Kermack-McKendrick Model Research Articles

Bogdanov–Takens Bifurcation of Kermack–McKendrick Model with Nonlinear Contact Rates Caused by Multiple Exposures

In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system.

FINAL SIZE RELATIONS FOR SOME COMPARTMENTAL MODELS IN EPIDEMIOLOGY

The summary measurement of an epidemic such as the final size is an important quantity that allows us to approximate the impact of the disease on the affected region. In recent years, final size measurements for vector-transmitted diseases have acquired importance because of the increased concern for mosquito-borne diseases like dengue, Zika, Chikungunya, etc. However, analytical expressions for this estimate in the stage-structured models applicable to vector-borne diseases are less, mostly focused on the classical Kermack–McKendrick model. In this paper, we first calculate the final size expressions for an SIR model with carrier state and a SEIR–SI host–vector model. Then we extend this for the SEIR–SI host–vector model with treatment class in the host, as well as for the model with vertical transmission in the vector population. Finally, we verify our final size expression with some real scenarios.

Fundamental bound on epidemic overshoot in the SIR model.

We derive an exact upper bound on the epidemic overshoot for the Kermack-McKendrick SIR model. This maximal overshoot value of 0.2984 · · · occurs at [Formula: see text]. In considering the utility of the notion of overshoot, a rudimentary analysis of data from the first wave of the COVID-19 pandemic in Manaus, Brazil highlights the public health hazard posed by overshoot for epidemics with R0 near 2. Using the general analysis framework presented within, we then consider more complex SIR models that incorporate vaccination.

An analytical approach to evaluate the impact of age demographics in a pandemic.

The time required to identify and confirm risk factors for new diseases and to design an appropriate treatment strategy is one of the most significant obstacles medical professionals face. Traditionally, this approach entails several clinical studies that may last several years, during which time strict preventative measures must be in place to contain the epidemic and limit the number of fatalities. Analytical tools may be used to direct and accelerate this process. This study introduces a six-state compartmental model to explain and assess the impact of age demographics by designing a dynamic, explainable analytics model of the SARS-CoV-2 coronavirus. An age-stratified mathematical model taking the form of a deterministic system of ordinary differential equations divides the population into different age groups to better understand and assess the impact of age on mortality. It also provides a more accurate and effective interpretation of the disease evolution, specifically in terms of the cumulative numbers of infected cases and deaths. The proposed Kermack-Mckendrick model is incorporated into a non-linear least-squares optimization curve-fitting problem whose optimized parameters are numerically obtained using the Levenberg-Marquard algorithm. The curve-fitting model's efficiency is proved by testing the age-stratified model's performance on three U.S. states: Connecticut, North Dakota, and South Dakota. Our results confirm that splitting the population into different age groups leads to better fitting and forecasting results overall as compared to those achieved by the traditional method, i.e., without age groups. By using comprehensive models that account for age, gender, and ethnicity, regional public health authorities may be able to avoid future epidemics from inflicting more fatalities and establish a public health policy that reduces the burden on the elderly population.

Modeling the influence of the information domain on countermeasure effectiveness in case of COVID-19

A common way to model an epidemic — restricted to contagion aspects only — is a modification of the Kermack-McKendrick SIR Epidemic model (SIR model) with differential equations. (Mis-)Information about epidemics may influence the behavior of the people and thus the course of epidemics as well. We have thus coupled an extended SIR model of the COVID-19 pandemic with a compartment model of the (mis-)information-based attitude of the population towards epidemic countermeasures. The resulting combined model is checked concerning basic plausibility properties like positivity and boundedness. It is calibrated using COVID-19 data from RKI and attitude data provided by the COVID-19 Snapshot Monitoring (COSMO) study. The values of parameters without corresponding observation data have been determined using an L2 -fit under mild additional assumptions. The predictions of the calibrated model are essentially in accordance with observations. An uncertainty analysis of the model shows, that our results are in principle stable under measurement errors. We also assessed the scale, at which specific parameters can influence the evolution of epidemics. Another result of the paper is that in a multi-domain epidemic model, the notion of controlled reproduction number has to be redefined when being used as an indicator of the future evolution of epidemics.

A Kermack–McKendrick model with age of infection starting from a single or multiple cohorts of infected patients

The infectiousness of infected individuals is known to depend on the time since the individual was infected, called the age of infection. Here, we study the parameter identifiability of the Kermack–McKendrick model with age of infection which takes into account this dependency. By considering a single cohort of individuals, we show that the daily reproduction number can be obtained by solving a Volterra integral equation that depends on the flow of newly infected individuals. We test the consistency of the method by generating data from deterministic and stochastic numerical simulations. Finally, we apply our method to a dataset from SARS-CoV-1 with detailed information on a single cluster of patients. We stress the necessity of taking into account the initial data in the analysis to ensure the identifiability of the problem.

Separable mixing: The general formulation and a particular example focusing on mask efficiency.

The aim of this short note is twofold. First, we formulate the general Kermack-McKendrick epidemic model incorporating static heterogeneity and show how it simplifies to a scalar Renewal Equation (RE) when separable mixing is assumed. A key general feature is that all information about the heterogeneity is encoded in one nonlinear real valued function of a real variable. Next, we specialize the model ingredients so that we can study the efficiency of mask wearing as a non-pharmaceutical intervention to reduce the spread of an infectious disease. Our main result affirms that the best way to protect the population as a whole is to protect yourself. This qualitative insight was recently derived in the context of an SIR network model. Here, we extend the conclusion to proportionate mixing models incorporating a general function describing expected infectiousness as a function of time since infection.

S-I-R Model and COVID-19 Data-Based Numerical Ro Estimation for Pandemic Modeling

A contagious disease transmits from human to human or animal to human. At present world is encountered with such a disease, known as COVID-19. More than half a million people have died due to this pandemic. The pandemic started in China and spread within no time to other parts of the world. Italy and USA are the most unfortunate countries as a large number of deaths occurred in these two countries. No doubt this contagious disease has created social as well as economic problems all over the world, especially in underdeveloped countries. The disease easily transmits to a healthy person during social contact. An epidemic model was developed known as the Kermack-McKendrick model described as SIR (Susceptible Infected and Recovered) model, it deals with the rate of transmission of disease and rate of infection. It gives a trend of infectious disease in a large population. The model helps epidemiologists and health policymakers to understand the probable transmission of disease and to take possible and effective measures to control or reduce the spread of the virus. The factor Ro, known as the reproductive number, can be considered as a threshold value for the disease to be an epidemic. In this study, we used the SIR model to study the effect of COVID-19 in Pakistan. Three coupled differential equations of the SIR model have been solved by numerically using COVID-19 data for Pakistan. The Ro estimated by the current Pakistan COVID-19 data is found to be 2.656 from which control measures will cause a decrease in Ro. Due to the reduction in Ro, the apex of the infected population curve predicted to be range from 26 % to 3 %, and the time to reach the apex ranges from 161 to 710 days. Also, the current data is compared with the numerical values by solving the SIR model. However, the model has limitations due to which parameters can be approximately calculated that might match the actual values to some extent. The application of the model is simple and students can easily learn about the computational techniques used to solve the coupled differential equations.

Asymptotic behavior and semi-analytic solution of a novel compartmental biological model

This study proposes a novel mathematical model of COVID-19 and its qualitative properties. Asymptotic behavior of the proposed model with local and global stability analysis is investigated by considering the Lyapunov function. The mentioned model is globally stable around the disease-endemic equilibrium point conditionally. For a better understanding of the disease propagation with vaccination in the population, we split the population into five compartments: susceptible, exposed, infected, vaccinated, and recovered based on the fundamental Kermack-McKendrick model. He's homotopy perturbation technique is used for the semi-analytical solution of the suggested model. For the sake of justification, we present the numerical simulation with graphical results.

Critical periodic traveling waves for a Kermack-McKendrick epidemic model with diffusion and seasonality

This paper is concerned with the existence and exponential decay of traveling waves with the critical speed for a periodic and diffusive Kermack-McKendrick epidemic model. By a delicate analysis of periodic traveling waves with super-critical speeds and strong maximum principle of parabolic equations, we obtain the existence of positive periodic traveling waves with the critical speed. Thus, the open problem [29] is solved completely. In addition, this approach is also valid for the model in [30] without any limitation condition. Besides the existence, the main part of the paper is devoted to the exponential decay of the infected component when the critical periodic traveling waves approach the initial disease-free steady state.

A mathematical model of an epidemic with an arbitrary law of recovery

The paper proposes a mathematical model of the epidemic process, taking into account the dependence of the rates of cure and loss of immunity on time. Today, mathematical models of epidemics based on the basic Kermack–McKendrick model have become widespread. The most famous models are Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR). These models are based on dividing the population into separate groups that are in different epidemic conditions. The description of the models is based on differential equations similar to the equations of birth and death in the process of radioactive transformations of elements in a radioactive chain. However, this approach does not take into account the dependence of the probabilities of the transition of the population from group to group on time spent in the treatment process and in the process of loss of acquired immunity. The known models do analyze the nature of the course of the epidemic for long periods of time, when the process can enter a stationary state. The paper proposes a mathematical model based on dividing the population into separate groups. The first group consists of healthy people susceptible to infection due to contact with members of the second group, which includes the infected population. Members of the third group are being treated, the fourth group includes members of the society who have recovered with antibodies and are vaccinated. The fifth group consists of deceased members of society. In contrast to the SIR and SEIR models, the proposed approach takes into account that immunity is lost over time, and people who survived again move to the group susceptible to infection. The dependences of the probabilities of transition from group to group on time spent both during the treatment process and in the loss of acquired immunity have been taken into account. Thus, the proposed mathematical model is based on five integro-differential equations, two of which are partial differential equations. A new mathematical model has been formulated that makes it possible to take into account the dependence of the cure rate and the probability of transition from the vaccinated state to the initial state on time spent in the corresponding state. It is shown that the proposed model is autocatalytic. With increasing time, a state of bistability is observed, when, under the same boundary conditions, two stable states coexist. Switching between states is determined by the epidemic spread control parameter found in the work. One of the stable states is stationary and leads to the end of the epidemic, the second one leads to the population’s extinction. It has been shown that, for the stationary regime, the form of the distribution function in terms of treatment time and exposure time in the vaccinated state does not affect the final result in any way. The conditions for suppressing the epidemic for managing the process of its development are formulated. Numerical experiments were carried out to simulate the epidemic spreading process, taking into account the constancy of all transition probabilities. Integration of the original system of equations was carried out using the Radaus algorithm for stiff differential equations. The results of numerical simulations have confirmed that the experimental data agree with the theory of the control parameter. The results of the work can be used to organize the management of the epidemic spreading process in order to suppress it as soon as possible by changing the value of the control parameter.

Mathematical modeling of the effect of quarantine rate on controlling the infection of COVID19 in the population of Saudi Arabia

With the development of communications and transportation worldwide, the challenge of controlling epidemiological diseases becomes higher. The COVID19 has put all nations in a lethal confront with a severe disease that needed serious and painful actions. The sooner the actions, the less destructive the impact. In this paper, we incorporate what we believe is crucial but applicable to control the spread of COVID19 in the populations, that is, quarantine. We keep the model as simple as SI Kermack-McKendrick model with an additional compartment of quarantined patients. We established the system’s basic properties and studied the stability of the disease-free equilibrium and its relation to the basic reproduction number R0 in which we calculated its formula. The focus of our study is to measure the effect of quarantine rate on controlling the spread of COVID19. We use the data collected from the Ministry of Health in Saudi Arabia. We studied three different values of the quarantine rate where newly infectious patients are detected and isolated within 14, 7, and 5 days. The simulations show a significant effect of the quarantine where COVID19 can be fully controlled if the newly infected patient enters the quarantine within five days. These results were proposed to the Public Health Authority in Saudi Arabia and approved by the Ministry of Health in which they applied promptly.

The Kermack-McKendrick epidemic model with variable infectivity modified

We propose a deterministic mathematical model for an epidemic disease of a variable infectivity transmission in a population in which some infective individuals recover from the disease with immunity and the others without it. The demographic changes such as migration, births, and deaths due to natural causes and the chronological age of individuals are disregarded. The model is based on a coupled system of integro-ordinary and partial differential equations subject to conditions of the integral type. We study two forms of the infection rate and prove the existence and uniqueness of the global solution of the model in both cases of the infection rate form. We also study the long-time behaviour of the solution.

A computational propagation model for malware based on the SIR classic model

The main goal of this work is to reformulate the compartmental and deterministic global SIR Kermack-McKendrick model in terms of stochastic and individual-based techniques. Specifically, the novel model proposed is based on the use of a probabilistic cellular automaton. Specific local transition functionsendowed with appropriate epidemiological coefficients are considered with the aim to replicate the simulation results obtained from the global and continuous approach. Moreover, this new model exhibits important improvements with respect to the original Kermack-McKendrick model: different contact topologies can be considered (not only complete networks but also small-world networks and scale-free networks) and also specific and differentiating characteristics of the devices (resistance to infection, number of adjacent infectious nodes, detection and removal coefficients, etc.) and the specimen of malware (virulence) are taken into account. A comparison between both models is introduced by showing that scale-free networks accelerate the propagation process.

The discrete-time Kermack-McKendrick model: A versatile and computationally attractive framework for modeling epidemics.

The COVID-19 pandemic has led to numerous mathematical models for the spread of infection, the majority of which are large compartmental models that implicitly constrain the generation-time distribution. On the other hand, the continuous-time Kermack-McKendrick epidemic model of 1927 (KM27) allows an arbitrary generation-time distribution, but it suffers from the drawback that its numerical implementation is rather cumbersome. Here, we introduce a discrete-time version of KM27 that is as general and flexible, and yet is very easy to implement computationally. Thus, it promises to become a very powerful tool for exploring control scenarios for specific infectious diseases such as COVID-19. To demonstrate this potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and the same initial growth rate, compartmental models systematically predict lower peak sizes than models in which the latent and the infectious period have fixed duration.

Modeling the COVID-19 pandemic: a primer and overview of mathematical epidemiology

Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have.

Mandatory Social Distancing for Covid-19 Outbreak and Its Economic Consequences in West Sumatera - Indonesia

The Governor of West Sumatra Province of Indonesia imposed a mandatory social distancing (PSBB) to slow down the Covid-19 outbreak from 22 April to 7 June 2020. The cons argued that PSBB was ineffective because of the impossibility to limit people’s movements. The pros, on the other hand, viewed PSBB as a training facility to increase people’s awareness about the pandemic. Our research aims at evaluating the effectiveness of PSBB by using the Kermack-McKendrick SIR pandemic model and the NLSUR non-linear least square estimation. Using daily data published by the Covid-19 task force, we found that PSBB had a positive impact on flattening the curve. Assuming that vulnerable people were around 30% of the population, PSBB has reduced the rate of spread of Covid-19 from around 2 persons to less than 1 person for every infected. We also found that the economic consequences of PSBB on commodities varied by their demand and supply characteristics. We, therefore, suggest that policy interventions related to price control and subsidy should consider these characteristics. For future research, as data more available, the effect of PSBB on broader economic variables such as poverty and growth in the province needs to be examined.

Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

Predicting the Size and Duration of the COVID-19 Pandemic

This article explores the ongoing COVID-19 pandemic, asking how long it might last. Focusing on Bahrain, which has a finite population of 1.7M, it aimed to predict the size and duration of the pandemic, which is key information for administering public health policy. We compare the predictions made by numerical solutions of variations of the Kermack-McKendrick SIR epidemic model and Tsallis-Tirnakli model with the curve-fitting solution of the Bass model of product adoption. The results reiterate the complex and difficult nature of estimating parameters, and how this can lead to initial predictions that are far from reality. The Tsallis-Tirnakli and Bass models yield more realistic results using data-driven approaches but greatly differ in their predictions. The study discusses possible sources for predictive inaccuracies, as identified during our predictions for Bahrain, the United States, and the world. We conclude that additional factors such as variations in social network structure, public health policy, and differences in population and population density are major sources of inaccuracies in estimating size and duration.

Evaluación de las medidas de distanciamiento social sobre la transmisibilidad de COVID-19 en el medio rural. Estudio longitudinal retrospectivo de casos posibles

AimTo evaluate the social distance effect on the daily frequency of possible SARS-CoV-2 cases in Primary Care, in relation to the predictive model Kermack-McKendrick. MethodsLongitudinal retrospective study in 2 rural populations of Aragon (13,579h). A time series evaluation with a t-Student analysis was carry on, during the first 70 days of the pandemic. A simple Kermack-McKendrick predictive model was compared with the possible COVID-19 cases. Complementary ANOVA analysis to assess the before-after number of daily cases, follow-up days and days from symptoms onset to first contact with Primary Health Care. ResultsThree hundred and fifty-nine cases were detected (53.4% women; 70.7% under 60). Primary Care followed 95.3% of cases. The number of cases during the first social distancing strategies was higher in comparison with the model (P=.004, P=.006 and P=.004) with a media of decreases of 6.7 possible cases by series. In relation to the lockdown period the model and cases are close (P=.608 and P=.093), with an average decrease of 1.8 cases per series. During post-containment, the number of cases per day (P<.001) and days of follow-up (P<.001) increased. ConclusionsSocial distancing and containment measures were effective in reducing the number of possible COVID-19 cases in rural areas. Primary Care followed most of the cases.