A Heuristic Approach for Determining Efficient Vaccination Plans under a SARS-CoV-2 Epidemic 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.

We present a combined epidemiological and economic model for control of diseases spreading on local and small-world networks. The disease is characterized by a pre-symptomatic infectious stage that makes detection and control of cases more difficult. The effectiveness of local (ring-vaccination or culling) and global control strategies is analysed by comparing the net present values of the combined cost of preventive treatment and illness. The optimal strategy is then selected by minimizing the total cost of the epidemic. We show that three main strategies emerge, with treating a large number of individuals (global strategy, GS), treating a small number of individuals in a well-defined neighbourhood of a detected case (local strategy) and allowing the disease to spread unchecked (null strategy, NS). The choice of the optimal strategy is governed mainly by a relative cost of palliative and preventive treatments. If the disease spreads within the well-defined neighbourhood, the local strategy is optimal unless the cost of a single vaccine is much higher than the cost associated with hospitalization. In the latter case, it is most cost-effective to refrain from prevention. Destruction of local correlations, either by long-range (small-world) links or by inclusion of many initial foci, expands the range of costs for which the NS is most cost-effective. The GS emerges for the case when the cost of prevention is much lower than the cost of treatment and there is a substantial non-local component in the disease spread. We also show that local treatment is only desirable if the disease spreads on a small-world network with sufficiently few long-range links; otherwise it is optimal to treat globally. In the mean-field case, there are only two optimal solutions, to treat all if the cost of the vaccine is low and to treat nobody if it is high. The basic reproduction ratio, R(0), does not depend on the rate of responsive treatment in this case and the disease always invades (but might be stopped afterwards). The details of the local control strategy, and in particular the optimal size of the control neighbourhood, are determined by the epidemiology of the disease. The properties of the pathogen might not be known in advance for emerging diseases, but the broad choice of the strategy can be made based on economic analysis only.

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.

The pharmaceutical interventions of emerging infectious diseases are constrained by the available medical resources such as drugs, vaccines, hospital beds, isolation places and the efficiency of the treatment. The awareness of the population also plays an important role in reducing contacts and consequently, reducing the disease transmission rate. In this paper, we propose a multi-group Susceptible, Infected and Recovered (SIR) epidemic model incorporating the awareness of population and the saturated treatment function that describes the effects of the availability of medical resources for treatment. We assume that the treatment of the infected individuals of a group is affected by the medical resources for the treatment of each group. We calculate the basic reproduction number [Formula: see text] in the term of the awareness parameter using the next generation approach. We determine the local and global stabilities of equilibrium (disease free equilibrium and endemic equilibrium) in terms of [Formula: see text] and the availability of medical resources for treatment. We obtain that backward bifurcation occurs at [Formula: see text] along with the existence of multiple endemic equilibria when [Formula: see text] Further, we consider the special case with a single group epidemic system and ensure the existence of multiple endemic equilibria. We showed a necessary condition on the parameter related to the availability of medical resources when backward bifurcation occurs. This situation indicates that reducing the basic reproduction number below unity is not sufficient to remove the disease when the medical resources for treatment are scarce. We used numerical simulations to support and counterpart our theoretical results and discussed the impacts of the awareness of susceptible population and availability of medical resources for treatment in each group, on the epidemic size of each group. Our findings suggest that in the case of limited medical resources, the high treatment rate and awareness of the population are very helpful to control the disease (to reduce the prevalence of infection) and the eradication of disease also depends on initial population sizes. More importantly, it is also obtained that sufficient medical resources for every group are required to eradicate the disease from an entire population.

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.

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.

Stability analysis of an HIV/AIDS epidemic model with treatment

INTRODUCTION Dengue is a disease caused by any one of five virus serotypes and transmitted to humans by the Aedes aegypti mosquito. Climate change and health conditions have combined to make dengue a global public health problem. The situation is especially serious in Colombia, where by week 36 of 2018, dengue incidence was 96 cases per 100,000 population, with a total of 111 deaths. Different mathematical and statistical models have been proposed to understand the dynamics of transmission and consequently to apply control strategies to reduce the number of dengue cases. OBJECTIVE Forecast the number of dengue cases expected in Colombia from 2018 through 2022 with the stochastic Auto-Regressive Integrated Moving Average (ARIMA) model and use the results to adjust the parameters of an ordinary differential equations model in order to determine the disease's basic reproduction number in the year presenting the highest number of dengue cases. METHODS An ecological time series study was conducted to forecast dengue incidence in Colombia from 2018 through 2022. The data were compiled from Colombia's National Health Institute series on dengue cases reported by epidemiological week from 2009 to 2017. The stochastic ARIMA time series model was applied. Forecasts were then analyzed, and the year with the highest number of predicted cases was used to adjust the parameters of an ordinary differential equations model (ODE) through nonlinear least squares regression to calculate the vectorial capacity of the transmitting mosquito. RESULTS Forecasts of the total number of dengue cases per year in Colombia for the following five years were: 32,411 (2018); 88,221 (2019); 56,392 (2020); 47,940 (2021); and 77,344 (2022). The highest number of cases was forecast for 2019. Values for the parameters affecting dengue transmission that year (by the year's four quarters), such as recovery rate (0.0992, 0.0838, 0.1177, and 0.1535, respectively), vectorial capacity of the transmitting mosquito (0.1720, 0.1705, 0.1204, and 0.2147, respectively) and the basic dengue reproduction number (1.73, 2.03, 1.02, and 1.40, respectively) were estimated, indicating that most cases would occur in the second quarter and, since the basic reproduction number values were >1, the disease would persist in the country throughout the entire year. CONCLUSIONS ARIMA model forecasts for 2018 through 2022 predicted the highest incidence of dengue cases in Colombia would occur in 2019. Comparison of ARIMA model forecasts and the ODE model allowed projections of possible variations in dengue cases reported, and the basic reproduction number predicted that dengue would persist throughout 2019. KEYWORDS Arboviruses, climate, dengue, models, theoretical, basic reproduction number, prognosis, Colombia.

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.

A discrete epidemic model with stage structure☆

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.

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.

Actualmente uno de los activos más valiosos en las redes informáticas son los datos que viajan a través de los diversos nodos que componen las topologías de redes. En la mayoría de los casos, la presencia de grandes volúmenes de datos en las redes, permiten que estos se dañen debido a la alteración de la información por algún tipo de virus, malware o troyano lo que genera en una propagación de una enfermedad en la red. Ante esta situación, planteamos el uso de un modelo clásico de enfermedad infecciosa basado en SEIRS, el cual permite plantear un marco de espacio – tiempo discreto para la expansión de la enfermedad. Se consideran tres tipos de conectividad que se representan a través de una relación de vecindad para la que se utiliza una rejilla de tamaño n x n. Los tipos de vecindario considerados son los de Von Neumann, Moore y L. Específicamente, el uso de reglas locales y el tipo de vecindario definen la dinámica de expansión de la enfermedad infecciosa. Luego presentamos un acercamiento a la evolución de modelos deterministas de ecuaciones diferenciales (ODE), como los modelos SIR y SEIRS para estimar los parámetros de este modelo discreto a partir de datos. Ilustramos el enfoque propuesto para los datos de las redes informáticas a partir de datos paramétricos. Más que eso, se modela con éxito, la Epidemia del SIDA en Cuba (1986-2000) y la Epidemia de la Peste bubónica en India (1908) desde nuestro enfoque. Este trabajo ayuda a graficar equivalencias entre dos modelos conceptualmente diferentes y destaca que dan resultados similares al tomar apropiadamente los valores de los parámetros.

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infected-susceptible) epidemic model with n patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. The basic reproduction number ${\cal R}_0$ for the model is determined. It is then shown that when the disease-free equilibrium is stable (${\cal R}_0 < 1$) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (${\cal R}_0 > 1$) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (${\cal R}_0 < 1$) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (${\cal R}_0 > 1$). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.

A profound understanding of influenza transmission mechanisms is crucial for developing effective control strategies. This study first establishes a deterministic SAIV influenza model incorporating vaccination and asymptomatic carriers. The basic reproduction number $ R_0 $ is derived using the next-generation matrix method, followed by an analysis of equilibrium existence and stability. The existence of a transcritical bifurcation in the deterministic model is rigorously demonstrated through bifurcation theory. Subsequently, a stochastic SAIV model incorporating both white noise and Lévy jumps is formulated based on the deterministic framework. By employing Lyapunov function theory, the existence, uniqueness, and non-negative boundedness of the global positive solution are proven, ensuring biological plausibility. Furthermore, threshold conditions for disease extinction and persistence are analytically derived, providing a theoretical delineation of influenza fade-out or endemic transmission. Finally, model parameters are estimated using empirical influenza data, validating the accuracy of the deterministic model and the reliability of stochastic dynamical predictions. The results indicate that limited medical resources facilitate sustained transmission and reduce eradication probability, while environmental stochasticity further emphasizes the critical role of optimized resource allocation in influenza containment.