We consider the qualitative behavior of a mathematical model for transmission dynamics with two nonlinear stages of contagion. The proposed model is inspired by phenomena occurring in epidemiology (spread of infectious diseases) or social dynamics (spread of opinions, behaviors, ideas), and described by a compartmental approach. Upon contact with a promoter (contagious individual), a naive (susceptible) person can either become promoter himself or become $\textit{weakened}$, hence more vulnerable. Weakened individuals become contagious when they experience a second contact with members of the promoter group. After a certain time in the contagious compartment, individuals become inactive (are insusceptible and cannot spread) and are removed from the chain of transmission. We combine this two-stage contagion process with renewal of the naive population, modeled by means of transitions from the weakened or the inactive status to the susceptible compartment. This leads to rich dynamics, showing for instance coexistence and bistability of equilibria and periodic orbits. Properties of (nontrivial) equilibria are studied analytically. In addition, a numerical investigation of the parameter space reveals numerous bifurcations, showing that the dynamics of such a system can be more complex than those of classical epidemiological ODE models.
In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune int…
In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is…
Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability…
In this work we have investigated the evolutionary dynamics of a generalist pathogen, e.g. a virus population, that evolves towards specialisation in an environment with multiple host types. We have particularly explored under which conditions generalist viral strains may rise in frequency and coexist with specialist strains or even dominate the population. By means of a nonlinear mathematical model and bifurcation analysis, we have determined the theoretical conditions for stability of nine identified equilibria and provided biological interpretation in terms of the infection rates for the vi…
Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infectious individual. Here, we attempt to investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. In the framework of multi-agent systems, we propose a particle stochastic model describing the infection transmission through interactions among agents and the individual phys…
We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD4+ T cells by a logistic function and the infected CD4+ T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some pa…
In this paper, we derive and analyze a compartmental model for the control of arboviral diseases which takes into account an imperfect vaccine combined with individual protection and some vector control strategies already studied in the literature. After the formulation of the model, a qualitative study based on stability analysis and bifurcation theory reveals that the phenomenon of backward bifurcation may occur. The stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the reproduction number, R 0 , is less than unity. Using Lyapunov function theory, w…
Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions…
Multiscale feedback drives viral evolution and epidemic dynamics
Departament de Física Teòrica, Universitat de València（瓦伦西亚大学理论物理系）；Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学数学系）；Institute of Mathematics, UPC-BarcelonaTech (IMTech)（加泰罗尼亚理工大学-巴塞罗那理工大学数学研究所）；Centre de Recerca Matemàtica (CRM)（数学研究中心）；Santa Fe Institute（圣塔菲研究所）
This paper considers a mathematical model based on the transmission dynamics of hepatitis C virus (HCV) infection. In addition to the usual compartments for susceptible, exposed, and infected individuals, this model includes compartments for individuals who are under treatment and those who have had vaccination against HCV infection. It is assumed that the immunity provided by the vaccine fades with time. The basic reproduction number, $R_0$, and the equilibrium solutions of the model are determined. The model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibr…
The Covid-19 pandemic has taken millions of lives, demonstrating the tragedy and disruption of respiratory diseases, and how difficult they can be to manage. However, there is still significant debate in the scientific community as to which transmission pathways are most significant and how settings and behaviour affect risk of infection, which all have implications for which mitigation strategies are most effective. This study presents a general model to estimate the rate of viral transfer between individuals, objects, and the air. The risk of infection to individuals in a setting is then com…
In this paper, we derive and analyse a model for the control of arboviral diseases which takes into account an imperfect vaccine combined with some other mechanisms of control already studied in the literature. We begin by analyse the basic model without controls. We prove the existence of two disease-free equilibrium points and the possible existence of up to two endemic equilibrium points (where the disease persists in the population). We show the existence of a transcritical bifurcation and a possible saddle-node bifurcation and explicitly derive threshold conditions for both, including def…
Polymerase-mediated quasispecies dynamics: bifurcations, complementation, and robustness of τ-tipping
Universidad Viña del Mar（维尼亚德尔马大学）；Embry-Riddle Aeronautical University（埃姆布里-里德尔航空大学）；Institute for Integrative Systems Biology (I2SysBio), CSIC-Universitat de València（综合系统生物学研究所（I2SysBio），西班牙瓦伦西亚大学与西班牙国家研究委员会）；The Santa Fe Institute（圣菲研究所）；Facultad de Ingeniería, Universidad San Sebastián（圣塞巴斯蒂安大学工程学院）；Centre de Recerca Matemàtica (CRM)（数学研究中心）
We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.
We propose a mathematical model to investigate the effects of information-dependent vaccination behavior on meningitis transmission. The information is represented by means of information index as early proposed in (d'Onofrio et al., Theor. pop. biol., 2007). We perform a qualitative analysis based on stability theory, focusing to the global stability of the disease free equilibrium (DFE) and the related transcritical bifurcation taking place at the threshold for the DFE. Finally, we assess the role of epidemiological and information parameters in the model dynamics through numerical simulatio…
Threshold Dynamics and Integrated Control of Cholera Transmission with Sanitation and Vaccination
Delaware State University（特拉华州立大学）；University of the Western Cape（西开普敦大学）；National Center for Research, Institute of Space Research and Aerospace（国家研究中心，空间研究与航空航天研究所）；Georgia State University（佐治亚州立大学）；Necmettin Erbakan University（杰姆特汀埃尔巴坎大学）；Kyrgyz-Turkish Manas University（吉尔吉斯-土耳其玛纳斯大学）
In this paper, a mathematical analysis of the global dynamics of a partial differential equation viral infection cellular model is carried out. We study the dynamics of a hepatitis C virus (HCV) model, under therapy, that considers both absorption phenomenon and diffusion of virions, infected and uninfected cells in liver. Firstly, we prove boundedness of the potential solutions, global existence, uniqueness and positivity of the solutions to the obtained initial value and boundary problem. Then, the dynamical behavior of the model is completely determined by a threshold parameter called the b…
A simple nonlinear transmission-line model of the cochlea with longitudinal coupling is introduced that can reproduce Basilar membrane response and neural tuning in the chinchilla. It is found that the middle ear has little effect on cochlear resonances, and hence conclude that the theory of coherent reflections is not applicable to the model. The model also provides an explanation of the emergence of spontaneous otoacoustic emissions (SOAEs). It is argued that SOAEs arise from Hopf bifurcations of the transmission-line model and not from localized instabilities. The paper shows that emissions…
