ABSTRACT
Inapparent infection plays an important role in the disease spread, which is an infection by a pathogen that causes few or no signs or symptoms of infection in the host. Many pathogens, including HIV, typhoid fever, and coronaviruses such as COVID-19 spread in their host populations through inapparent infection. In this paper, we formulated a degenerated reaction-diffusion host-pathogen model with multiple infection period. We split the infectious individuals into two distinct classes: apparent infectious individuals and inapparent infectious individuals, coming from exposed individuals with a ratio of (1-p) and p, respectively. Some preliminary results and threshold-type results are achieved by detailed mathematical analysis. We also investigate the asymptotic profiles of the positive steady state (PSS) when the diffusion rate of susceptible individuals approaches zero or infinity. When all parameters are all constants, the global attractivity of the constant endemic equilibrium is established. It is verified by numerical simulations that spatial heterogeneity of the transmission rates can enhance the intensity of an epidemic. Especially, the transmission rate of inapparent infectious individuals significantly increases the risk of disease transmission, compared to that of apparent infectious individuals and pathogens in the environment, and we should pay special attentions to how to regulate the inapparent infectious individuals for disease control and prevention, which is consistent with the result on the sensitive analysis to the transmission rates through the normalized forward sensitivity index. We also find that disinfection of the infected environment is an important way to prevent and eliminate the risk of environmental transmission.
ABSTRACT
We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.
ABSTRACT
In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.
ABSTRACT
A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine. [ FROM AUTHOR]
ABSTRACT
As the COVID-19 epidemic has entered the normalization stage, the task of prevention and control remains very arduous. This paper constructs a time delay reaction-diffusion model that is closer to the actual spread of the COVID-19 epidemic, including relapse, time delay, home quarantine and temporal-spatial heterogeneous environment that affect the spread of COVID-19. These factors increase the number of equations and the coupling between equations in the system, making it difficult to apply the methods commonly used to discuss global dynamics, such as the Lyapunov function method. Therefore, we use the global exponential attractor theory in the infinite-dimensional dynamic system to study the spreading trend of the COVID-9 epidemic with relapse, time delay, home quarantine in a temporal-spatial heterogeneous environment. Using our latest results of global exponential attractor theory, the global asymptotic stability and the persistence of the COVID-19 epidemic are discussed. We find that due to the influence of relapse in the in temporal-spatial heterogeneity environment, the principal eigenvalue λ * can describe the spread of the epidemic more accurately than the usual basic reproduction number R 0 . That is, the non-constant disease-free equilibrium is globally asymptotically stable when λ * < 0 and the COVID-19 epidemic is persisting uniformly when λ * > 0 . Combine with the latest official data of the COVID-19 and the prevention and control strategies of different countries, some numerical simulations on the stability and global exponential attractiveness of the spread of the COVID-19 epidemic in China and the USA are given. The simulation results fully reflect the impact of the temporal-spatial heterogeneous environment, relapse, time delay and home quarantine strategies on the spread of the epidemic, revealing the significant differences in epidemic prevention strategies and control effects between the East and the West. The results of this study provide a theoretical basis for the current epidemic prevention and control.