Mathematical Biology

We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright-Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability. We also give asymptotically sharp bounds for the fixation time distribution.

We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright-Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability. We also give asymptotically sharp bounds for the fixation time distribution.

The safety of using meat and bone meal (MBM) in mammal feed was studied in view of BSE, by quantifying the risk of BSE transmission through different infection routes. This risk is embodied in the basic reproduction ratio R 0 of the infection, i.e. the average number of new infections induced by one initial infection. Only when R 0 is below 1, will the disease die out with certainty and the population will become free from BSE. Unfortunately this is a slow process due to the slow progression of the disease. We calculate R 0 explicitly from basic ingredients taking several different transmission routes into account. Several of the basic ingredients are functions of age or of infection-age. We also calculate the exponential growth rate r in terms of the same basic ingredients. Next we quantify the ingredients from available data and compute the effects on R 0 of various scenario's for controlling BSE, with examples for the UK and the Netherlands.

If a system of several populations of microorganisms compete exploitatively for a single nonreproducing limiting nutrient which is introduced into and washed out of the system at a cnnstant rate, then competitive exclusion results, Provided that there are no inhibition effects, the population requiring the lowest "break-even" concentration of nutrient will be the winner. This outcome can be changed if a predator population is introduced into the system. In this paper we explore some of the possiblities of coexistence and competition reversal that may arise.

We study a model of the chemostat with two species competing for two perfectly substitutable resources in the case of linear functional response. Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the coexistence equilibrium. Then, using compound matrix techniques, we provide a global analysis in a subset of parameter space. In particular, we show that each solution converges to an equilibrium, even in the case that the coexistence equilibrium is a saddle. Finally, we provide a bifurcation analysis based on the dilution rate. In this context, we are able to provide a geometric interpretation that gives insight into the role of the other parameters in the bifurcation sequence.

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R 0 , is less than 1, the disease free steady state is stable. When R 0 > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.

The Lagrangian formalism based on the standard Lagrangians, which are characterized by the presence of the kinetic and potential energy-like terms, is established for selected population dynamics models. A general method that allows for constructing such Lagrangians is developed, and its specific applications are presented and discussed. The obtained results are compared with the previously found Lagrangians, whose forms were different as they did not allow for identifying the energy-like terms. It is shown that the derived standard Lagrangians for the population dynamics models can be used to study the oscillatory behavior of the models and the period of their oscillations, which may have ecological and environmental implications. Moreover, other physical and biological insights that can be gained from the constructed standard Lagrangians are also discussed.

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment, focusing on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global in-all-patch extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and its stochastic analog. The deterministic model has either two or four attractors. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either an in-all-patch expansion or an in-all-patch extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the in-all-patch extinction and the basin of attraction of the in-all-patch expansion. Above this threshold, while increasing the intensity of the dispersal, both deterministic and stochastic models predict a synchronization of the patches resulting in either a global expansion or a global extinction: for the deterministic model, two of the four attractors present when the dispersal is weak are lost, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee threshold as the global population size in the deterministic and the stochastic two-patch models evolves as dictated by the single-patch counterparts. For all values of the dispersal parameter, Allee effects promote in-all-patch extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.

Motivated by our study of infiltrating dynamics of immune cells into tumors, we propose a stochastic model in terms of Ito stochastic differential equations to study how two parameters, the chemoattractant production rate and the chemotactic coefficient, influence immune cell migration and how these parameters distinguish two types of gliomas. We conduct a detailed analysis of the stochastic model and its deterministic counterpart. The deterministic model can differentiate two types of gliomas according to the range of the chemoattractant production rate as two equilibrium solutions, while the stochastic model also can differentiate two types of gliomas according to the ranges of the chemoattractant production rate and chemotactic coefficient with thresholds as one non-zero ergodic invariant measure and one weak persistent state when the noise intensities are small. When the noise intensities are large comparing with the chemotactic coefficient, there is only one type of glioma that corresponds to a non-zero ergodic invariant measure. Using our experimental data, numerical simulations are carried out to demonstrate properties of our models, and we give medical interpretations and implications for our analytical results and numerical simulations. This study also confirms some of our results about IDH gliomas.

Diabetes is a public health problem affected pregnant rats associated with
developmental defects of their growing fetuses and histopathological abnormalities of their
body organs. The traditional application of phytotherapy encourages author to develop the
more safety plants which exerts antidiabetic activity and improve the histological structure.
The present study aimed to evaluate the intensity of lesions induced in liver, kidney, heart
and lingual mucosa of 18-day old fetuses of diabetic mother. Also, how can cinnamon-extract
supplementation exert antiapoptic activity and improved the histological picture during in
utero treatment. Twenty pregnant rats were used in the present work. They were categorized
into four groups (n = 5); control, cinnamon extract group, diabetes, diabetes and cinnamon
supplementation. Diabetes was developed by single i.p. administration of streptozotocin (60
mg/kg in citrate buffer pH 4.5 plus 100mg/kg nicotinamide). Cinnamon watery extract (300mg/
kg body weight) was daily orally administrered from 6th day of gestation until 18th day of
gestation. At the end of treatment, the mother was sacrificed, and their fetuses were removed
and liver, kidney, heart and tongue were dissected and preserved in 10% phosphate buffered
formalin pH 7.4. Also, immunohistochemistry of caspase 3 and P53 were carried out. At 18th
day of gestation, maternal blood glucose levels were monitored in the investigated groups. The
present findings revealed that diabetes induced damage of hepatocytes, deformation of renal
tubules and renal corpuscles, fragility of myocardial muscles and damage of epithelium lining
the lingual mucosa and retarded the differentiation of lingual papillae especially fungiform
papillae. Increase average of apoptic cells were detected in the examined tissues of diabetic
mother. Cinnamon-treatment reduced the incidence of apoptosis and improved the histological
picture of liver, kidney, heart and tongue of fetuses maternally diabetic compared to the control.
Image analysis revealed overexpression of immunohistochemical reaction of caspase 3 in liver,
kidney and heart as well as caspase 3 and p53 in heart of fetuses of diabetic mother compared
to those of diabetic mother supplemented cinnamon extract and control. The authors finally
concluded that cinnamon extract showed a hypoglycaemic activity, reduced the streptozotocin
associated diabetes and ameliorated the fetal liver, kidney, heart and tongue histological and
immunohistochemical picture.
Keywords: Cinnamo

A spatially explicit plant-herbivore model composed of planktonic herbivores, algal preys and nutrients was constructed to examine the effects of consumer-driven nutrient recycling (CNR) on the algal species richness with and without spatial structure. The model assumed that either of two essential nutrients (N and P) limited growth of algal populations and that consumer individuals moved randomly in the lattice and grazed all the algal species with the same efficiency. The results showed that when there was no CNR, the number of persistent algal species was affected by neither supply rates of external nutrients nor spatial structure and was consistently low. When consumers recycled nutrients according to their stoichiometry, the algal species richness changed with supply rates of external nutrients depending on spatial structure: the algal species richness decreased with increasing nutrient loadings when there were no spatial structure because CNR increased the probability of stochastic extinction of algal species by amplifying the oscillation of algae-consumer dynamics. However, when spatial structures were created by the migration of consumers, CNR increased the algal species richness in a range of nutrient loadings because spatial variation of grazing pressure functioned to stabilize the algal-consumer dynamics. The present study suggests that through grazing and nutrient recycling, consumer individuals can create ephemeral heterogeneity in growth environments for algal species and that this ephemerality is one of the keys to understanding algal species in nature.

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of phenotypic distribution. The main result of the paper is a theorem on asymptotic stability of trait distribution. This theorem is applied to models with the offspring trait distribution given by additive and multiplicative random perturbations of the parental mean trait.

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses.

We extend the theory of neural fields which has been developed in a deterministic framework by considering the influence spatio-temporal noise. The outstanding problem that we here address is the development of a theory that gives rigorous meaning to stochastic neural field equations, and conditions ensuring that they are well-posed. Previous investigations in the field of computational and mathematical neuroscience have been numerical for the most part. Such questions have been considered for a long time in the theory of stochastic partial differential equations, where at least two different approaches have been developed, each having its advantages and disadvantages. It turns out that both approaches have also been used in computational and mathematical neuroscience, but with much less emphasis on the underlying theory. We present a review of two existing theories and show how they can be used to put the theory of stochastic neural fields on a rigorous footing. We also provide general conditions on the parameters of the stochastic neural field equations under which we guarantee that these equations are well-posed. In so doing we relate each approach to previous work in computational and mathematical neuroscience. We hope this will provide a reference that will pave the way for future studies (both theoretical and applied) of these equations, where basic questions of existence and uniqueness will no longer be a cause for concern.

A deterministic model for controlling the neglected tropical filariasis disease known as elephantiasis, caused by a filarial worm, is developed. The model incorporates drug resistance in human and insecticide-resistant vector populations. An investigation into whether the model is of biological importance reveals that it is positively invariant, mathematically well posed, and tractable for epidemiological studies. The filariasis-free and filariasis-present equilibrium points were obtained. The next-generation matrix technique is used to derive the basic reproduction number R 0 , which is then used to determine the local stability analysis of the model. It is established that the system is locally asymptotically stable when R 0 < 1. The technique by Castillo-Chavez and a Lyapunov function were employed to prove the global stability of the model's fixed points. The results of this analysis of filariasis-free equilibrium show that the system is globally asymptotically stable when R 0 < 1 and unstable when R 0 > 1. Similarly, the filariasis-present equilibrium point is proved to be globally asymptotically stable when R 0 > 1 and unstable otherwise. This indicates that the fight against the spread of the disease is achievable. It is observed that increasing human-infected mosquito contacts or mosquito-infected human contacts raises the value of R 0 , whereas decreasing the progression of micro-filaria into infective larva and killing more mosquitoes will decrease the R 0 value according to the sensitivity analysis of the model. The variable precision arithmetic technique executed in MATLAB R2014a was used to determine the elasticity indices of the parameters of R 0 , which showed that the value of R 0 = 0.94639. Further investigations revealed that ω 2 has a significant influence on the reproduction number, suggesting that treatment of acute infections is crucial in the control of the disease. Pontryagin's Maximum Principle (PMP) is used for optimal control analysis. The numerical result revealed that strategy D is the most effective based on the infection averted ratio (IAR) value.

We analyze the optimal harvesting problem for an ecosystem of species that experience environmental stochasticity. Our work generalizes the current literature significantly by taking into account non-linear interactions between species, state-dependent prices, and species injections. The key generalization is making it possible to not only harvest, but also 'seed' individuals into the ecosystem. This is motivated by how fisheries and certain endangered species are controlled. The harvesting problem becomes finding the optimal harvesting-seeding strategy that maximizes the expected total income from the harvest minus the lost income from the species injections. Our analysis shows that new phenomena emerge due to the possibility of species injections. It is well-known that multidimensional harvesting problems are very hard to tackle. We are able to make progress, by characterizing the value function as a viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equations. Moreover, we provide a verification theorem, which tells us that if a function has certain properties, then it will be the value function. This allows us to show heuristically, as was shown in Lungu and Øksendal (Bernoulli '01), that it is almost surely never optimal to harvest or seed from more than one population at a time. It is usually impossible to find closed-form solutions for the optimal harvesting-seeding strategy. In order to by-pass this obstacle we approximate the continuous-time systems by Markov chains. We show that the optimal harvesting-seeding strategies of the Markov chain approximations converge to the correct optimal harvesting strategy. This is used to provide numerical approximations to the optimal harvesting-seeding strategies and is a first step towards a full understanding of the intricacies of how one should harvest and seed interacting species. In particular, we look at three examples: one species modeled by a Verhulst-Pearl diffusion, two competing species and a two-species predator-prey system.

We propose and analyze a family of epidemiological models that extend the classic Susceptible-Infectious-Recovered/Removed (SIR)-like framework to account for dynamic heterogeneity in infection risk. The family of models takes the form of a system of reaction-diffusion equations given populations structured by heterogeneous susceptibility to infection. These models describe the evolution of population-level macroscopic quantities S, I , R as in the classical case coupled with a microscopic variable f , giving the distribution of individual behavior in terms of exposure to contagion in the population of susceptibles. The reaction terms represent the impact of sculpting the distribution of susceptibles by the infection process. The diffusion and drift terms that appear in a Fokker-Planck type equation represent the impact of behavior change both during and in the absence of an epidemic. We first study the mathematical foundations of this system of reaction-diffusion equations and prove a number of its properties. In particular, we show that the system will converge back to the unique equilibrium distribution after an epidemic outbreak. We then derive a simpler system by seeking self-similar solutions to the reaction-diffusion equations in the case of Gaussian profiles. Notably, these self-similar solutions lead to a system of ordinary differential equations including classic SIR-like compartments and To the memory of Fred Brauer.

This paper presents a seven-dimensional ordinary differential equation of mathematical model of zika virus between humans and mosquitoes population with non-linear forces of infection in form of saturated incidence rate. Vertical transmission is introduced into the model. These incidence rates produce antibodies in response to the presence of parasite-causing zika virus in both human and mosquito populations. The existence of region where the model is epidemiologically feasible is established (invariant set) and the positivity of the models is also established. The basic properties of the model are determined including the reproduction number of both cases, 0 and 0 0 |  q p R respectively .Stability analysis of the disease-free equilibrium is investigated via the threshold parameter (reproduction number 00 | pq R ) obtained using the next generation matrix technique. The special case model results shown that the disease-free equilibrium is locally asymptotical stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Under specific conditions on the model parameters, the global dynamics of the special case model around the equilibra are explored using Lyapunov functions. For a threshold parameter less than unity, the disease-free equilibrium is globally asymptotically stable. While the endemic equilibrium is shows to be globally asymptotically stable at threshold parameter greater than unity. Numerical simulations are carried out to confirm the analytic results and explore the possible behavior of the formulated model. The result shows that, horizontal and vertical transmission contributes a higher percentage of infected individuals in the population than only horizontal transmission.

We introduce a methodology to study the possible matter flows of an ecosystem defined by observational biomass data and realistic biological constraints. The flows belong to a polyhedron in a multi dimensional space making statistical exploration difficult in practice; instead, we propose to solve a convex optimization problem. Five criteria corresponding to ecological network indices have been selected to be used as convex goal functions. Numerical results show that the method is fast and can be used for large systems. Minimum flow solutions are analyzed using flow decomposition in paths and circuits. Their consistency is also tested by introducing a system of differential equations for the biomasses and examining the stability of the biomass fixed point. The method is illustrated and explained throughout the text on an ecosystem toy model. It is also applied to realistic food models.

Research has shown that undergraduate students benefit from seeing examples of mathematics applied to real-world situations. This article describes three different paradigms for how math and discipline partner faculty worked together to create mathematical activities that illustrate applications of the topics being studied in precalculus and calculus. All three examples are discussed within the framework of PDSA cycles to describe the process by which the teams collaborated to plan, enact, study, and refine their lessons. Findings discuss both the difficulties of creating integrated activities (differences in terms and definitions between mathematics and science faculty, different foregrounding of mathematics versus science among faculty), and the value of the resultant lessons, such as increased level of student engagement, higher cognitive demand, and the role that relevant applications can play in piquing student interest in STEM.

The fact that physical phenomena are modelled, mostly, by nonlinear differential equations underlines the importance of having reliable methods to solve them. In this work, we present a comparison of homotopy perturbation method (HPM), nonlinearities distribution homotopy perturbation method (NDHPM), Picard, and Picard-Pade´methods to solve Michaelis-Menten equation. The results show that NDHPM possesses the smallest average absolute relative error 1.51(À2) of all tested methods, in the range of r 2 ½0; 5. Also, we introduce the combination of Picard's iterative method and Pade´approximants as an alternative to reduce complexity of Picard's solutions and increase accuracy.

It is well known that non-linear systems share the universal property of presenting symmetry-breaking bifurcations. Often, these processes lead to spatial periodic patterns that are both complicated and robust. In basic atomic processes, non-linear interactions could be important when forming a crystal or an amorphous solid from the melt. Furthermore, one is often interested in building periodic nano-structures. In this paper I present two different non-linear systems that produce periodic patterns. This theoretical study suggests that new experimental techniques could be devised to build solid state structures with specifically tailored geometrical properties.

The Lagrangian formalism based on the standard Lagrangians, which are characterized by the presence of the kinetic and potential energy-like terms, is established for selected population dynamics models. A general method that allows for constructing such Lagrangians is developed, and its specific applications are presented and discussed. The obtained results are compared with the previously found Lagrangians, whose forms were different as they did not allow for identifying the energy-like terms. It is shown that the derived standard Lagrangians for the population dynamics models can be used to study the oscillatory behavior of the models and the period of their oscillations, which may have ecological and environmental implications. Moreover, other physical and biological insights that can be gained from the constructed standard Lagrangians are also discussed.

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC.

This investigation studies the dynamic behaviour of a population in which there is a geographic variation in fitness of a one-locus, two allele trait. The results derived here complement certain results by W. H. Fleming who described the existence of a stable polymorphism when the ratio of selection intensity to dispersion rate exceeds a critical value. Here an approximation to the polymorphic state is obtained, and the evolution of other initial states to it is studied.

Patterned growth of bacteria created by interactions between the cells and moving gradients of nutrients and chemical buffers is observed frequently in laboratory experiments on agar pour plates. This has been investigated by several microbiologists and mathematicians usually focusing on some hysteretic mechanism, such as dependence of cell uptake kinetics on pH. We show here that a simpler mechanism, one based on cell torpor, can explain patterned growth. In particular, we suppose that the cell population comprises two subpopulationsone actively growing and the other inactive. Cells can switch between the two populations depending on the quality of their environment (nutrient availability, pH, etc.) We formulate here a model of this system, derive and analyze numerical schemes for solving it, and present several computer simulations of the system that illustrate various patterns formed. These compare favorably with observed experiments.

The powerful mathematical tools developed for the study of large scale reaction networks have given rise to applications of this framework beyond the scope of biochemistry. Recently, reaction networks have been suggested as an alternative way to model social phenomena. In this "socio-chemical metaphor" molecular species play the role of agents' decisions and their outcomes, and chemical reactions play the role of interactions among these decisions. From here, it is possible to study the dynamical properties of social systems using standard tools of biochemical modelling.

Even in the absence of sensory stimulation the brain is spontaneously active. This background "noise" seems to be the dominant cause of the notoriously high trial-to-trial variability of neural recordings. Recent experimental observations have extended our knowledge of trial-to-trial variability and spontaneous activity in several directions: 1. Trial-to-trial variability systematically decreases following the onset of a sensory stimulus or the start of a motor act. 2. Spontaneous activity states in sensory cortex outline the region of evoked sensory responses. 3. Across development, spontaneous activity aligns itself with typical evoked activity patterns. 4. The spontaneous brain activity prior to the presentation of an ambiguous stimulus predicts how the stimulus will be interpreted. At present it is unclear how these observations relate to each other and how they arise in cortical circuits. Here we demonstrate that all of these phenomena can be accounted for by a deterministic self-organizing recurrent neural network model (SORN), which learns a predictive model of its sensory environment. The SORN comprises recurrently coupled populations of excitatory and inhibitory threshold units and learns via a combination of spike-timing dependent plasticity (STDP) and homeostatic plasticity mechanisms. Similar to balanced network architectures, units in the network show irregular activity and variable responses to inputs. Additionally, however, the SORN exhibits sequence learning abilities matching recent findings from visual cortex and the network's spontaneous activity reproduces the experimental findings mentioned above. Intriguingly, the network's behaviour is reminiscent of sampling-based probabilistic inference, suggesting that correlates of sampling-based inference can develop from the interaction of STDP and homeostasis in deterministic networks. We conclude that key observations on spontaneous brain activity and the variability of neural responses can be accounted for by a simple deterministic recurrent neural network which learns a predictive model of its sensory environment via a combination of generic neural plasticity mechanisms.

The use of Trojan Y chromosomes has been proposed as a genetic strategy for the eradication of invasive species. The strategy is particularly relevant to invasive fish species that have XY sex determination system and are amenable to sexreversal. In this paper we study the dynamics of an invasive fish population occupying a dendritic domain in which Trojan individuals bearing multiple Y chromosomes have been released as a means of eradication. We demonstrate the existence of a bounded absorbing set that represents extinction of the invasive species irrespective of the dendritic configuration. The method of analysis used to obtain global estimates could be applied to other population problems and other geometries.

This study analyses measles transmission vertically with vaccination failure and delay of vaccination. The effect of infected newborns as a time delay is analysed. Time delay is considered as a loss of maternal immunity amongst newborns. The system of non-linear differential equations for the proposed problem is formulated. The next generation matrix method is used to find the reproduction number, and to obtain the stability of the infection free as well as the endemic equilibrium. Effect of time delay in vaccination has been studied for disease-free equilibrium. The local and global stability of the system is analysed. Sensitivity of the key parameters is measured using numerical simulation and observed that it supports the analytical results.

We consider the problem of a subnetwork of observed nodes embedded into a larger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these hidden states given information about the subnetwork dynamics. The biochemical networks underlying many cellular and metabolic processes are important realizations of such a scenario as typically one is interested in reconstructing the time evolution of unobserved chemical concentrations starting from the experimentally more accessible ones. We present an application to this problem of a novel dynamical mean field approximation, the Extended Plefka Expansion, which is based on a path integral description of the stochastic dynamics. As a paradigmatic model we study the stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings. The resulting joint distribution is known to be Gaussian and this allows us to fully characterize the posterior statistics of the hidden nodes. In particular the equal-time hidden-to-hidden variance-conditioned on observations-gives the expected error at each node when the hidden time courses are predicted based on the observations. We assess the accuracy of the Extended Plefka Expansion in predicting these single node variances as well as error correlations over time, focussing on the role of the system size and the number of observed nodes.

The human anatomy is very complex structure created by God in which various organs and systems play their different roles. These processes are so complex that they cannot be easily studied wholly. Biomathematics is a field of study that helps indevelopment of predictive and analytical models of biological and medical systems. It has been used in many areas, including: Cellular neurobiology, Epidemic modelling and Population genetics. Innovation in Bio-mathematics can be a powerful tool for biological sciences. It can be useful for testing hypotheses, especially when a direct experiment cannot be conducted.

In this paper, we present a mathematical model to analyze the dynamics of leptospirosis and COVID-19 co-infection. The model used actual data, and estimation of the parameters via the MLE method is performed, which includes the rates of disease transmission, progression of the disease, disease-related death, and recovery rates for each disease and their co-infection. Key parameters: β 1 , β 2 , φ 1 , φ 2 , and μ c are used to characterize the dynamics of the co-infection burden of leptospirosis and COVID-19. The results reveal a notable contrast in transmission dynamics and clinical outcomes between the two diseases, with leptospirosis demonstrating a higher transmission rate and increased morbidity. Co-infections showed more severe clinical outcomes, with higher mortality and delayed recovery than single infections. These findings highlight the importance of targeted public health strategies for managing co-infected populations.

Sewall Wright first encountered the complex systems characteristic of gene combinations while a graduate student at Harvard's Bussey Institute from 1912 to 1915. In Mendelian breeding experiments, Wright observed a hierarchical dependence of the organism's phenotype on dynamic networks of genetic interaction and organization. An animal's physical traits, and thus its autonomy from surrounding environmental constraints, depended greatly on how genes behaved in certain combinations. Wright recognized that while genes are the material determinants of the animal phenotype, operating with great regularity, the special nature of genetic systems contributes to the animal phenotype a degree of spontaneity and novelty, creating unpredictable trait variations by virtue of gene interactions. As a result of his experimentation, as well as his keen interest in the philosophical literature of his day, Wright was inspired to see genetic systems as conscious, living organisms in their own right. Moreover, he decided that since genetic systems maintain ordered stability and cause unpredictable novelty in their organic wholes (the animal phenotype), it would be necessary for biologists to integrate techniques for studying causally ordered phenomena (experimental method) and chance phenomena (correlation method). From 1914 to 1921 Wright developed his ''method of path coefficient'' (or ''path analysis''), a new procedure drawing from both laboratory experimentation and statistical correlation in order to analyze the relative influence of specific genetic interactions on phenotype variation. In this paper I aim to show how Wright's philosophy for understanding complex genetic systems (panpsychic organicism) logically motivated his 1914-1921 design of path analysis.

Many urban phenomena exhibit remarkable regularity in the form of nonlinear scaling behaviors, but their implications on a system of networked cities has never been investigated. Such knowledge is crucial for our ability to harness the complexity of urban processes to further sustainability science. In this paper, we develop a dynamical modeling framework that embeds population-resource dynamics-a generalized Lotka-Volterra system with modifications to incorporate the urban scaling behaviors-in complex networks in which cities may be linked to the resources of other cities and people may migrate in pursuit of higher welfare. We find that isolated cities (i.e., no migration) are susceptible to collapse if they do not have access to adequate resources. Links to other cities may help cities that would otherwise collapse due to insufficient resources. The effects of inter-city links, however, can vary due to the interplay between the nonlinear scaling behaviors and network structure. The long-term population level of a city is, in many settings, largely a function of the city's access to resources over which the city has little or no competition. Nonetheless, careful investigation of dynamics is required to gain mechanistic understanding of a particular city-resource network because cities and resources may collapse and the scaling behaviors may influence the effects of inter-city links, thereby distorting what topological metrics really measure.

Io non mi sono mai preso sul serio e quindi non mi avvilisco se le mie teorie sono ridicolizzate. Bevo con moderazione un bicchiere di Nero d’Avola regolarmente coi pasti.
A volte, per festeggiare se qualcuno mette un “mi piace” su FaceBook bevo un grappino.

The strategic placement of beehives on a flower farm plays a crucial role in determining pollination efficiency and bee colonies' productivity to enhance crop yields, support healthy bee populations, and maximize honey production. This investigation aims to develop a mathematical model for hive placement to optimize both ecological and economic outcomes through the perspective of Flower Constancy Theory and Streaker Bees Theory. Using extensive mathematical logic computations, a mathematical model has been created to offer complementary perspectives on how bees interact with their environment and utilize flower clusters, thereby increasing agricultural productivity in the long run.