Narrow Results

The aim of this work is to develop a model of alcohol that takes drug features into account. We assess the model feasibility and explain its formulation in terms of a nonlinear differential equation. Using the subsequent matrix generation technique, we ascertain the reproductive number in order to assess the dynamics of the model. We also examine the system equilibrium points, namely the positive and free alcohol equilibrium points. To gain insights into the stability properties of the model, we utilize the Lyapunov function and the Routh–Hurwitz criterion. Through these methods, we investigate both the local stability and global stability of the considered model. Furthermore, we employ numerical simulations to complement and illustrate the theoretical results obtained. These simulations provide visual representations that enhance the understanding of the model dynamics and behavior.

Calculation of the final infection size has become a topic of significant interest in recent years. Despite considerable progress, determining the final infection size in a heterogeneous infectious disease model with nonlinear incidence rate on short-time scales remains a challenging problem. In this paper, we investigate a heterogeneous $\mathrm{\text{SEIR}}$ epidemic model with nonlinear incidence rate. We establish both the existence and uniqueness of the solution regarding final size, and based on which, we are able to introduce a computational algorithm to calculate the final infection size. Furthermore, we apply our findings to study the early phase of the COVID-19 endemic in New York County and present a numerical simulation to illustrate the practical implications of our approach.

We propose new mathematical models based on ordinary differential equations to investigate the transmission dynamics of infectious diseases by adding a hospitalization compartment $H$ into the classic $SIS$ and $SIR$ models. The models incorporate a general incidence rate between susceptibles and infected individuals and a nonlinear recovery rate for hospitalized individuals. We rigorously analyze the existence, local stabilities, and global stabilities of equilibria and derive threshold conditions determining whether the disease dies out or persists. As an application of our $SIH$ model, we perform a case study with an endemic of COVID-19, conduct data fitting and sensitivity analysis for model parameters, and present simulation results to emphasize the role of the hospitalization rate.

Some modified versions of susceptible-infected-recovered-susceptible (SIRS) model are defined on small-world networks. Latency, incubation and variable susceptibility are separately included. Phase transitions in these models are studied. Then inhomogeneous models are introduced. In some cases, the application of the models to small-world networks is shown to increase the epidemic region.

Let y(x) be a smooth sigmoidal curve, y⁽ⁿ⁾ be its nth derivative and {x_m,i} and {x_a,i}, i = 1,2,…, be the set of points where respectively the derivatives of odd and even order reach their extreme values. We argue that if the sigmoidal curve y(x) represents a phase transition, then the sequences {x_m,i} and {x_a,i} are both convergent and they have a common limit x_c that we characterize as the critical point of the phase transition. In this study, we examine the logistic growth curve and the Susceptible-Infected-Removed (SIR) epidemic model as typical examples of symmetrical and asymmetrical transition curves. Numerical computations indicate that the critical point of the logistic growth curve that is symmetrical about the point (x₀, y₀) is always the point (x₀, y₀) but the critical point of the asymmetrical SIR model depends on the system parameters. We use the description of the sol–gel phase transition of polyacrylamide-sodium alginate (SA) composite (with low SA concentrations) in terms of the SIR epidemic model, to compare the location of the critical point as described above with the "gel point" determined by independent experiments. We show that the critical point t_c is located in between the zero of the third derivative t_a and the inflection point t_m of the transition curve and as the strength of activation (measured by the parameter k/η of the SIR model) increases, the phase transition occurs earlier in time and the critical point, t_c, moves toward t_a.

In this work, we study a simple mathematical model to analyze the emergence and control of radicalization phenomena, motivated by the recent far-right extremist events in Brazil, occurred in 8 January 2023. For this purpose, we considered a compartmental SIS-like model that takes into account only the right electors, for simplicity. The model considers radical and moderated right electors, and the transitions between the two compartments are ruled by probabilities, taking into account pairwise social interactions and the important influence of social media through the dissemination of fake news. The role of the Brazilian Federal Supreme Court on the control of such violent activities is also considered in a simple way. The analytical and numerical results show that the influence of social media is essential for the spreading and prevalence of radicalism in the population. In the presence of such social media, we show that radicalism can be controlled, but not extincted, by an external influence, that models the acting of the Federal Supreme Court over the violent activities of radicals. If the social media effect is absent, the radicalism can disappear of the population, and this phenomenon is associated with an active-absorbing nonequilibrium phase transition, like the one that occurs in the standard SIS model.

In this work we propose a simple model for the emergence of drug dealers. For this purpose, we built a compartmental model considering four subpopulations, namely susceptibles, passive supporters, drug dealers and arrested drug dealers. The target is to study the influence of the passive supporters on the long-time prevalence of drug dealers. Passive supporters are people who are passively consenting to the drug trafficking cause. First we consider the model on a fully connected network, in such a way that we can write a rate equation for each subpopulation. Our analytical and numerical results show that the emergence of drug dealers is a consequence of the rapid increase in the number of passive supporters. Such increase is associated with a nonequilibrium active-absorbing phase transition. After that, we consider the model on a two-dimensional square lattice, in order to compare the results in the presence of a simple social network with the previous results. The Monte Carlo simulation results suggest a similar behavior in comparison with the fully connected network case, but the location of the critical point of the transition is distinct, due to the neighbors’ correlations introduced by the presence of the lattice.

Lattice models describing the spatial spread of rabies among foxes are studied. In these models, the fox population is divided into three-species: susceptible (S), infected or incubating (I), and infectious or rabid (R). They are based on the fact that susceptible and incubating foxes are territorial while rabid foxes have lost their sense of direction and move erratically. Two different models are investigated: a one-dimensional coupled-map lattice model, and a two-dimensional automata network model. Both models take into account the short-range character of the infection process and the diffusive motion of rabid foxes. Numerical simulations show how the spatial distribution of rabies, and the speed of propagation of the epizootic front depend upon the carrying capacity of the environment and diffusion of rabid foxes out of their territory.

In this paper, we present two epidemic models with a nonlinear incidence and transfer from infectious to recovery. For epidemic models, the basic reproductive number is calculated. A dynamic system based on threshold, using LaSalle’s invariance principle and Lyapunov function, is structured completely by the basic reproductive number. By studying the SIR and SIRS models under the nonlinear condition, the general validity of the method is verified.

In this study, we focus on exploring the propagation characteristics of particle swarms in social networks and analyze the diffusion process of viruses among populations based on system dynamics. The article mainly discusses three propagation influence mechanisms, including individual attributes, group attributes, and particle swarm attributes, and delves into the modeling of diffusion processes based on network structures. Firstly, we adopt the main roads in the transportation network (hub nodes) as the initial network backbone. On this basis, by introducing branch networks with small-world characteristics and scale-free characteristics, we construct a transportation network that integrates multiple properties. Using this network, we conducted a detailed simulation and analysis of the COVID-19 transmission process and compared and verified it with the infection dynamic data of COVID-19 in Shanghai from March to September 2022. The verification results reveal that our proposed model can significantly improve prediction accuracy. Compared with other existing dynamic models, our model demonstrates excellent performance, possessing high practical application value. This study provides robust theoretical support for the propagation characteristics of particle swarms in social networks and lays the foundation for further research and application in related fields.

A mathematical model for a nonsterilizing vaccine is studied. The model considers a vaccination policy represented by the vaccine application rate, waning and an index of reduction of viral load. The model also incorporates the possibility of escape mutants that avoid vaccine action. The main result is that we can show the existence of an endemic equilibrium point when R₀ is less than one. The reason behind it is the existence of escape mutants that promote an increased rate of infection large enough to trigger an increase in the density of infected people even in the subthreshold case.

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.

In this paper, we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non-commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration dynamics. A construction based on the transport formalism of kinetic theory allows to give a clear model interpretation to the interactions between infected and susceptible in compartmental space-dependent models. In addition, in a suitable scaling limit, our approach permits to couple the two populations through a consistent diffusion model acting at the urban scale. A discretization of the system based on finite volumes on unstructured grids, combined with an asymptotic preserving method in time, shows that the model is able to describe correctly the main features of the spatial expansion of an epidemic. An application to the initial spread of COVID-19 is finally presented.

The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources such as the amount of infectious individuals. In this paper, we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale. Through a diffusive rescaling, the model allows us to recover classical diffusion equations related to commuting dynamics. The numerical solution of the resulting multiscale hyperbolic system with uncertainty is then tackled using a stochastic collocation approach in combination with a finite volume Implicit–Explicit (IMEX) method. The ability of the model to correctly describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is confirmed with a study of the outbreak of COVID-19 in Italy and its spread in the Lombardy Region.

In this work, using a detailed dataset furnished by National Health Authorities concerning the Province of Pavia (Lombardy, Italy), we propose to determine the essential features of the ongoing COVID-19 pandemic in terms of contact dynamics. Our contribution is devoted to provide a possible planning of the needs of medical infrastructures in the Pavia Province and to suggest different scenarios about the vaccination campaign which possibly help in reducing the fatalities and/or reducing the number of infected in the population. The proposed research combines a new mathematical description of the spread of an infectious diseases which takes into account both age and average daily social contacts with a detailed analysis of the dataset of all traced infected individuals in the Province of Pavia. These information are used to develop a data-driven model in which calibration and feeding of the model are extensively used. The epidemiological evolution is obtained by relying on an approach based on statistical mechanics. This leads to study the evolution over time of a system of probability distributions characterizing the age and social contacts of the population. One of the main outcomes shows that, as expected, the spread of the disease is closely related to the mean number of contacts of individuals. The model permits to forecast thanks to an uncertainty quantification approach and in the short time horizon, the average number and the confidence bands of expected hospitalized classified by age and to test different options for an effective vaccination campaign with age-decreasing priority.

In this paper, we use modified versions of the SIAR model for epidemics to propose two ways of understanding and quantifying the effect of non-compliance to non-pharmaceutical intervention measures on the spread of an infectious disease. The SIAR model distinguishes between symptomatic infected (I) and asymptomatic infected (A) populations. One modification, which is simpler, assumes a known proportion of the population does not comply with government mandates such as quarantining and social-distancing. In a more sophisticated approach, the modified model treats non-compliant behavior as a social contagion. We theoretically explore different scenarios such as the occurrence of multiple waves of infections. Local and asymptotic analyses for both models are also provided.

The importance of exogenous reinfection versus endogenous reactivation for the resurgence of tuberculosis (TB) has been a matter of ongoing debate. Previous mathematical models of TB give conflicting results on the possibility of multiple stable equilibria in the presence of reinfection, and hence the failure to control the disease even when the basic reproductive number is less than unity. The present study reconsiders the effect of exogenous reinfection, by extending previous studies to incorporate a generalized rate of reinfection as a function of the number of actively infected individuals. A mathematical model is developed to include all possible routes to the development of active TB (progressive primary infection, endogenous reactivation, and exogenous reinfection). The model is qualitatively analyzed to show the existence of multiple equilibria under realistic assumptions and plausible range of parameter values. Two examples, of unbounded and saturated incidence rates of reinfection, are given, and simulation results using estimated parameter values are presented. The results reflect exogenous reinfection as a major cause of TB emergence, especially in high prevalence areas, with important public health implications for controlling its spread.

We consider an SEIR epidemic model for an infectious disease that spreads in the human host population through both horizontal and vertical transmission. A periodically varying contact rate is introduced to simulate recurrent outbreaks. We use the optimal control theory to assess the disease control. Optimal vaccination strategies to minimize both the disease burden and the intervention costs are analyzed. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella under several scenarios, by using epidemiological parameters obtained by real data.

In the beginning of fall semester 2009, over 2,000 students contacted the student health service at Washington State University to report symptoms of influenza. The epidemic in Pullman, WA made national news, and many speculated on the severity and extent of the disease spread. Analysis of data from the influenza A(H1N1)pdm09 epidemic in Pullman, WA offers an opportunity to gain insights into characteristics of this rural campus community outbreak. In this study, an individual-based stochastic epidemic simulation model was used with the data to estimate infection parameters and make projections of the number of symptomatic individuals that would result given a variety of plausible scenarios. The parameters that were estimated include the number of individuals initially infected and the basic reproductive ratio (R₀). The model was then used to predict the magnitude of infection with vaccination, isolation and quarantine. The results show that the best single intervention strategy is vaccination, and the reduction in infection is greatest when vaccination, isolation and quarantine are used simultaneously.

In this paper, we analyze some epidemic models by considering a time-varying transmission rate in complex heterogeneous networks. The transmission rate is assumed to change in time, due to a switching signal, and since the spreading of the disease also depends on connections between individuals, the population is modeled as a heterogeneous network. We establish some stability results related to the behavior of the time-weighted average Basic Reproduction Number (BRN).

Later, a Susceptible–Exposed–Infectious–Recovered (SEIR) model which describes the measles disease is proposed and we show that its dynamics is determined by a threshold value, below which the disease dies out. Moreover, compared with models without the Exposed compartment, we can find weaker stability results. A control strategy with an imperfect vaccine is applied, to determine how it could effect the size of the peak. Due to the periodic behavior of the switching rule, we compare the results with the BRN of the model. Some simulations are given, using a scale-free network, to illustrate the threshold conditions found.