A discrete epidemic model with stage structure☆

ABSTRACTA discrete SIRS epidemic model with vaccination and a general infection probability function is investigated in this paper. The basic reproduction number of the model is calculated. It is proven that the disease-free equilibrium is locally asymptotically stable if and globally asymptotically stable for some additional conditions, and the endemic equilibrium is locally asymptotically stable if . Numerical simulations are presented to illustrate the stability results, and the sensitivity analysis of the basic reproduction number shows that the proportionality coefficient of unvaccinated susceptible, the probability that infected individuals do not recover, and the probability of survival have much more effect on .

In this paper we consider a discrete epidemic SIR/COVID model.Since models can be used for early warning and to forecast the behavior of an epidemic and develop intervention strategies it is critical to be able to effectively predict transmission and recovery rates.Based on available daily infection and death data from South Carolina for the period December 1, 2020, to June 1, 2021 we develop a discrete model and analyze evolution of the model using optimization, artificial neural network, machine learning and Grey model inferring the daily, transmission, and reproduction rates, and recovery for each day of the period.The models and results are consistent with the observations.The models developed using data help us understand the recovery and transmission rates, hence the evolution of the epidemic.The infection and recovery increasing in South Carolina do not show improvement in the period covered.The number of dead people tends to increase although by small amount.Forecasting data for a short time in the future can be used to judge the possible evolution of the epidemic and intervention.Models were developed based on the available data.For the period December to June there were no available data on recovered populations and we have to determine them as well as transmission and recovery rates based on data of infected populations and dead population using artificial neural networks and optimization methodologies where transmission, recovery, relapsation immunity and death rates from infection are considered as decision variables.From the data from CDC we see that the number of infected population is increasing.We have also data for the number of dead population due to the virus.Our models are consistent with the data we have available for the infected and dead population.However, there were no data for recovered population in South Carolina for the entire period December 1 to June 1.We have to use our model to come up with recovered population number.One thing we observe is that the number of infected population was increasing.One of the control measures that are believed to be reliable methods of curbing the spread of the virus is quarantine.We include a model that includes quarantine in our work.In our quarantine we see that if 100,000 susceptible people in the whole state were quarantined there would have been a considerable decrease in the number of infected population.

STABILITY OR CHAOS IN DISCRETE EPIDEMIC MODELS

Discrete epidemic models are applied to describe the physical phenomena of spreading infectious diseases in a household. In this paper, an attempt has been made to develop a modified epidemic chain model by assuming a beta distribution of third kind for the probability of being infected by contact with a given infective from the same household with closed population. This paper emphasizes mainly on developing the probabilities of all possible epidemic chains with one introductory case for three, four and five member household. The key phenomenon towards developing this paper is to provide an alternative model of chain binomial model.

The epidemic growth model is an important tools used in predicting the future of a population and the spread of disease in the population. An epidemic model is usually formed in a differential equation or a system consisting several differential equations. The biological complexity in the underlying population affects the complexity of the epidemic model. One example of biological complexity is the Allee effect which reflects the critical density dependent of the population growth. In this paper we discuss a Logistic epidemic by considering this Allee effect on the population. Dynamic analysis is performed by determining fixed point and its stability analysis in crisp condition. We found the Basic Reproduction Ratio (BRR) for the model. The properties of the solution of the model are explored by the use of its numerical solution. Since we also consider the fuzziness of parameters and variables in the model, the numerical solution is generated using a modified Runge-Kutta method. This is done to explore the effect of inaccuracy and uncertainty which often occur in epidemiological problems.

To investigate the impacts of demographics on the spread of infectious diseases, a susceptible-infectious-recovered (SIR) pairwise model on heterogeneous networks is established. This model is reduced by using the probability generating function and moment closure approximations. The basic reproduction number of the low-dimensional model is derived to rely on the recruitment and death rate, the first and second moments of newcomers’ degree distribution. Sensitivity analysis for the basic reproduction number is performed, which indicates that a larger variance of newcomers’ degrees can lead to an epidemic outbreak with a smaller transmission rate, and contribute to a slight decrease of the final density of infectious nodes with a larger transmission rate. Besides, stochastic simulations indicate that the low-dimensional model based on the log-normal moment closure assumption can well capture important properties of an epidemic. And the authors discover that a larger recruitment rate can inhibit the spread of disease.

Patterns of contact in social behaviour and seasonality due to environmental influences often affect the spread and persistence of diseases. Models of epidemics with seasonality and patterns in the contact rate include time-periodic coefficients, making the systems nonautonomous. No general method exists for calculating the basic reproduction number, the threshold for disease extinction, in nonautonomous epidemic models. However, for some epidemic models with periodic coefficients and constant population size, the time-averaged basic reproduction number has been shown to be a threshold for disease extinction. We extend these results by showing that the time-averaged basic reproduction number is a threshold for disease extinction when the population demographics are periodic. The results are shown to hold in epidemic models with periodic demographics that include temporary immunity, isolation, and multiple strains.

This paper aims at the study of the delimitation of the sets of feasible trajectories (discrete trajectories having a biological sense) generated by a family of two-dimensional map Trelated to a discrete epidemic model. The inverse of Thas vanishing denominator giving rise to non-classical singularities: a non definition line, a focal point and a prefocal line. In particular, we give an answer to one of the open problems proposed by Kocic and Ladas.

We consider a class of system of nonlinear difference equations arising from mathematical models describing a discrete epidemic model. Sufficient conditions are established that guarantee the existence of positive solutions, the existence of a unique nonnegative equilibrium, and the convergence of the positive solutions to the nonnegative equilibrium of the system of difference equations. The obtained results are new and they complement previously known results.

This chapter examines mathematical models that describe disease spread, beginning with the classical Kermack–McKendrick SIR model, which divides populations into Susceptible, Infected, and Removed compartments. It explains how the basic reproduction number (R0) determines whether an epidemic grows or dies out, and explores strategies to reduce R0, such as vaccination and social distancing. The chapter extends to SIR model variations, including the SEIR and SIS models, which account for incubation periods and diseases without lasting immunity, respectively. It also analyses real-world applications, such as Covid-19 lockdown measures and foot-and-mouth disease control, where mathematical models were used to guide policy decisions. It introduces structured models, where parameters may vary depending on the age of an individual or on their behaviour. Understanding how drugs are absorbed in the body and how they tackle disease is the subject of the area of Pharmacokinetic/Pharmacodynamic (PK/PD) modelling and the key building blocks of this approach are presented. The chapter also introduces within-host models, which aim to capture how a virus grows in the host. Coupling such models with those at the population level can then capture key aspects of the multiscale complexity of disease spread.

The SIR model is one of the epidemic models to describe the spread of infectious diseases with healing and without immunity to these infections. Environmental changes can affect changes in disease patterns that can cause endemic. One of the diseases that cause endemic is Measles (Measles). Therefore, it is necessary to take preventive measures to reduce the rate of spread of the disease, the most effective measure to prevent the spread of the disease is vaccination. Measles transmission prevention events that occur in a population can be modeled in a mathematical form, one of which is the SIR model. The SIR model is divided into four subpopulations, namely the susceptible population or a subpopulation of susceptible individuals to the disease, the infected subpopulation or a subpopulation of infected individuals and can transmit the disease and the recovary subpopulation or individual subpopulation recovering from the disease. Vaccination in this case is the addition of controls to the SIR model, where before being controlled, Measles was only treated normally without vaccines, so that the disease is still common in the community. Giving the right vaccine will reduce the number of infected subpopulations, so that the recovery subpopulation will increase. In this study, the SIR model was developed with the addition of controls. The control in this model is a vaccination given to infected subpopulations, so that the recovery subpopulation has increased, because the number of infected subpopulations has decreased. Abstrak Model SIR merupakan salah satu model epidemik untuk menggambarkan penyebaran penyakit infeksi dengan adanya penyembuhan dan tanpa adanya kekebalan terhadap infeksi tersebut. Perubahan lingkungan hidup dapat mempengaruhi perubahan pola penyakit yang dapat menimbulkan endemik. Salah satu penyakit yang menyebabkan endemi yaitu penyakit Campak (Measles). Oleh karena itu perlu adanya tindakan pencegahan untuk mengurangi laju penyebaran penyakit tersebut, tindakan yang dinilai paling efektif untuk mencegah penyebaran penyakit adalah dengan cara vaksinasi. Kejadian pencegahan penularan penyakit Campak yang terjadi pada suatu populasi dapat dimodelkan ke dalam bentuk matematis, salah satunya adalah model SIR. Model SIR dibagi menjadi empat subpopulasi yaitu populasi susceptible atau subpopulasi individu rentan terhadap penyakit, subpopulasi infected atau subpopulasi individu terinfeksi serta dapat menularkan penyakit dan subpopulasi recovary atau subpopulasi individu sembuh dari penyakit. Vaksinasi dalam hal ini adalah penambahan kontrol pada model SIR, dimana sebelum dikontrol, penyakit Campak hanya diobati biasa tanpa pemberian vaksin, sehingga penyakit tersebut masih banyak dijumpai di masyarakat. Pemberian vaksin yang tepat, akan menurunkan jumlah subpopulasi terinfeksi, sehingga subpopulasi recovery akan mengalami kenaikan. Pada penelitian ini mengembangkan model SIR dengan penambahan kontrol. Kontrol pada model tersebut merupakan vaksinasi yang diberikan kepada subpopulasi infected, sehingga subpopulasi recovery mengalami kenaikan, kerena jumlah subpopulasi infected menurun.

Stability analysis of an HIV/AIDS epidemic model with treatment

An SIRS model is developed to account for precautionary measures and immunity loss. The existence and stability of equilibria are studied. It is shown that precautionary measures can induce Hopf bifurcations leading to the occurrence of cyclical behavior. The model is then extended to a two-patch scenario to explore how disease spread patterns are influenced by dispersal (travel). Theoretical analyses establish the stability of the disease-free equilibrium, the basic reproduction number, and thresholds related to travel rates. The persistence of the system, as well as the existence of boundary and endemic equilibria, are also discussed. Using Hopf bifurcation theory, we further examine the interaction between nonlinear incidence functions, travel rates, and precaution delay effects in shaping the stability of the endemic equilibrium. The findings reveal a strong connection between reduced infection rates due to precaution and the emergence of Hopf bifurcations, emphasizing the importance of timely and accurate disease information in curbing the spread of diseases. Additionally, the study highlights the significant impact of different infection force functions on equilibrium stability, underscoring the critical role of precautionary measures in disease transmission mechanisms. The results also show the diverse effects of travel rates on disease spread, suggesting that restricting travel may not always lead to favorable outcomes. This underscores the necessity for governments to consider multiple factors comprehensively in their efforts to prevent and control diseases.

Due to the negative impact of infectious diseases on population growth, it is important to understand the dynamic behavior of such diseases. Mathematical deterministic SIR models are widely used to study the spread of infectious diseases. In real life, there is a lot of randomness and stochasticity such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. in the considered models. In that case, we must introduce Poisson noises into the population models for describing such discontinuous systems. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous SIR model disturbed by white noise, centered and non-centered Poisson noises. In the deterministic case there is a threshold of the system for an epidemic to occur, so called the basic reproduction number. Depending on the value of the reproduction number there is the disease-free equilibrium, or there is an endemic equilibrium, which implies the disease always remains. In the case of the autonomous stochastic SIR model, we study the asymptotic behavior of the solution to the corresponding system of stochastic differential equations around these points of equilibrium.

This research presents an advanced fractional-order compartmental model designed to delve into the complexities of COVID-19 transmission dynamics, specifically accounting for the influence of environmental pathogens on disease spread. By enhancing the classical compartmental framework, our model distinctively incorporates the effects of order derivatives and environmental shedding mechanisms on the basic reproduction numbers, thus offering a holistic perspective on transmission dynamics. Leveraging fractional calculus, the model adeptly captures the memory effect associated with disease spread, providing an authentic depiction of the virus’s real-world propagation patterns. A thorough mathematical analysis confirming the existence, uniqueness and stability of the model’s solutions emphasizes its robustness. Furthermore, the numerical simulations, meticulously calibrated with real COVID-19 case data, affirm the model’s capacity to emulate observed transmission trends, demonstrating the pivotal role of environmental transmission vectors in shaping public health strategies. The study highlights the critical role of environmental sanitation and targeted interventions in controlling the pandemic’s spread, suggesting new insights for research and policy-making in infectious disease management.