Diffusive Epidemic Model Research Articles

Traveling wave solutions for Zika transmission model with nonlocal diffusion

The problem of traveling wave solutions (TWS) for a nonlocal diffusive Zika transmission model with bilinear incidence was studied. More specifically, the system admits a nontrivial TWS if the threshold ℜ > 1 and c>c⁎, where c⁎ is the critical wave speed; the system admits no nontrivial TWS for ℜ > 1 and 0<c<c⁎ or ℜ ≤ 1. The proofs are based on sub- and super-solution methods, Schauder's fixed point theorem and Laplace transformation. We also explain the influence of model parameters on the transmission of Zika virus. Compared with the previous studies on nonlocal diffusive epidemic models, a major difficulty is the boundedness of TWS arising from the bilinear incidence. The main result helps us to provide some new perspectives for the TWS of the nonlocal diffusion epidemic model with multiple infectious compartments.

On the effect of lowering population's movement to control the spread of an infectious disease

We study the asymptotic behavior of endemic equilibrium solutions of a diffusive epidemic model in spatially heterogeneous environment when the diffusion rates dS of the susceptible and dI of the infected groups approach zero. Our results indicate that when dI and dS are sufficiently small, the size of dIdS plays a crucial role in the dynamics of the disease in the sense that: (i) if dIdS is small, the disease may persist and the total size of the infected group will be maximized; (ii) if dIdS is large, then the total size of the susceptible group is maximized while the total size of the infected group is minimized. Hence, our results suggest that lowering the movement rate of the population in an attempt to limit infection is an effective control strategy if the susceptible hosts' movement rate is kept sufficiently smaller than that of the infected individuals.

A New Numerical Scheme for Time Fractional Diffusive SEAIR Model with Non-Linear Incidence Rate: An Application to Computational Biology

In this paper, we propose a modified fractional diffusive SEAIR epidemic model with a nonlinear incidence rate. A constructed model of fractional partial differential equations (PDEs) is more general than the corresponding model of fractional ordinary differential equations (ODEs). The Caputo fractional derivative is considered. Linear stability analysis of the disease-free equilibrium state of the epidemic model (ODEs) is presented by employing Routh–Hurwitz stability criteria. In order to solve this model, a fractional numerical scheme is proposed. The proposed scheme can be used to find conditions for obtaining positive solutions for diffusive epidemic models. The stability of the scheme is given, and convergence conditions are found for the system of the linearized diffusive fractional epidemic model. In addition to this, the deficiencies of accuracy and consistency in the nonstandard finite difference method are also underlined by comparing the results with the standard fractional scheme and the MATLAB built-in solver pdepe. The proposed scheme shows an advantage over the fractional nonstandard finite difference method in terms of accuracy. In addition, numerical results are supplied to evaluate the proposed scheme’s performance.

Front Propagation of Exponentially Truncated Fractional-Order Epidemics

The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns, prevents them from making arbitrarily large displacements. Their random movements indeed occur in a finite space with an upper bound. In order to make realistic models, by introducing exponentially truncated Lévy flights, such an upper bound can thus be taken into account in the reaction-diffusion models. In this work, we have investigated the influence of the λ-truncated fractional-order diffusion operator on the spatial propagation of the epidemics caused by infectious diseases, where λ is the truncation parameter. Analytical and numerical simulations show that depending on the value of λ, different asymptotic behaviours of the travelling-wave solutions can be identified. For small values of λ (λ≳0), the tails of the infective waves can decay algebraically leading to an exponential growth of the epidemic speed. In that case, the truncation has no impact on the superdiffusive epidemics. By increasing the value of λ, the algebraic decaying tails can be tamed leading to either an upper bound on the epidemic speed representing the maximum speed value or the generation of the infective waves of a constant shape propagating at a minimum constant speed as observed in the classical models (second-order diffusion epidemic models). Our findings suggest that the truncated fractional-order diffusion equations have the potential to model the epidemics of animals performing Lévy flights, as the animal diseases can spread more smoothly than the exponential acceleration of the human disease epidemics.

Optimal control of pattern formations for an SIR reaction–diffusion epidemic model

Patterns arising from the reaction–diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction–diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, “normal serious prevalent area (NSPA)”, is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to NSPA. To effectively reduce NSPA of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction–diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.

A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

&lt;abstract&gt; &lt;p&gt;This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.&lt;/p&gt; &lt;/abstract&gt;

Nonconstant positive steady states and pattern formation of a diffusive epidemic model

It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zero-flux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey's functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.

Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

Methodology of emergency medical logistics for multiple epidemic areas in public health emergency.

Coronavirus disease 2019(COVID-19) has brought great disasters to humanity, and its influence continues to intensify. In response to the public health emergencies, prompt relief supplies are key to reduce the damage. This paper presents a method of emergency medical logistics to quick response to emergency epidemics. The methodology includes two recursive mechanisms: (1) the time-varying forecasting of medical relief demand according to a modified susceptible-exposed-infected- Asymptomatic- recovered (SEIAR) epidemic diffusion model, (2) the relief supplies distribution based on a multi-objective dynamic stochastic programming model. Specially, the distribution model addresses a hypothetical network of emergency medical logistics with considering emergency medical reserve centers (EMRCs), epidemic areas and e-commerce warehousing centers as the rescue points. Numerical studies are conducted. The results show that with the cooperation of different epidemic areas and e-commerce warehousing centers, the total cost is 6% lower than without considering cooperation of different epidemic areas, and 9.7% lower than without considering cooperation of e-commerce warehousing centers. Particularly, the total cost is 20% lower than without considering any cooperation. This study demonstrates the importance of cooperation in epidemic prevention, and provides the government with a new idea of emergency relief supplies dispatching, that the rescue efficiency can be improved by mutual rescue between epidemic areas in public health emergency.

Analysis on critical waves for a diffusive epidemic model with saturating incidence rate and infinitely distributed delay

The purpose of this paper is to obtain an existence theorem of critical waves for a diffusive Susceptible‐Infected‐Removed (SIR) epidemic model with saturating incidence rate and infinitely distributed delay. Our results completely answer an aforementioned unsolved questions by using the Schauder's fixed point theorem and constructing a perfect pair of upper‐lower solutions.

Qualitative analysis on an SIRS reaction–diffusion epidemic model with saturation infection mechanism

In this paper, we deal with an SIRS reaction–diffusion epidemic model with saturation infection mechanism. Based on the uniform boundedness of the parabolic system, we investigate the extinction and persistence of the infectious disease in terms of the basic reproduction number. To better investigate the effects of infection mechanism and individual diffusion, we further analyze the asymptotic profiles of the endemic equilibrium for small or large motility rate and large saturation rate. In particular it is shown that large saturation may cause the elimination of disease. Our study may provide some significant useful insight on disease control and prevention.

Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic models. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number R0 greater than 1, there exists a critical wave speed c* > 0, such that for each c >= c * the system admits a nontrivial TWS and for c < c* there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behaviour of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.

Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism

&lt;p style='text-indent:20px;'&gt;In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.&lt;/p&gt;

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

Design and analysis of a discrete method for a time‐delayed reaction–diffusion epidemic model

In this work, we propose a time‐delayed reaction–diffusion model to describe the propagation of infectious viral diseases like COVID‐19. The model is a two‐dimensional system of partial differential equations that describes the interactions between disjoint groups of a human population. More precisely, we assume that the population is conformed by individuals who are susceptible to the virus, subjects who have been exposed to the virus, members who are infected and show symptoms, asymptomatic infected individuals, and recovered subjects. Various realistic assumptions are imposed upon the model, including the consideration of a time‐delay parameter which takes into account the effects of social distancing and lockdown. We obtain the equilibrium points of the model and analyze them for stability. Moreover, we examine the bifurcation of the system in terms of one of the parameters of the model. To simulate numerically this mathematical model, we propose a time‐splitting nonlocal finite‐difference scheme. The properties of the model are thoroughly established, including its capability to preserve the positivity of solutions, its consistency, and its stability. Some numerical experiments are provided for illustration purposes.

Long-time dynamics of a diffusive epidemic model with free boundaries

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number $ R_0 $ obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if $ R_0&gt;1 $, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

Analysis of a general multi-group reaction–diffusion epidemic model with nonlinear incidence and temporary acquired immunity

Considering the individual differences, spatial environment and the temporary acquired immunity, a general multi-group reaction–diffusion epidemic model with nonlinear incidence and temporary acquired immunity is proposed in this paper. The well-posedness of the solution including the existence of global solutions and the ultimate boundedness of the solutions are obtained, and then we define the basic reproduction number R0. Further, the threshold dynamics of the disease are obtained in terms of R0. Moreover, by constructing suitable Lyapunov functions, we obtain the endemic steady state is globally asymptotically stable in homogeneous space and heterogeneous case when R0>1. Based on theoretical analysis, we conclude that the existence of immunity loss rate and the difference of diffusion rate may bring great difficulties to control the disease. Finally, we find the spatial heterogeneity cannot always enhance the infectious risk of the disease and simply increasing the diffusion coefficient cannot effectively control the spread of the disease via some numerical simulations.

Numerical simulation for solution of SEIR models by meshless and finite difference methods

Abstract The transmission of influenza has been explained by analyzing a diffusive epidemic model. The Operating splitting based on finite difference (OSBFD), explicit formula based on meshless method (EFBMM), Operator splitting based on meshless method (OSBMM) are applied to obtain numerical solutions of equations under varied initial distribution of dense population. The specific role of diffusion and distribution has been accentuated in spread of ailment. It is also presented that how the transmission of disease is specifically reduced by the medicative and non-medicative innovations. The numerical solutions involved in stability of all the equilibria are also stated.

A reaction–diffusion epidemic model with incubation period in almost periodic environments

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent λ* for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of λ* It is shown that the disease-free almost periodic solution is globally attractive if λ* &lt; 0, while the disease is persistent if λ* &gt; 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

Optimising data diffusion while reducing local resources consumption in Opportunistic Mobile Crowdsensing

The combination of Mobile Crowdsensing (MCS) with Opportunistic Networking (OppNet) allows mobile users to share sensed data easily and conveniently without the use of fixed infrastructure. OppNet is based on intermittent connectivity among wireless mobile devices, in which mobile nodes may store, carry and forward messages (sensing information) by taking advantage of wireless ad hoc communication opportunities. A common approach for the diffusion of this sensing data in OppNet is the epidemic protocol, which carries out a fast data diffusion at the expense of increasing the usage of local buffers on mobile nodes and also the number of transmissions, thereby limiting scalability.A way to reduce this consumption of local resources is to set a message expiration time that forces the removal of old messages from local buffers. Since dropping messages too early may reduce the speed of information diffusion, we propose a dynamic expiration time setting to limit this effect. Moreover, we introduce an epidemic diffusion model for evaluating the impact of the expiration time. This model allows us to obtain optimal expiration times that achieve performances similar to those other approaches where no expiration is considered, with a significant reduction of local buffer and network usage. Furthermore, in our proposed model, the buffer utilisation remains steady with the number of nodes, whereas in other approaches it increases sharply. Finally, our approach is evaluated and validated in a mobile crowdsensing scenario, where students collect and broadcast information regarding a university campus, showing a significant reduction on buffer usage and nodes message transmissions, and therefore, decreasing battery consumption.