Global dynamics of two-strain epidemic model with single-strain vaccination in complex networks.

Contagious pathogens, such as influenza and COVID-19, are known to be represented by multiple genetic strains. Different genetic strains may have different characteristics, such as spreading more easily, causing more severe diseases, or even evading the immune response of the host. These facts complicate our ability to combat these diseases. There are many ways to prevent the spread of infectious diseases, and vaccination is the most effective. Thus, studying the impact of vaccines on the dynamics of a multi-strain model is crucial. Moreover, the notion of complex networks is commonly used to describe the social contacts that should be of particular concern in epidemic dynamics. In this paper, we investigate a two-strain epidemic model using a single-strain vaccine in complex networks. We first derive two threshold quantities, R 1 and R 2 , for each strain. Then, by using the basic tools for stability analysis in dynamical systems (i.e., Lyapunov function method and LaSalle's invariance principle), we prove that if R 1 < 1 and R 2 < 1 , then the disease-free equilibrium is globally asymptotically stable in the two-strain model. This means that the disease will die out. Furthermore, the global stability of each strain dominance equilibrium is established by introducing further critical values. Under these stability conditions, we can determine which strain will survive. Particularly, we find that the two strains can coexist under certain condition; thus, a coexistence equilibrium exists. Moreover, as long as the equilibrium exists, it is globally stable. Numerical simulations are conducted to validate the theoretical results.

A diffusive virus model with a fixed intracellular delay and combined drug treatments.

The method of administration of an effective drug treatment to eradicate viruses within a host is an important issue in studying viral dynamics. Overuse of a drug can lead to deleterious side effects, and overestimating the efficacy of a drug can result in failure to treat infection. In this study, we proposed a reaction-diffusion within-host virus model which incorporated information regarding spatial heterogeneity, drug treatment, and the intracellular delay to produce productively infected cells and viruses. We also calculated the basic reproduction number [Formula: see text] under the assumptions of spatial heterogeneity. We have shown that the infection-free periodic solution is globally asymptotically stable when [Formula: see text], whereas when [Formula: see text] there is an infected periodic solution and the infection is uniformly persistent. By conducting numerical simulations, we also revealed the effects of various parameters on the value of [Formula: see text]. First, we showed that the dependence of [Formula: see text] on the intracellular delay could be monotone or non-monotone, depending on the death rate of infected cells in the immature stage. Second, we found that the mobility of infected cells or virions could facilitate the virus clearance. Third, we found that the spatial fragmentation of the virus environment enhanced viral infection. Finally, we showed that the combination of spatial heterogeneity and different sets of diffusion rates resulted in complicated viral dynamic outcomes.

Within-Host Viral Dynamics in a Multi-compartmental Environment.

The discrepancy in the turnover of cells and virus in different organs or viral reservoirs necessitates the investigation of multiple compartments within a host. Establishing a multi-compartmental structure that describes the complexity of various organs, where viral infection comprehensively proceeds, provides a modeling framework for exploring the effect of spatial heterogeneity on viral dynamics. To successfully suppress within-host viral replication, it is imperative to determine drug administration during therapy, particularly for a combination of antiretroviral drugs. The proposed model provides quantitative insights into pharmacokinetics and the resulting virus population, which substantially relates to environmental heterogeneity. The main results are the following: (1) A model incorporating drug treatment admits threshold dynamics, driving to either viral extinction or uniform persistence, regardless of non-trivial initial infection, in the entire system. (2) Viral infection may be underestimated if a well-mixed (single-compartmental) model is used. (3) Optimal drug administration depends not only on the drug distribution over various compartments but also on the timing, described by phase shifts, of the administration of different drugs in a combined therapy.

Adaptive dispersal effect on the spread of a disease in a patchy environment.

During outbreaks of a communicable disease, people intensely follow the media coverage of the epidemic. Most people attempt to minimize contact with others, and move themselves to avoid crowds. This dispersal may be adaptive regarding the intensity of media coverage and the population numbers in different patches. We propose an epidemic model with such adaptive dispersal rates to examine how appropriate adaption can facilitate disease control in connected groups or patches. Assuming dependence of the adaptive dispersal on the total population in the relevant patches, we derived an expression for the basic reproduction number R 0 to be related to the intensity of media coverage, and we show that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 , and it becomes unstable if R 0 > 1 . In the unstable case, we showed a uniform persistence of disease by using a perturbation theory and the monotone dynamics theory. Specifically, when the disease mildly affects the dispersal of infectious individuals and rarely induces death, a unique endemic equilibrium exists in the model, which is globally asymptotically stable in positive states. Moreover, we performed numerical calculations to explain how the intensity of media coverage causes competition among patches, and influences the final distribution of the population.

Age-structured viral dynamics in a host with multiple compartments.

Several studies have reported dual pathways for HIV cell infection, namely the binding of free virions to target cell receptors (cell-free), and direct transmission from infected cells to uninfected cells through virological synapse (cell-to-cell). Furthermore, understanding spread of the infection may require a relatively in-depth comprehension of how the connection between organs, each with characteristic cell composition and infection kinetics, affects viral dynamics. We propose a virus model consisting of multiple compartments with cell populations subject to distinct infectivity kernels as a function of cell infection-age, in order to imitate the infection spread through various organs. When the within-host structure is strongly connected, we formulate the basic reproduction number to be the threshold value determining the viral persistence or extinction. On the other hand, in non-strongly connected cases, we also formulate a sequence of threshold values to find out the infection pattern in the whole system. Numerical results of derivative examples show that: (1) In a strongly connected system but lacking some directional connection between compartments therein, the migration of cells certainly affects the viral dynamics and it may not monotonously depend on the value of migration rate. (2) In a non-strongly connected structure, increasing migration rate may first change persistence of the virus to extinction in the whole system, and then for even larger migration rate, trigger the infection in a subset of compartments. (3) For data-informed cases of intracellular delay and gamma-distributed cell infectivity kernels, compartments with faster kinetics representative of cell-to-cell transmission mode, as opposed to cell-free, can promote persistence of the virus.

On an age structured population model with density-dependent dispersals between two patches.

Motivated by an age-structured population model over two patches that assumes constant dispersal rates, we derive a modified model that allows density-dependent dispersal, which contains both nonlinear dispersal terms and delayed non-local birth terms resulted from the mobility of the immature individuals between the patches. A biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population enables us investigate the model theoretically. Well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. Also addressed are a positive invariant set and global convergence of solutions under certain conditions. Although the levels of the density- dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal- spatial patterns and the final population sizes.

Coexistence and extinction for two competing species in patchy environments.

A system of two competing species µ and ν that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.

Multistability for Delayed Neural Networks via Sequential Contracting.

In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.

Scale-limited activating sets and multiperiodicity for threshold-linear networks on time scales.

The existing results for multiperiodicity of threshold-linear networks (TLNs) are scale-free on time evolution and hence exhibit some restrictions. Due to the nature of the scale-limited activating set, it is interesting to study the dynamical properties of neurons on time scales. In this paper we analyze and obtain results concerning nondivergence, attractivity, and multiperiodic dynamics of TLNs on time scales. Using the notion of exponential functions on time scales, we obtain results for scale-limited type criteria for boundedness and global attractivity of TLNs. Moreover, by constructing simple algebraic inequalities over scale-limited activating sets, we achieve results regarding multiperiodicity of TLNs. This will show that each scale-limited activating set depends on scale-synchronous self-excitation, and the existence of inactive neurons will slow down convergence of TLNs. At the end of the paper, we perform computer simulations to illustrate the obtained new theories.

Complete stability in multistable delayed neural networks.

We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3n equilibria and 2n positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.