Dynamics of virus infection models with density‐dependent diffusion and Beddington–DeAngelis functional response

Abstract

In this work, we integrate both density-dependent diffusion process and Beddington-DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single-strain and multi-strain viruses and find that, for the multi-strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice.The classical Fickian diffusion equation assumes that random walkers in a system do not have interactions and has been used in a variety of models. However, individuals in real-world systems may interact, and as such, the diffusivity varies accordingly, known as the density-dependent diffusion [19]. Pao [20][21][22][23][24] performed a thorough study on a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions and proved that the systems are well posed.Mathematical systems modeling the propagation of multiple virus strains have been considered in the literature to investigate the competitive exclusion and coexistence of pathogens. Bremerman and Thieme constructed an Susceptibles-Infectives-Recovers model with multiple viral strains. The authors investigated the model analytically and obtained its competitive exclusion [25]. Andreasen and Pugliese established a disease model and investigated the coexistence of two competing infectious diseases. The authors studied how the host density affects the viral transmission [26]. Andreasen et al. integrated a finite number of viral strains into a disease model to reveal the immunity structure of the host population [27]. Feng et al. established an SIR model with two viral strains to consider the spread of a vector-borne disease [28]. Martcheva and Pilyugin established a multi-disease model and performed investigations on its bistability and bifurcation [29]. Castillo-Chavez et al. designed an susceptible-infective-susceptible sexually transmitted disease model to describe the transmission of two competing viral strains [30]. Iggidr et al. established an age-structured Malaria intra-host model with n viral strains and analyzed its global property [31]. Leenheer and Pilyugin studied the effects of viral mutation on the behaviors of a within-host multi-virus model [32]. The authors used a Lyapunov function to prove the global stability of the model's steady state.Dai and Zou established an age-structured within-host model and studied its global stability to investigate the interplay between two viral strains with viral mutations [33].The diffusion of virions is responsible for the short-range infections. Such vira...

Threshold Dynamics and Competitive Exclusion in a Virus Infection Model with General Incidence Function and Density-Dependent Diffusion

“…To explore the mechanisms of these diseases, scientists have proposed plenty of mathematical models describing the transmission of disease such as the infectious disease model (SI, SIR, SEI, and SIS) and the within-host virus model (HBV, HCV, HIV, Ebola, and Zika). Since Nowak et al [1,2] has established the basic within-host virus dynamics model, more and more scholars have devoted to investigate the within-host virus dynamics model and obtained so many significant dynamical results [3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”

“…So, parabolic or elliptic systems with density-dependent diffusion have been widely studied by many researchers [18][19][20][21]. More recently, Wang et al have investigated a virus infection model with density-dependent diffusion and Holling II type [6] or Beddington-DeAngelis type incidence function [7] subjected to the homogeneous Neumann boundary conditions. And, they have obtained the significant results of threshold dynamics and competitive exclusion.…”

“…One crucial component is incidence function, which can be classified into many different types, such as bilinear type [2], Holling II type [22], Beddington-DeAngelis type [23], and Crowley-Martin type [24]. Many researchers have concentrated on these withinhost models or epidemic models and have obtained some interesting results such as global threshold dynamics [8,9], delay-induced Hopf bifurcation [10,11], competitive exclusion [6,7], traveling wave [12,13], and pattern formation [14][15][16]. Recently, Korobeinikov [25] studied an ordinary differential equation model with general incidence function and dose-dependent parasite reproduction and virulence.…”

“…Iggidr et al [34] analyzed globally a new malaria within-host model through a competitive exclusion principle. Wang et al investigated the competitive exclusion in the virus infection model with density-dependent diffusion and two types of incidence functions [6,7]. Duan et al [36] considered the competitive exclusion in the diffusive virus infection model with age of infection and general incidence function, and so on.…”

“…Here, we have to stress that the Neumann boundary conditions, initial conditions, and steady-state system of Model (7) are described in Section 3. However, to the best of our knowledge, there are no results on the above models (1) and (7) such as global threshold dynamics and competitive exclusion.…”