Kermack-McKendrick Epidemic Model Research Articles

Pulsating waves in a multidimensional reaction-diffusion system of epidemic type

In this work we investigate the existence of front propagation for a two-component reaction -diffusion system of epidemic type posed in a multi-dimensional periodic medium. This system is a spatially heterogeneous version of the well-known Kermack–McKendrick epidemic model with Fickian diffusion. We derive sufficient conditions for the existence of pulsating travelling waves propagating in an arbitrarily given unit direction. Specifically, we prove that the set of admissible wave speeds contains a semi-infinite interval. Then, for each direction of propagation and each admissible speed, there exists a pulsating travelling wave solution of the system which is globally bounded.

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.

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.

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.

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.

Undetected infectives in the Covid-19 pandemic

ObjectivesEpidemiological investigations and mathematical models have revealed that the rapid diffusion of Covid-19 can mostly be attributed to undetected infective individuals who continue to circulate and spread the disease: finding their number would be of great importance in the control of the epidemic. MethodsThe dynamics of an infection can be described by the SIR model, which divides the population into susceptible (S), infective I, and removed R subjects. In particular, we exploited the Kermack-McKendrick epidemic model, which can be applied when the population is much larger than the fraction of infected subjects. ResultsWe proved that the fraction of undetected infectives, compared to the total number of infected subjects, is given by 1−1R0, where R0 is the basic reproduction number. The mean value R0=2.10 2.09-2.11 for the Covid-19 epidemic in three Italian regions yielded a percentage of undetected infectives of 52.4% (52.2%–52.6%) compared to the total number of infectives. ConclusionsOur results, straightforwardly obtained from the SIR model, highlight the role of undetected carriers in the transmission and spread of the SARS-CoV-2 infection. Such evidence strongly recommends careful monitoring of the infective population and ongoing adjustment of preventive measures for disease control until a vaccine becomes available for most of the population.

Traveling waves in a nonlocal dispersal SIR model with non-monotone incidence

It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder’s fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator.

On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model

A recent paper [Y.-Y. Chen, J.-S. Guo, F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359] presented a discrete diffusive Kermack-McKendrick epidemic model, and the authors proved the existence of traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium. However, the boundary asymptotic behavior of the traveling waves converge to the endemic equilibrium at $ +\infty $ is still an open problem. In this paper, we investigate the above open problem and completely solve it by constructing suitable Lyapunov functional and employing Lebesgue dominated convergence theorem.

Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality

In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number $$R_0$$ of the corresponding periodic ordinary differential model and the minimal wave speed $$c^*$$ , we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with $$R_0$$ and $$c^*$$ .

Low‐density barriers for controlling plant crowd diseases: How far and fast can pathogens spread?

AbstractWe explore numerically the possibility of controlling the spread of plant diseases characterized by relatively low dispersal (crowd diseases) through the introduction of a spatial barrier with low density of susceptible hosts. We use the diffusion approximation to Kendall's spatially extended version of the Kermack–McKendrick epidemic model and illustrate our findings within the context of a representative viral disease that affects cocoa trees.RECOMMENDATIONS FOR MANAGERS: Our numerical results suggest that using low‐density barriers of hosts in crowd plant diseases might be an effective way of halting the spatial dispersal of pathogens. The introduction of these barriers may reduce the economic impact when compared with other methods of controlling the disease spread. Before using the model to approximate suitable sizes of barriers, it is necessary to execute an exhaustive assessment of the model appropriateness for any particular disease under consideration. Our results suggest that to improve the efficiency of low‐density barriers it is important to explore their use in combination of current alternative control methods.

Traveling waves in the Kermack–McKendrick epidemic model with latent period

We study traveling waves for a diffusive susceptible–infected–recovery model, due to Kermack and McKendrick, of an epidemic with standard incidence and latent period included. In contrast to the classical case where the mass action incidence is employed, the total population is varied in the present model. It turns out that the governing equation for the recovery species cannot be decoupled from the other two equations for the susceptible and the infected species, and hence that the present model cannot be reduced to a two-component system as the classical one does. The existence of traveling waves of the model in this study can be completely characterized by the basic reproduction number of the system of ordinary differential equations associated with the present model. The model admits a continuum of traveling waves parameterized by wave speed c when waves do exist. Our approach is based on the fixed point theory and a delicately designed pair of super-/sub-solutions. This set of super-/sub-solutions also allows us to completely answer two unsolved questions in the existing literatures where the latent period is zero: (i) the existence of the minimal-speed wave which is believed to play a key role in the evolution of epidemic diseases and (ii) the existence of traveling waves does not depend on the relative ratio of the diffusivity of the infected species to the one of the recovery species.

Traveling waves in a nonlocal dispersal SIR model with critical wave speed

This paper is concerned with the existence of traveling waves for a nonlocal dispersal Kermack–McKendrick epidemic model. As we know, due to the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity, it is difficult to investigate traveling waves with the critical wave speed. Furthermore, the nonlocal dispersal and bilinear incidence bring additional difficulties to get the boundedness of traveling waves. In the present paper, we overcome these difficulties to obtain the boundedness of traveling waves by analysis technique firstly and then prove the existence of traveling waves under the critical wave speed.

Traveling waves of a nonlocal dispersal Kermack–McKendrick epidemic model with delayed transmission

In this paper, we obtain the information about the existence and nonexistence of traveling waves for the nonlocal Kermack–McKendrick epidemic model with delayed transmission. We find that the information is also determined by the reproduction number and the wave speed is explicitly effected by the delay. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. Here, the compactness of the supported set of dispersal kernel is needed in the proof of the existence of the traveling waves.

Traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with spatio-temporal delay

This paper mainly discusses the existence and non-existence of traveling wave solutions for the nonlocal Kermack-McKendrick epidemic model with nonlocal delayed transmission. We get that the existence and non-existence of traveling wave solutions are determined by the reproduction number and the wave speed. We also obtain that the spreading speed of the traveling wave solutions depends on the nonlocal delayed interaction and spatial movement patterns of the individuals. To prove these results, we apply the Schauders xed point theorem and the properties of Laplace transform.

Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence

Kermack-McKendrick epidemic model is considered as the basis from which many other compartmental models were developed. But the development of fractional calculus applied to mathematical epidemiology is still ongoing and relatively recent. We provide, in this article, some interesting and useful properties of the Kermack-McKendrick epidemic model with nonlinear incidence and fractional derivative order in the sense of Caputo. In the process, we used the generalized mean value theorem (Odibat and Shawagfeh in Appl. Math. Comput. 186:286-293, 2007) extended to fractional calculus to conclude some of the properties. A model of the Kermack-McKendrick with zero immunity is also investigated, where we study the existence of equilibrium points in terms of the nonlinear incidence function. We also establish the condition for the disease free equilibrium to be asymptotically stable and provide the expression of the basic reproduction number. Finally, numerical simulations of the monotonic behavior of the infection are provided for different values of the fractional derivative order α ( ). Comparing to the model with first-order derivative, there is a similar evolution for close values of α. The results obtained may help to analyze more complex fractional epidemic models.

Traveling waves in a Kermack–Mckendrick epidemic model with diffusion and latent period

This paper is devoted to the study of a Kermack–Mckendrick epidemic model with diffusion and latent period. We first consider the well-posedness of solutions of the model. Furthermore, using the Schauder fixed point theorem and Laplace transform, we show that if the threshold value R0>1, then there exists c∗>0 such that for every c>c∗, the model admits a traveling wave solution, and if R0<1 and c≥0; or R0>1 and c∈(0,c∗), then the model admits no traveling wave solutions. Hence, the existence and non-existence of traveling wave solutions is determined completely by R0, and the constant c∗ is the minimum speed for the existence of traveling wave solutions of the model.

On a Pandemic Threshold Theorem of the Early Kermack–McKendrick Model with Individual Heterogeneity

A pandemic threshold theorem of the Kermack–McKendrick epidemic system with individual heterogeneity is proved from the definition of R 0 by Diekmann, Heesterbeek, and Metz. The early Kermack–McKendrick epidemic model is extended to recognize individual heterogeneity, where the state variable indicates an epidemiological state or genetic, physiological, or behavioral characteristics such as risk of infection. With the basic reproduction number R 0 for the heterogeneous population, the final size equation of the limit epidemic starting from a completely susceptible steady state at t = −∞ has a unique positive solution if and only if R 0 > 1. The main result is that the positive solution of the final size equation gives the lower bound of the intensity of any epidemic starting from a host population composed of susceptible and a few infected individuals who spread on a noncompact domain of the trait variable.

Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

In this paper, we consider a Kermack-McKendrick epidemic model withnonlocal dispersal. We find that the existence and nonexistence oftraveling wave solutions are determined by the reproduction number.To prove the existence of nontrivial traveling wave solutions, weconstruct an invariant cone in a bounded domain with initialfunctions being defined on, and apply Schauder's fixed point theoremas well as limitingargument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, thenonexistence of traveling wave solutions is obtained by Laplace transform if the speed isless than the critical velocity.

Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission

We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack–McKendrick SIR models with non-local and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder’s fixed point theorem.

The Kermack–McKendrick epidemic model revisited

The Kermack–McKendrick epidemic model of 1927 is an age of infection model, that is, a model in which the infectivity of an individual depends on the time since the individual became infective. A special case, which is formulated as a two-dimensional system of ordinary differential ordinary differential equations, has often been called the Kermack–McKendrick model. One of the products of the SARS epidemic of 2002–2003 was a variety of epidemic models including general contact rates, quarantine, and isolation. These models can be viewed as age of infection epidemic models and analyzed using the approach of the full Kermack–McKendrick model. All these models share the basic properties that there is a threshold between disappearance of the disease and an epidemic outbreak, and that an epidemic will die out without infecting the entire population.