model for measles transmission, which is incorporating logistic growth rate and vaccination, is formulated and rigorously analyzed. The certain epidemiological threshold, known as the basic reproduction number, is derived. The proposed model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number, is less than unity. Further, the proposed model exhibits the phenomenon of backward bifurcation, where stable disease-free equilibrium of the model coexists with a stable endemic equilibrium, whenever the basic reproduction number is less than unity. This study is suggested that decreasing the basic reproduction number is insufficient for disease eradication due to schedule vaccination is the cause of the occurrence of backward bifurcation. Furthermore, the study results are shown that the backward bifurcation in the formulated model is removed if increasing the efficacy of vaccine, coverage of primary vaccination, boosting second dose vaccination and decreasing waning of vaccine. When the basic reproduction number is greater than unity, the models have a unique endemic equilibrium which is globally asymptotically stable. The study results can be helpful in providing the information to public health authorities and policy maker in controlling the spread of measles by vaccination.
•There is a negative correlation between the final number of infections and their rate of growth.•There is both theoretical arguments and experimental evidence for the existence of such a ...correlation.•Flattening the curve causes an increased number of infections.•By mid-May 2020 the diffusion of COVID-19 in 25 countries had followed an S-shaped logistic pattern.•The earlier a country locked down, the smaller the number of its infections was.
A negative correlation between the final ceiling of the logistic curve and its slope, established long time ago via a simulation study, motivated this closer look at flattening the curve of COVID-19. The diffusion of the virus is analyzed with S-shaped logistic-curve fits on the 25 countries most affected in which the curve was more than 95% completed at the time of the writing (mid-May 2020.) A negative correlation observed between the final number of infections and the slope of the logistic curve corroborates the result obtained long time ago via an extensive simulation study. There is both theoretical arguments and experimental evidence for the existence of such correlations. The flattening of the curve results in a retardation of the curve's midpoint, which entails an increase in the final number of infections. It is possible that more lives are lost at the end by this process. Our analysis also permits evaluation of the various governments’ interventions in terms of rapidity of response, efficiency of the actions taken (the amount of flattening achieved), and the number of days by which the curve was delayed. Not surprisingly, early decisive response—such as countrywide lockdown—proves to be the optimum strategy among the countries studied.
Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability ...analysis that had not been discussed in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e. if the basic reproduction number less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.
The number of deaths from car accidents and from the unlawful use of guns can be described by logistic growth curves. The annual rates of both have traced completed logistic trajectories following ...which they have been self-regulated for many decades at what seems to be a homeostatic equilibrium level through legislative actions. Exception constitutes the number of deaths from mass shootings, which has been so far tracing an exponential trajectory. Despite the fact that mass-shooting deaths represent only 0.1 % of all gun deaths today, they are poised to continue growing exponentially until they become the major cause of gun deaths, short of unprecedented action by society.
•The number of deaths from car accidents and from the unlawful use of guns can be described by logistic growth curves.•Annual rates of deaths from guns and car accidents (but not from mass shootings) have been self-regulated for many decades.•Mass-shooting deaths represent only 0.1 % of all gun deaths in the US today but they are growing exponentially.•Mass-shootings will continue growing exponentially short of unprecedented action by society.
From the beginning of the usage of radiotherapy (RT) for cancer treatment, mathematical modeling has been integral to understanding radiobiology and for designing treatment approaches and schedules. ...There has been extensive modeling of response to RT with the inclusion of various degrees of biological complexity. In this study, we compare three models of tumor volume dynamics: (1) exponential growth with RT directly reducing tumor volume, (2) logistic growth with direct tumor volume reduction, and (3) logistic growth with RT reducing the tumor carrying capacity with the objective of understanding the implications of model selection and informing the process of model calibration and parameterization. For all three models, we: examined the rates of change in tumor volume during and RT treatment course; performed parameter sensitivity and identifiability analyses; and investigated the impact of the parameter sensitivity on the tumor volume trajectories. In examining the tumor volume dynamics trends, we coined a new metric - the point of maximum reduction of tumor volume (MRV) - to quantify the magnitude and timing of the expected largest impact of RT during a treatment course. We found distinct timing differences in MRV, dependent on model selection. The parameter identifiability and sensitivity analyses revealed the interdependence of the different model parameters and that it is only possible to independently identify tumor growth and radiation response parameters if the underlying tumor growth rate is sufficiently large. Ultimately, the results of these analyses help us to better understand the implications of model selection while simultaneously generating falsifiable hypotheses about MRV timing that can be tested on longitudinal measurements of tumor volume from pre-clinical or clinical data with high acquisition frequency. Although, our study only compares three particular models, the results demonstrate that caution is necessary in selecting models of response to RT, given the artifacts imposed by each model.
Invasive plant species pose a significant threat to biodiversity and the economy, yet their management is often resource-intensive and expensive, and further research is required to make control ...measures more efficient. Evidence suggests that roads can have an important effect on the spread of invasive plant species, although little is known about the underlying mechanisms at play. We have developed a novel mathematical model to analyse the impact of roads on the propagation of invasive plants. The integro-difference equation model is formulated for stage-structured population and incorporates a road sub-domain in the spatial domain. The results of our study reveal, that, depending on the definition of the growth function in the model, there are three distinct types of behaviour in front of the road. Roads can act as barriers to invasion, lead to a formation of a beachhead in front of the road, or act as corridors allowing the invasive species to invade the domain in front of the road. Analytical and computational findings on how roads can impact the spread of invasive species show that a small change in conditions of the environment favouring the invasive species can change the case for the road, allowing the invasive species to invade the domain in front of the road where it previously could not spread.
We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different ...end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L2 and L∞ convergence rates are obtained. We first convert the model into a 2×2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.
•New stochastic model of heterogeneous cell migration and proliferation.•Continuum limit related to a novel generalisation of classical logistic growth.•Parameterise model with experimentally-derived ...heterogeneous proliferation rates.•New quantitative framework to explore the implication of neglecting heterogeneity.•Perturbation solutions provide analytical insight into the role of heterogeneity.
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Cell proliferation is the most important cellular-level mechanism responsible for regulating cell population dynamics in living tissues. Modern experimental procedures show that the proliferation rates of individual cells can vary significantly within the same cell line. However, in the mathematical biology literature, cell proliferation is typically modelled using a classical logistic equation which neglects variations in the proliferation rate. In this work, we consider a discrete mathematical model of cell migration and cell proliferation, modulated by volume exclusion (crowding) effects, with variable rates of proliferation across the total population. We refer to this variability as heterogeneity. Constructing the continuum limit of the discrete model leads to a generalisation of the classical logistic growth model. Comparing numerical solutions of the model to averaged data from discrete simulations shows that the new model captures the key features of the discrete process. Applying the extended logistic model to simulate a proliferation assay using rates from recent experimental literature shows that neglecting the role of heterogeneity can, at times, lead to misleading results.
•Growth patterns in a tripartite relay intercrop were analyzed with logistic growth curves.•Relative yields indicate substantial system level yield advantages.•Yield increase per plant in wheat ...occurred only in the outer rows.•Root barriers did not affect growth patterns or yield.•Growth responses were shaped by aboveground size-asymmetric competition.
Intercropping is a promising model for ecological intensification of modern agriculture. Little information is available on how species growth patterns are affected by size-asymmetric above- and belowground competitive interactions, especially in intercrops with more than two species. We studied plant growth and competitive interactions in a novel intercropping system with three species: wheat, watermelon and maize. Wheat and maize are grown sequentially (as a double cropping system) in narrow strips while watermelon is grown between the cereal strips, with partial overlap in growing period with the two cereals. Growth patterns were monitored over two years and described with logistic growth curves. Root barriers were used to study the effect of belowground interactions. Wheat produced 31% greater yield per plant in the intercrop than in the sole crop but 24% lower yield per unit total (inter)crop area. Wheat yield increase per plant was associated with faster growth and substantial overyielding in the outer rows of wheat strips. Watermelon did not competitively affect wheat. Watermelon biomass was substantially reduced at the time of wheat harvest. However, compensatory growth after wheat harvest and greater allocation to fruits resulted in a good yield of intercropped watermelon, 92% of monoculture yields, at final harvest. Intercropped maize produced 32% lower grain yield per plant and per unit area than sole maize, as a consequence of later sowing and a changed plant configuration in the intercrop as compared to the sole crop, and competitive effects of watermelon, as shown by comparison with a skip-row maize system without watermelon. Root barriers did not affect yield of any of the species, indicating that aboveground competitive interactions in this case played a more important role in shaping the observed growth responses than belowground interactions. Plant interactions in this tripartite intercrop system are consistent with the hypothesis of size-asymmetric competition for light.
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