Global dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infections.

In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number R 0 and show that the infection-free steady state is globally asymptotically stable when R 0 ≤ 1 , while the disease is uniformly persistent when R 0 > 1 . In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of R 0 and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses.

Transmission dynamics of a reaction-advection-diffusion dengue fever model with seasonal developmental durations and intrinsic incubation periods.

In this paper, we propose a reaction-advection-diffusion dengue fever model with seasonal developmental durations and intrinsic incubation periods. Firstly, we establish the well-posedness of the model. Secondly, we define the basic reproduction number ℜ 0 for this model and show that ℜ 0 is a threshold parameter: if ℜ 0 < 1 , then the disease-free periodic solution is globally attractive; if ℜ 0 > 1 , the system is uniformly persistent. Thirdly, we study the global attractivity of the positive steady state when the spatial environment is homogeneous and the advection of mosquitoes is ignored. As an example, we use the model to investigate the dengue fever transmission case in Guangdong Province, China, and explore the impact of model parameters on ℜ 0 . Our findings indicate that ignoring seasonality may underestimate ℜ 0 . Additionally, the spatial heterogeneity of transmission may increase the risk of disease transmission, while the increase of seasonal developmental durations, intrinsic incubation periods and advection rates can all reduce the risk of disease transmission.

Threshold dynamics of a reaction-advection-diffusion schistosomiasis epidemic model with seasonality and spatial heterogeneity.

Most water-borne disease models ignore the advection of water flows in order to simplify the mathematical analysis and numerical computation. However, advection can play an important role in determining the disease transmission dynamics. In this paper, we investigate the long-term dynamics of a periodic reaction-advection-diffusion schistosomiasis model and explore the joint impact of advection, seasonality and spatial heterogeneity on the transmission of the disease. We derive the basic reproduction number R 0 and show that the disease-free periodic solution is globally attractive when R 0 < 1 whereas there is a positive endemic periodic solution and the system is uniformly persistent in a special case when R 0 > 1 . Moreover, we find that R 0 is a decreasing function of the advection coefficients which offers insights into why schistosomiasis is more serious in regions with slow water flows.

Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems.

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 [Formula: see text]where [Formula: see text] is a Hille-Yosida linear operator (possibly unbounded, non-densely defined) on a Banach space [Formula: see text], and the maps [Formula: see text], [Formula: see text] 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.

Threshold dynamics of an almost periodic vector-borne disease model.

Many infectious diseases cannot be transmitted from human to human directly, and the transmission needs to be done via a vector. It is well known that vectors' life cycles are highly dependent on their living environment. In order to investigate dynamics of vector-borne diseases under environment influence, we propose a vector-borne disease model with almost periodic coefficients. We derive the basic reproductive number [Formula: see text] for this model and establish a threshold type result on its global dynamics in terms of [Formula: see text]. As an illustrative example, we consider an almost periodic model of malaria transmission. Our numerical simulation results show that the basic reproductive number may be underestimated if almost periodic coefficients are replaced by their average values . Finally, we use our model to study the dengue fever transmission in Guangdong, China. The parameters are chosen to fit the reported data available for Guangdong. Numerical simulations indicate that the annual dengue fever case in Guangdong will increase steadily in the near future unless more effective control measures are implemented. Sensitivity analysis implies that the parameters with strong impact on the outcome are recovery rate, mosquito recruitment rate, mosquito mortality rate, baseline transmission rates between mosquito and human. This suggests that the effective control strategies may include intensive treatment, mosquito control, decreasing human contact number with mosquitoes (e.g., using bed nets and preventing mosquito bites), and environmental modification.

Basic reproduction ratio of a mosquito-borne disease in heterogeneous environment.

To explore the influence of spatial heterogeneity on mosquito-borne diseases, we formulate a reaction-diffusion model with general incidence rates. The basic reproduction ratio [Formula: see text] for this model is introduced and the threshold dynamics in terms of [Formula: see text] are obtained. In the case where the model is spatially homogeneous, the global asymptotic stability of the endemic equilibrium is proved when [Formula: see text]. Under appropriate conditions, we establish the asymptotic profiles of [Formula: see text] in the case of small or large diffusion rates, and investigate the monotonicity of [Formula: see text] with respect to the heterogeneous diffusion coefficients. Numerically, the proposed model is applied to study the dengue fever transmission. Via performing simulations on the impacts of certain factors on [Formula: see text] and disease dynamics, we find some novel and interesting phenomena which can provide valuable information for the targeted implementation of disease control measures.

Analysis of a COVID-19 Epidemic Model with Seasonality.

The statistics of COVID-19 cases exhibits seasonal fluctuations in many countries. In this paper, we propose a COVID-19 epidemic model with seasonality and define the basic reproduction number [Formula: see text] for the disease transmission. It is proved that the disease-free equilibrium is globally asymptotically stable when [Formula: see text], while the disease is uniformly persistent and there exists at least one positive periodic solution when [Formula: see text]. Numerically, we observe that there is a globally asymptotically stable positive periodic solution in the case of [Formula: see text]. Further, we conduct a case study of the COVID-19 transmission in the USA by using statistical data.

A reaction-advection-diffusion model of cholera epidemics with seasonality and human behavior change.

Cholera is a water- and food-borne infectious disease caused by V. cholerae. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of V. cholerae. We derive the basic reproduction number [Formula: see text] for this system and then establish a threshold type result on its global dynamics in terms of [Formula: see text]. Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of [Formula: see text] for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread.

Basic reproduction ratios for periodic and time-delayed compartmental models with impulses.

Much work has focused on the basic reproduction ratio [Formula: see text] for a variety of compartmental population models, but the theory of [Formula: see text] remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of [Formula: see text] for a class of such impulsive models. We first introduce [Formula: see text] and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of [Formula: see text]. Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.

Air quality index induced nonsmooth system for respiratory infection.

It has been observed that air pollution greatly affects respiratory infection and generates public health problem, but there are many challenges to quantifying the dynamics of air pollution and evaluating its impact on respiratory infections. A periodic Filippov system describing the state-dependent control strategy for air pollution, described by air quality index (AQI), is proposed. We theoretically analyze the non-autonomous Filippov subsystem for variation of AQI by converting into an autonomous Filippov system via increasing dimension of the system. We obtain that there is a unique periodic solution which is globally asymptotically stable. In particular, it shows that AQI stabilizes at either one of the periodic solutions of the free and control systems or a new periodic solution induced by the on-off control strategies, depending on the threshold levels. Then, a periodic system for respiratory infection with AQI-embedded transmission probability is formulated to examine the influence of air pollution on respiratory infection. We further obtain the control reproduction number (basic reproduction number with control strategies) of this periodic respiratory infection model. It is shown that the respiratory infection will go extinct if the control reproduction number is less than unity, while uniformly persists for larger than unity. The case study showed that the estimated threshold level coincides with the actual threshold which launches the traffic limitation measure in Xi'an, indicating the untimely density-dependent traffic limitation measure was ineffective in improving air quality. Numerically studies indicate that increasing the threshold level leads to an increase in the maximum value of the unique periodic solution for AQI and the control reproduction number for the AQI embedded SEIS model. These findings emphasize the importance of suitable threshold level to trigger interventions and suggest that untimely implementing control strategy may not effectively control air population.

A parasitism-mutualism-predation model consisting of crows, cuckoos and cats with stage-structure and maturation delays on crows and cuckoos.

In this paper, a parasitism-mutualism-predation model is proposed to investigate the dynamics of multi-interactions among cuckoos, crows and cats with stage-structure and maturation time delays on cuckoos and crows. The crows permit the cuckoos to parasitize their nestlings (eggs) on the crow chicks (eggs). In return, the cuckoo nestlings produce a malodorous cloacal secretion to protect the crow chicks from predation by the cats, which is apparently beneficial to both the crow and cuckoo population. The multi-interactions, i.e., parasitism and mutualism between the cuckoos (nestlings) and crows (chicks), predation between the cats and crow chicks are modeled both by Holling-type II and Beddington-DeAngelis-type functional responses. The existence of positive equilibria of three subsystems of the model are discussed. The criteria for the global stability of the trivial equilibrium are established by the Krein-Rutman theorem and other analysis methods. Moreover, the threshold dynamics for the coexistence and weak persistence of the model are obtained, and we show, both analytically and numerically, that the stabilities of the interior equilibria may change with the increasing maturation time delays. We find there exists an evident difference in the dynamical properties of the parasitism-mutualism-predation model based on whether or not we consider the effects of stage-structure and maturation time delays on cuckoos and crows. Inclusion of stage structure results in many varied dynamical complexities which are difficult to encompass without this inclusion.

Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD4[Formula: see text] T Cells Death: A Nonlocal Spatial Mathematical Model.

Caspase-1-mediated pyroptosis is the predominance for driving CD4[Formula: see text] T cells death. Dying infected CD4[Formula: see text] T cells can release inflammatory signals which attract more uninfected CD4[Formula: see text] T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1[Formula: see text] released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make [Formula: see text], that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4[Formula: see text] cells which are caused by pyroptosis can make [Formula: see text], that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.

A reaction-diffusion malaria model with seasonality and incubation period.

In this paper, we propose a time-periodic reaction-diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number [Formula: see text] is derived, and it is shown that the disease-free periodic solution is globally attractive if [Formula: see text], while there is an endemic periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

The effect of media coverage on threshold dynamics for a stochastic SIS epidemic model.

Media coverage is one of the important measures for controlling infectious diseases, but the effect of media coverage on diseases spreading in a stochastic environment still needs to be further investigated. Here, we present a stochastic susceptible-infected-susceptible (SIS) epidemic model incorporating media coverage and environmental fluctuations. By using Feller's test and stochastic comparison principle, we establish the stochastic basic reproduction number R 0 s , which completely determines whether the disease is persistent or not in the population. If R 0 s ≤ 1 , the disease will go to extinction; if R 0 s = 1 , the disease will also go to extinction in probability, which has not been reported in the known literatures; and if R 0 s > 1 , the disease will be stochastically persistent. In addition, the existence of the stationary distribution of the model and its ergodicity are obtained. Numerical simulations based on real examples support the theoretical results. The interesting findings are that (i) the environmental fluctuation may significantly affect the threshold dynamical behavior of the disease and the fluctuations in different size scale population, and (ii) the media coverage plays an important role in affecting the stationary distribution of disease under a low intensity noise environment.

Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches.

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species.

In this article, we are concerned with a nonlocal reaction-diffusion-advection model which describes the evolution of a single phytoplankton species in a eutrophic vertical water column where the species relies solely on light for its metabolism. The new feature of our modeling equation lies in that the incident light intensity and the death rate are assumed to be time periodic with a common period. We first establish a threshold type result on the global dynamics of this model in terms of the basic reproduction number R0. Then we derive various characterizations of R0 with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, which in turn give rather precise conditions to determine whether the phytoplankton persist or become extinct. Our theoretical results not only extend the existing ones for the time-independent case, but also reveal new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species, and thereby suggest important implications for phytoplankton growth control.

Density of voltage-gated potassium channels is a bifurcation parameter in pyramidal neurons.

Several types of intrinsic dynamics have been identified in brain neurons. Type 1 excitability is characterized by a continuous frequency-stimulus relationship and, thus, an arbitrarily low frequency at threshold current. Conversely, Type 2 excitability is characterized by a discontinuous frequency-stimulus relationship and a nonzero threshold frequency. In previous theoretical work we showed that the density of Kv channels is a bifurcation parameter, such that increasing the Kv channel density in a neuron model transforms Type 1 excitability into Type 2 excitability. Here we test this finding experimentally, using the dynamic clamp technique on Type 1 pyramidal cells in rat cortex. We found that increasing the density of slow Kv channels leads to a shift from Type 1 to Type 2 threshold dynamics, i.e., a distinct onset frequency, subthreshold oscillations, and reduced latency to first spike. In addition, the action potential was resculptured, with a narrower spike width and more pronounced afterhyperpolarization. All changes could be captured with a two-dimensional model. It may seem paradoxical that an increase in slow K channel density can lead to a higher threshold firing frequency; however, this can be explained in terms of bifurcation theory. In contrast to previous work, we argue that an increased outward current leads to a change in dynamics in these neurons without a rectification of the current-voltage curve. These results demonstrate that the behavior of neurons is determined by the global interactions of their dynamical elements and not necessarily simply by individual types of ion channels.

A PERIODIC ROSS-MACDONALD MODEL IN A PATCHY ENVIRONMENT.

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number [Formula: see text] and show that either the disease-free periodic solution is globally asymptotically stable if [Formula: see text] or the positive periodic solution is globally asymptotically stable if [Formula: see text]. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

Effect of avian influenza scare on transmission of zoonotic avian influenza: A case study of influenza A (H7N9).

Avian influenza scare is a human psychological factor that asserts both positive and negative effects on the transmission of zoonotic avian influenza. In order to study the dichotomous effect of avian influenza scare on disease transmission, taking H7N9 avian influenza as a typical case, a two-patch epidemic model is proposed. The global dynamics and the threshold criteria are established by LaSalle invariant principle and the theory of asymptotic autonomous system. To mitigate the negative effects and curb illegal poultry trade, a game-theoretic model is adopted to explore the optimal policy of culling subsidies to reasonably compensate stakeholders for their economic losses resulting from the scare. The optimal policy of culling subsidy is found to heavily depend on the penalty of illegal poultry trade, the stakeholders' income, the intensity of control measures, and the prevalence level of the disease. The negative effect of avian influenza scare on disease transmission is considerably more significant than the positive effect. In order to avoid a widespread outbreak of zoonotic avian influenza across the region, a comprehensive national global control strategy is essential and effective, even in the presence of the negative effect of the avian influenza scare.

A rigorous theoretical and numerical analysis of a nonlinear reaction-diffusion epidemic model pertaining dynamics of COVID-19.

The spatial movement of the human population from one region to another and the existence of super-spreaders are the main factors that enhanced the disease incidence. Super-spreaders refer to the individuals having transmitting ability to multiple pathogens. In this article, an epidemic model with spatial and temporal effects is formulated to analyze the impact of some preventing measures of COVID-19. The model is developed using six nonlinear partial differential equations. The infectious individuals are sub-divided into symptomatic, asymptomatic and super-spreader classes. In this study, we focused on the rigorous qualitative analysis of the reaction-diffusion model. The fundamental mathematical properties of the proposed COVID-19 epidemic model such as boundedness, positivity, and invariant region of the problem solution are derived, which ensure the validity of the proposed model. The model equilibria and its stability analysis for both local and global cases have been presented. The normalized sensitivity analysis of the model is carried out in order to observe the crucial factors in the transmission of infection. Furthermore, an efficient numerical scheme is applied to solve the proposed model and detailed simulation are performed. Based on the graphical observation, diffusion in the context of confined public gatherings is observed to significantly inhibit the spread of infection when compared to the absence of diffusion. This is especially important in scenarios where super-spreaders may play a major role in transmission. The impact of some non-pharmaceutical interventions are illustrated graphically with and without diffusion. We believe that the present investigation will be beneficial in understanding the complex dynamics and control of COVID-19 under various non-pharmaceutical interventions.