We study an epidemic patch model that describes the disease spread in population with variable latency due to the differences in immunologic tolerance between individuals. We focus on whether the disease can spread in space that leads to the emergence of epidemic wave, that is the travelling wave solution with constant speed. We first establish some properties of the linearized wave profile equations, which are helpful in obtaining the priori estimates of travelling waves and wave speeds. Then, applying the truncation method and limiting arguments, we can obtain threshold propagation dynamics of the epidemic model. Our result gives a complete characterization of the existence, nonexistence and minimal wave speed of travelling waves. To the best of our knowledge, this is the first time to characterize the propagation dynamics of epidemic patch model with variable latency, which contributes to the understanding of the transmission phenomenon of disease.

Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran–Kingman model. For the sharp-peak landscape, we perform several exact computations and derive several exact formulas. We also obtain an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and does not require a specific asymptotic expansion (such as sending the length of the macromolecules to $\infty$ or the mutation probability to 0). With the help of these formulas, we present an original proof for the well-known phenomenon of the error threshold. We recover the limiting quasispecies distribution in the long-chain regime. We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. Going beyond the sharp-peak landscape, we consider fitness landscapes having finitely many peaks and a plateau-type landscape. Finally, within this framework, we prove rigorously the possible occurrence of the survival of the flattest, a phenomenon which was previously discovered by Wilke et al. (Nature 412, 2001) and which has been investigated in several works (see e.g. Codoñer et al. (PLOS Pathogens 2, 2006), Franklin et al. (Artificial Life 25, 2019), Sardanyés et al. (J. Theoret. Biol. 250, 2008), and Tejero et al. (BMC Evolutionary Biol. 11, 2011)).

Cannibalism is often an extreme interaction in the animal species to quell competition for limited resources. To model this critical factor, we improve the predator–prey model with nonlocal competition effect by incorporating the cannibalism term, and different kernels for competition are considered in this model numerically. We give the critical conditions leading to the double Hopf bifurcation, in which the gestation time delay and the diffusion coefficient were selected as the bifurcation parameters. The innovation of the work lies near the double Hopf bifurcation point, and the stable homogeneous and inhomogeneous periodic solutions can coexist. The theoretical results of the extended centre manifold reduction and normal form method are in good agreement with the numerical simulation.

In this paper, we consider reaction-diffusion epidemic models with mass action or standard incidence mechanism and study the impact of limiting population movement on disease transmissions. We set either the dispersal rate of the susceptible or infected people to zero and study the corresponding degenerate reaction-diffusion model. Our main approach to study the global dynamics of these models is to construct delicate Lyapunov functions. Our results show that the consequences of limiting the movement of susceptible or infected people depend on transmission mechanisms, model parameters and population size.

Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.

We consider a generalization of the well-known nonlinear Nicholson blowflies model with stochastic perturbations. Stability in probability of the positive equilibrium of the considered equation is studied. Two types of stability conditions: delay-dependent and delay-independent conditions are obtained, using the method of Lyapunov functionals and the method of linear matrix inequalities. The obtained results are illustrated by numerical simulations by means of some examples. The results are new, and complement the existing ones.

In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.

It is standard in chemistry to represent a sequence of reactions by a single overall reaction, often called a complex reaction in contrast to an elementary reaction. Photosynthesis $6 \text{CO}_2+6 \text{H}_2\text{O} \longrightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{O}_2$ is an example of such complex reaction. We introduce a mathematical operation that corresponds to summing two chemical reactions. Specifically, we define an associative and non-communicative operation on the product space ${\mathbb{N}}_0^n\times {\mathbb{N}}_0^n$ (representing the reactant and the product of a chemical reaction, respectively). The operation models the overall effect of two reactions happening in succession, one after the other. We study the algebraic properties of the operation and apply the results to stochastic reaction networks (RNs), in particular to reachability of states, and to reduction of RNs.

This corrigendum corrects a result in [1].

We study a mathematical model proposed in the literature with the aim of describing the interactions between tumor cells and the immune system, when a periodic treatment of immunotherapy is applied. Combining some techniques from non-linear analysis (degree theory, lower and upper solutions, and theory of free-homeomorphisms in the plane), we give a detailed global analysis of the model. We also observe that for certain therapies, the maximum level of aggressiveness of a cancer, for which the treatment works (or does not work), can be computed explicitly. We discuss some strategies for designing therapies. The mathematical analysis is completed with numerical results and conclusions.

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝn. We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

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* < 0, while the disease is persistent if λ* > 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete-time Markov chain.

Each species is subject to various biotic and abiotic factors during growth. This paper formulates a deterministic model with the consideration of various factors regulating population growth such as age-dependent birth and death rates, spatial movements, seasonal variations, intra-specific competition and time-varying maturation simultaneously. The model takes the form of two coupled reaction–diffusion equations with time-dependent delays, which bring novel challenges to the theoretical analysis. Then, the model is analysed when competition among immatures is neglected, in which situation one equation for the adult population density is decoupled. The basic reproduction number $\mathcal{R}_0$ is defined and shown to determine the global attractivity of either the zero equilibrium (when $\mathcal{R}_0\leq 1$ ) or a positive periodic solution ( $\mathcal{R}_0\gt1$ ) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is neglected, the model is neither cooperative nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number $\widetilde{\mathcal{R}}_0$ as a threshold index. Furthermore, numerical simulations are implemented on the population growth of two different species for two different cases to validate the analytic results.

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account “natural parameters”, such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals.

The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a “stop-and-go” procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade.

We revisit the well-known chemostat model, considering that bacteria can be attached together in aggregates or flocs. We distinguish explicitly free and attached compartments in the model and give sufficient conditions for coexistence of these two forms. We then study the case of fast attachment and detachment and show how it is related to density-dependent growth functions. Finally, we give some insights concerning the cases of multi-specific flocs and different removal rates.

Mass migration of cells (via wave motion) plays an important role in many biological processes, particularly chemotaxis. We study the existence of travelling wave solutions for a chemotaxis model on a microscopic scale. The interaction between nutrients and chemoattractants are considered. Unlike previous approaches, we allow for diffusion of substrates, degradation of chemoattractants and cell growth (constant and linear growth rate). We apply asymptotic methods to investigate the behaviour of the solutions when cells are highly sensitive to extracellular signalling. Explicit solutions are demonstrated, and their biological implications are presented. The results presented here extend and generalize known results.