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The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014–2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number R 0 < 1 . For R 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model. [ABSTRACT FROM AUTHOR]

We propose a general framework for simultaneously calculating the threshold value for population growth and determining the sign of the growth bound of the evolution family generated by the problem below d v (t) d t = A v (t) + F (t) v (t) - V (t) v (t) , where A : D (A) ⊂ X → X is a Hille–Yosida linear operator (possibly unbounded, non-densely defined) on a Banach space (X , ‖ · ‖) , and the maps t ∈ R ↦ V (t) ∈ L (X 0 , X) , t ∈ R ↦ F (t) ∈ L (X 0 , X) are p-periodic in time and continuous in the operator norm topology. We give applications of our approach for two general examples of an age-structured model, and a delay differential system. Other examples concern the dynamics of a nonlocal problem arising in population genetics and the dynamics of a structured human-vector malaria model. [ABSTRACT FROM AUTHOR]

Releasing Wolbachia-infected male mosquitoes to suppress wild female mosquitoes through cytoplasmic incompatibility has shown great promise in controlling and preventing mosquito-borne diseases. To make the release logistically and economically feasible, we propose a saturated release strategy, which is only implemented during the epidemic season of mosquito-borne diseases. Under this assumption, the model becomes a seasonally switching ordinary differential equation model. The seasonal switch brings rich dynamics, including the existence of a unique periodic solution or exactly two periodic solutions, which are proved by using the qualitative property of the Poincaré map. Sufficient conditions are also obtained for determining the stability of the periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general viral infection model to incorporate two infection modes (virus-to-cell mode and cell-to-cell mode), the CTL immune response, and the distributed intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We investigate the existence, the uniqueness, and the global stability of three equilibria: infection-free equilibrium E 0 , immune-inactivated equilibrium E 1 and immune-activated equilibrium E 2 , respectively. We prove that the viral dynamics are determined by two threshold parameters: the basic reproduction number for infection R 0 and the basic reproduction number for immune response R IM . We also numerically explore the viral dynamics beyond stability. We use bifurcation diagrams to show that increasing the delay in CTL immune cell recruitment can induce a switch in viral load from a stable constant level to sustained oscillations, and then back to a stable equilibrium. We also compare the contributions of the two infection modes to the total infection level and identify the key parameters that would affect the percentages of virus-to-cell infection and cell-to-cell infection. Finally, we explore how Filippov control can be applied in antiretroviral therapy to reduce the viral loads. [ABSTRACT FROM AUTHOR]

Cancer is usually considered a genetic disease caused by alterations in genes that control cellular behaviors, especially growth and division. Cancer cells differ from normal tissue cells in many ways that allow them to grow out of control and become invasive. However, experiments have shown that aberrant growth in many tissues burdened with varying numbers of mutant cells can be corrected, and wild-type cells are required for the active elimination of mutant cells. These findings reveal the dynamic cellular behaviors that lead to a tissue homeostatic state when faced with mutational and nonmutational insults. The current study was motivated by these observations and established a mathematical model of how a tissue copes with the aberrant behavior of mutant cells. The proposed model depicts the interaction between wild-type and mutant cells through a system of two delay differential equations, which include the random mutation of normal cells and the active extrusion of mutant cells. Based on the proposed model, we performed qualitative analysis to identify the conditions of either normal tissue homeostasis or uncontrolled growth with varying numbers of abnormal mutant cells. Bifurcation analysis suggests the conditions of bistability with either a small or large number of mutant cells, the coexistence of bistable steady states can be clinically beneficial by driving the state of mutant cell predominance to the attraction basin of the state with a low number of mutant cells. This result is further confirmed by the treatment strategy obtained from optimal control theory. [ABSTRACT FROM AUTHOR]

Assuming that there are multiple batches of sterile males reared and released during the maturation period, we derive a switching delay differential model to study the fate of wild females under an impulsive and periodic release of sterile males. For the release magnitude of each batch c, we find two threshold values c 1 ∗ and c 2 ∗ , and prove that when c ∈ (0 , c 1 ∗ ] , the model admits exactly two periodic solutions, among which one is asymptotically stable and the other is unstable. The trivial equilibrium, corresponding to the elimination of wild females, is locally asymptotically stable, and it becomes globally asymptotically stable when c ≥ c 2 ∗ . One key step is to prove that every solution is sandwiched between two “good” solutions. [ABSTRACT FROM AUTHOR]

Based on the idea that only sexually active sterile mosquitoes are included in the modeling process, we study the dynamics of the interactive wild and sterile mosquito model with time delay, which consists of three sub-equations. Due to the fact that the maturation period of sterile mosquitoes bred in the lab or mosquito factories is almost the same time period of wild adult mosquitoes matured from larvae, we particularly assume that the waiting period for two consecutive releases of sterile mosquitoes equals the maturation period of wild mosquitoes, as a new practical sterile mosquito release strategy. We first ingeniously solve the delay model with the initial functions that are solutions of the corresponding equation without delay and we call them “good” solutions. Using these “good” solutions, we then surprisedly obtain sufficient and necessary conditions for the trivial solution and a unique periodic solution of the delay model to be globally asymptotically stable, respectively. We provide a numerical example to demonstrate the model dynamics and brief discussions of our findings as well. [ABSTRACT FROM AUTHOR]

Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response. [ABSTRACT FROM AUTHOR]

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications. [ABSTRACT FROM AUTHOR]

In this work, we consider an epidemic model in a two-layer network in which the dynamics of susceptible-infected-susceptible process in the physical layer coexists with that of a cyclic process of unaware-aware-unaware in the virtual layer. For such multiplex network, we shall define the basic reproduction number R0V in the virtual layer, which is similar to the basic reproduction number R0P defined in the physical layer. We show analytically that if R0P≤1 and R0V≤1, then the disease and information free equilibrium is globally stable and if R0P≤1 and R0V>1, then the disease free and information saturated equilibrium is globally stable for all initial conditions except at the origin. In the case of R0P>1, whether the disease dies out or not depends on the competition between how well the information is transmitted in the virtual layer and how contagious the disease is in the physical layer. In particular, it is numerically demonstrated that if the difference in R0V and R0P is greater than the product of R0P, the deviation of R0V from 1 and the relative infection rate for an aware susceptible individual, then the disease dies out. Otherwise, the disease breaks out. [ABSTRACT FROM AUTHOR]

In this paper, a very general model of impulsive delay differential equations in n-patches is rigorously derived to describe the impulsive control of population of a single species over n-patches. The model allows an age structure consisting of immatures and matures, and also considers mobility and culling of both matures and immatures. Conditions are obtained for extinction and persistence of the model system under three special scenarios: (1) without impulsive control; (2) with impulsive culling of the immatures only; and (3) with impulsive culling of the matures only, respectively. In the case of persistence, the persistence level is also estimated for the systems in the case of identical n patches, by relating the issue to the dynamics of multi-dimensional maps. Two illustrative examples and their numerical simulations are given to show the effectiveness of the results. Based on the theoretical results, some strategies of impulsive culling are provided to eradicate the population of a pest species. [ABSTRACT FROM AUTHOR]

In this paper, strongly connected and non-strongly connected multi-group viral models with time delays and general incidence functions are considered. Employing the Lyapunov functional method and a graph-theoretic approach, we show that the global dynamics of the strongly connected system are determined by the basic reproduction number under some reasonable conditions for incidence functions. In addition, we find a more complex and more interesting result for multi-group viral models with non-strongly connected networks because of the basic reproduction numbers corresponding to each strongly connected component. Finally, we provide simulations for non-strongly connected multi-group viral models to support our conclusion. [ABSTRACT FROM AUTHOR]

In this paper, based on an SIS model, we construct an epidemic model with infection age to investigate the disease transmission on complex networks. By analyzing the characteristic equations associated with the equilibria, we obtain the basic reproduction number $$R_0$$ . It is shown that if $$R_0<1$$ then the disease-free equilibrium is globally asymptotically stable while if $$R_0>1$$ then there is a unique endemic equilibrium, which is asymptotically stable. Our investigation indicates that if the maximal degree of the network is large enough then the endemic equilibrium always exists. Sensitivity analysis on the basic reproduction number $$R_0$$ in terms of the parameters is carried out to illustrate their effects on the disease transmission and to develop appropriate control strategies. [ABSTRACT FROM AUTHOR]

In this paper we characterize the stability boundary in the $$(\alpha _{1},\alpha _{2})$$ -plane, for fixed $$\alpha _{3}$$ with $$-1<\alpha _{3}<+1$$ , for the characteristic equation from the title. Subsequently we describe a nonlinear cell population model involving quiescence and show that this characteristic equation governs the (in)stability of the nontrivial steady state. By relating the parameters of the cell model to the $$\alpha _{i}$$ , we are able to derive some biological conclusions. [ABSTRACT FROM AUTHOR]

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE-shift system) is a limiting case of a system of two standard delay equations. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross-MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number $$\mathcal{R }_0$$ is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of $$\mathcal{R }_0$$ . It is shown that the disease free equilibrium is asymptotically stable if $$\mathcal{R }_0<1$$ , and it is unstable if $$\mathcal{R }_0>1$$ ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for $$\mathcal{R }_0$$ are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for $$\mathcal{R }_0$$ in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, $$\mathcal{R }_0$$ is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned. [ABSTRACT FROM AUTHOR]

There is considerable interest in the management of insecticide resistance in mosquitoes. One possible approach to slowing down the evolution of resistance is to use late-life-acting (LLA) insecticides that selectively kill only the old mosquitoes that transmit malaria, thereby reducing selection pressure favoring resistance. In this paper we consider an age-structured compartmental model for malaria with two mosquito strains that differ in resistance to insecticide, using an SEI approach to model malaria in the mosquitoes and thereby incorporating the parasite developmental times for the two strains. The human population is modeled using an SEI approach. We consider both conventional insecticides that target all adult mosquitoes, and LLA insecticides that target only old mosquitoes. According to linearised theory the potency of the insecticide affects mainly the speed of evolution of resistance. Mutations that confer resistance can also affect other parameters such as mean adult life span and parasite developmental time. For both conventional and LLA insecticides the stability of the malaria-free equilibrium, with only the resistant mosquito strain present, depends mainly on these other parameters. This suggests that the main long term role of an insecticide could be to induce genetic changes that have a desirable effect on a vital parameter such as adult life span. However, when this equilibrium is unstable, numerical simulations suggest that a potent LLA insecticide can slow down the spread of malaria in humans but that the timing of its action is very important. [ABSTRACT FROM AUTHOR]

We propose a patch type model for mosquitoes that have aquatic larvae inhabiting ponds. Partial differential equations (PDEs) model the larvae on each of several disconnected patches representing the ponds, with conditions varying in each patch, coupled via the adults in the air. From the PDEs a scalar delay differential equation, with multiple delays, for the total adult mosquito population is derived. The various delays represent the larval development times in the patches. The coefficients contain all the relevant information about the sizes and geometry of the individual patches inhabited by the larvae, the boundary conditions applicable to those patches and the diffusivity of the larvae in each patch. For patches of general shapes and sizes, and without the need to specify the criteria by which an adult mosquito selects an oviposition patch, the modern theory of monotone dynamical systems and persistence theory enables a complete determination of the conditions for the mosquito population to go extinct or to persist. More detailed biological insights are obtained for the case when the patches are squares of various sizes, which allows a detailed discussion of the effects of scale, and for two particular criteria by which mosquitoes might select patches for oviposition, being (i) selection based solely on patch area, and (ii) selection based both on area and expected larval survival probability for each patch. In some parameter regimes, counterintuitive phenomena are predicted. [ABSTRACT FROM AUTHOR]

In this paper we study the effects that woody plant chemical defenses may have on interactions between boreal hares that in winter feed almost entirely on twigs. We focus particularly on the fact that toxin concentration often varies with the age of twig segments. The model incorporates the fact that the woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that these hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modeled using age-structured model equations which are reduced to a system of delay differential equations involving multiple delays in the woody plant-hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. This showed that age dependence in plant chemical defenses has the capacity to cause hare-plant population cycles, a new result. [ABSTRACT FROM AUTHOR]

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases. [ABSTRACT FROM AUTHOR]

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by themobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R0 is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R0 < 1, and becomes unstable if R0 > 1. In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases. [ABSTRACT FROM AUTHOR]

We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L1-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. [ABSTRACT FROM AUTHOR]