A stochastic SIRI epidemic model with relapse and media coverage

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Faculty of Sciences (Ibn Tofail University

Modelling the effect of mass media on influenza transmission and vaccine uptake

Influenza causes annual epidemics and occasional pandemics that have claimed millions of lives throughout history. Media reports affect social behaviour during epidemics and pandemics. Changes in social behaviour, in turn, effect key epidemic measurements such as peak magnitude, time to peak, and the beginning and end of an epidemic. The extent of this effect has not been realized. Mathematical models can be employed to study the effects of mass media. In this work, previous mathematical models concerning epidemics and mass media are studied. A novel inclusion of mass media is developed through the addition of a mass media compartment in a Susceptible-Exposed-Infected-Recovered (SEIR) model to look at the effect of mass media on an epidemic. Multiple levels of social distancing are considered in the framework of an ODE model. Vaccination is included in various models for susceptible individuals. Systems of stochastic differential equation models for each of the different scenarios have been derived. An Agent-Based Monte Carlo (ABMC) simulation is used to determine the variability in these key epidemic measurements, so as to provide some insight in to the effects of mass media on epidemic data. Data can help to provide an epidemic outcome that is seen at the population level. Data is used in order to inform parameter values and the novel inclusion of media. A look to future work is also included

MODEL EPIDEMI DISCRETE-TIME MARKOV CHAINS SUSCEPTIBLE EXPOSED INFECTED SUSCEPTIBLE (DTMC SEIS) PENYAKIT TUBERKULOSIS PADA DUA DAERAH

Model epidemi susceptible-exposed-infected-susceptible (SEIS) merupakan pengembangan terhadap model epidemi susceptible-infected-susceptible (SIS) yang menggambarkan pola penyebaran penyakit dengan individu sembuh dapat terinfeksi kembali. Populasi model epidemi SEIS terbagi dalam tiga kelompok, yaitu susceptible (S), exposed (E), dan infected (I). Model epidemi SEIS yang ditinjau dalam interval waktu diskrit dan mengikuti proses Markov dapat digambarkan dengan model epidemi discrete-time Markov chain (DTMC). Model epidemi DTMC SEIS dapat dikembangkan pada lebih dari satu daerah dikarenakan adanya individu yang berpindah dari daerah satu menuju daerah lain. Tujuan penelitian ini adalah mengonstruksikan dan menyimulasikan model epidemi DTMC SEIS penyakit tuberkulosis pada dua daerah. Penelitian ini menggunakan parameter laju kontak beta_1=beta_2=0.1211, laju infeksi sigma_1=sigma_2=0.9024, laju kesembuhan gamma_1=gamma_2=0.0124, dan laju kematian delta_b=0. Terdapat dua proses pada penelitian ini, yaitu yaitu proses infeksi dan proses dispersal. Berdasarkan simulasi model diperoleh bahwa pada masing-masing daerah banyaknya individu susceptible semakin lama semakin menurun, sedangkan banyaknya individu exposed dan infected semakin lama semakin meningkat

An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

An SIRS epidemic model incorporating media coverage with time delay is proposed. The positivity and boundedness are studied firstly. The locally asymptotical stability of the disease-free equilibrium and endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcate are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number R0<1. However, when R0>1, the stability of the endemic equilibrium will be affected by the time delay; there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay increases through a critical value. Finally, some examples for numerical simulations are also included

Modelling crowding effects in infectious disease transmission

Crowding is synonymous with patchy distributions, where some population units, called patches, contain more individuals than others. Lloyd's mean crowding index is a measure of crowding that has been used in differential equation models in ecology. In this thesis, a new mathematical justification of these models is provided. The models are then adapted for use in infectious disease modelling. Two forms of Lloyd's mean crowding are proposed for use in an infectious disease modelling context - the number of susceptible individuals per infected individual per patch, I*IS, and the number of infected individuals per infected individual per patch, I*. It is shown that the value of I*IS, at the start of an epidemic gives the maximum number of transmission events per patch. Over the course of the epidemic, the value of I* increases towards this limiting value. The ratio of I*IS, and I*, Ï I, is therefore proposed as a measure of how efficiently infections are transmitted. As available transmission events reduce with increasing values of I*, disease becomes easier to eliminate and the coexistence of competing infections is facilitated. In response to these results, a vaccination threshold that accounts for patchy distributions of infected individuals is developed, which results in lower proportions of the population needing to be vaccinated when I* increases in value. Human Papillomavirus, a multi-strain sexually transmitted infection with a patchy distribution, is used to explore the implications of these findings in the real world. It is shown that vaccination targeting one strain can result in increases in infection with another, but that a limited degree of cross protection against the non-target strain can eliminate it, in keeping with the fact that patchy distributions make infections easier to eliminate. Finally, the relationship between patch migration and crowding is shown. Changes in migration can either result in crowds of infected individuals and limited spread of infection, or the uniform spread of infection throughout the population. This final result demonstrates that understanding the movement of individuals is critical to controlling epidemics

Exact solutions and superposition rules for Hamiltonian systems generalizing stochastic SIS epidemic models with variable infection rates

Using the theory of Lie-Hamilton systems, formal generalized stochastic Hamiltonian systems that enlarge a recently proposed stochastic SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamitonian models based on the book and oscillator algebras, denoted respectively by $\mathfrak{b}_2$ and $\mathfrak{h}_4$. The last generalization corresponds to a SIS system possessing the so-called two-photon algebra symmetry $\mathfrak{h}_6$, according to the embedding chain $\mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6$, for which an exact solution cannot generally be found, but a nonlinear superposition rule is explicitly given.Comment: 24 page

Epidemic processes in complex networks

In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.Comment: 62 pages, 15 figures, final versio

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina