Analysis of SIR epidemic models with nonlinear incidence rate and treatment

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.     

Revealing the role of predator interference in a predator–prey system with disease in prey population

Predation on a species subjected to an infectious disease can affect both the infection level and the population dynamics. There is an ongoing debate about the act of managing disease in natural populations through predation. Recent theoretical and empirical evidence shows that predation on infected populations can have both positive and negative influences on disease in prey populations. Here, we present a predator–prey system where the prey population is subjected to an infectious disease to explore the impact of predator on disease dynamics. Specifically, we investigate how the interference among predators affects the dynamics and structure of the predator–prey community. We perform a detailed numerical bifurcation analysis and find an unusually large variety of complex dynamics, such as, bistability, torus and chaos, in the presence of predators. We show that, depending on the strength of interference among predators, predators enhance or control disease outbreaks and population persistence. Moreover, the presence of multistable regimes makes the system very sensitive to perturbations and facilitates a number of regime shifts. Since, the habitat structure and the choice of predators deeply influence the interference among predators, thus before applying predators to control disease in prey populations or applying predator control strategy for wildlife management, it is essential to carefully investigate how these predators interact with each other in that specific habitat; otherwise it may lead to ecological disaster.     

Hopf and torus bifurcations,torus destruction and chaos in population biology

One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed.However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig–MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of a Hopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity.Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle.Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case.     

Modeling experimental time series with ordinary differential equations

Recently some methods have been presented to extract ordinary differential equations (ODE) directly from an experimental time series. Here, we introduce a new method to find an ODE which models both the short time and the long time dynamics. The experimental data are represented in a state space and the corresponding flow vectors are approximated by polynomials of the state vector components. We apply these methods both to simulated data and experimental data from human limb movements, which like many other biological systems can exhibit limit cycle dynamics. In systems with only one oscillator there is excellent agreement between the limit cycling displayed by the experimental system and the reconstructed model, even if the data are very noisy. Furthermore, we study systems of two coupled limit cycle oscillators. There, a reconstruction was only successful for data with a sufficiently long transient trajectory and relatively low noise level.     

On evolution under symmetric and asymmetric competitions

This paper considers the coevolution of phenotypic traits in a community comprising two competitive species subject to strong Allee effects. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy under symmetric competition. Secondly, we find that evolutionary suicide is impossible when the two species undergo symmetric competition, however, evolutionary suicide can occur in an asymmetric competition model with strong Allee effects. Thirdly, it is found that evolutionary bistability is a likely outcome of the process under both symmetric and asymmetric competitions, which depends on the properties of symmetric and asymmetric competitions. Fourthly, under asymmetric competition, we find that evolutionary cycle is a likely outcome of the process, which depends on the properties of both intraspecific and interspecific competition. When interspecific and intraspecific asymmetries vary continuously, we also find that the evolutionary dynamics may admit a stable equilibrium and two limit cycles or two stable equilibria separated by an unstable limit cycle or a stable equilibrium and a stable limit cycle.     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

Spatially extended host-parasite interactions: the role of recovery and immunity

Techniques for determining the long-term dynamics of host-parasite systems are well established for mixed populations. The field of spatial modelling in ecology is more recent but a number of key advances have been made. In this paper, we use state-of-the-art approximation techniques, supported by simulations, in order to investigate the role of recovery and immunity in spatially structured populations. Our approach is to use correlation models, namely pair-wise models, to capture the spatial relationships of contacts and interactions between individuals. We use the pair-wise framework to address a number of key ecological questions; including, the persistence of endemic limit cycles and regions of parasite-driven extinction--features which differentiate spatial from non-spatial models--and the effects on invasion fitness. We demonstrate a loss of limit cycle behaviour, in addition to an increase in the critical transmissibility and extinction thresholds, when recovery is included. This approach allows for a better analytical understanding of the dynamics of host-parasite interactions and demonstrates the importance of recovery and immunity in local interactions.     

Transmission Dynamics of an SIS Model with Age Structure on Heterogeneous Networks

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).     

Erratic flu vaccination emerges from short-sighted behavior in contact networks

The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior.     

Ecoepidemic models with prey group defense and feeding saturation

In this paper we consider a model for the herd behavior of prey, that are subject to attacks by specialist predators. The latter are affected by a transmissible disease. With respect to other recently introduced models of the same nature, we focus here our attention to the possible feeding satiation phenomenon. The system dynamics is thoroughly investigated, to show the occurrence of several types of bifurcations. In addition to the transcritical and Hopf bifurcation that occur commonly in predator–prey system also a zero-Hopf and a global bifurcation occur. The Hopf and the global bifurcation occur only in the disease-free (so purely demographic) system. The latter is a heteroclinic connection for the between saddle equilibrium points where a stable limit cycle is disrupted and where the system disease-free collapses while in a parameter space region the endemic system exists stably.     

Backward bifurcations and multiple equilibria in epidemic models with structured immunity

Many disease pathogens stimulate immunity in their hosts, which then wanes over time. To better understand the impact of this immunity on epidemiological dynamics, we propose an epidemic model structured according to immunity level that can be applied in many different settings. Under biologically realistic hypotheses, we find that immunity alone never creates a backward bifurcation of the disease-free steady state. This does not rule out the possibility of multiple stable equilibria, but we provide two sufficient conditions for the uniqueness of the endemic equilibrium, and show that these conditions ensure uniqueness in several common special cases. Our results indicate that the within-host dynamics of immunity can, in principle, have important consequences for population-level dynamics, but also suggest that this would require strong non-monotone effects in the immune response to infection. Neutralizing antibody titer data for measles are used to demonstrate the biological application of our theory.     

The dynamics of an epidemic model with targeted antiviral prophylaxis

Due to the increasing risk of drug resistance and side effects with large-scale antiviral use, it has been suggested to provide antiviral drugs only to susceptibles who have had contacts with infectives. This antiviral distribution strategy is referred to as 'targeted antiviral prophylaxis'. The question of how effective this strategy is in infection control is of great public heath interest. In this paper, we formulate an ordinary differential equation model to describe the transmission dynamics of infectious disease with targeted antiviral prophylaxis, and provide the analysis of dynamical behaviours of the model. The control reproduction number ?( c ) is derived and shown to govern the disease dynamics, and the stability analysis is carried out. The local bifurcation theory is applied to explore the variety of dynamics of the model. Our theoretical results show that the system undergoes two Hopf bifurcations due to the existence of multiple endemic equilibria and the switch of their stability. Numerical results demonstrate that the system may have more complex dynamical behaviours including multiple periodic solutions and a homoclinic orbit. The results of this study suggest that the possibility of complex disease dynamics can be driven by the use of targeted antiviral prophylaxis, and the critical level of prophylaxis which achieves ?(c)=1 is not enough to control the prevalence of a disease.     

Dynamics of an SIQS epidemic model with transport-related infection and exit-entry screenings

Population dispersal, as a common phenomenon in human society, may cause the spreading of many diseases such as influenza, SARS, etc. which are easily transmitted from one region to other regions. Exit and entry screenings at the border are considered as effective ways for controlling the spread of disease. In this paper, the dynamics of an SIQS model are analyzed and the combined effects of transport-related infection enhancing and exit-entry screenings suppressing on disease spread are discussed. The basic reproduction number is computed and proved to be a threshold for disease control. If it is not greater than the unity, the disease free equilibrium is globally asymptotically stable. And there exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is greater than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Exit screening and entry screening are shown to be helpful for disease eradication since they can always have the possibility to eradicate the disease endemic led by transport-related infection and furthermore have the possibility to eradicate disease even when the isolated cites are disease endemic.     

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.     

Autocatalytic networks with translation

We consider the kinetics of an autocatalytic reaction network in which replication and catalytic actions are separated by a translation step. We find that the behaviour of such a system is closely related to second-order replicator equations, which describe the kinetics of autocatalytic reaction networks in which the replicators act also as catalysts. In fact, the qualitative dynamics seems to be described almost entirely be the second-order reaction rates of the replication step. For two species we recover the qualitative dynamics of the replicator equations. Larger networks show some deviations, however. A hypercyclic system consisting of three interacting species can converge toward a stable limit cycle in contrast to the replicator equation case. A singular perturbation analysis shows that the replication-translation system reduces to a second-order replicator equation if translation is fast. The influence of mutations on replication-translation networks is also very similar to the behavior of selection-mutation equations.     

Effect of media-induced social distancing on disease transmission in a two patch setting

We formulate an SIS epidemic model on two patches. In each patch, media coverage about the cases present in the local population leads individuals to limit the number of contacts they have with others, inducing a reduction in the rate of transmission of the infection. A global qualitative analysis is carried out, showing that the typical threshold behavior holds, with solutions either tending to an equilibrium without disease, or the system being persistent and solutions converging to an endemic equilibrium. Numerical analysis is employed to gain insight in both the analytically tractable and intractable cases; these simulations indicate that media coverage can reduce the burden of the epidemic and shorten the duration of the disease outbreak.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

Excitatory and inhibitory interactions in localized populations of model neurons

Coupled nonlinear differential equations are derived for the dynamics of spatially localized populations containing both excitatory and inhibitory model neurons. Phase plane methods and numerical solutions are then used to investigate population responses to various types of stimuli. The results obtained show simple and multiple hysteresis phenomena and limit cycle activity. The latter is particularly interesting since the frequency of the limit cycle oscillation is found to be a monotonic function of stimulus intensity. Finally, it is proved that the existence of limit cycle dynamics in response to one class of stimuli implies the existence of multiple stable states and hysteresis in response to a different class of stimuli. The relation between these findings and a number of experiments is discussed.     

Forward hysteresis and backward bifurcation caused by culling in an avian influenza model

The emerging threat of a human pandemic caused by the H5N1 avian influenza virus strain magnifies the need for controlling the incidence of H5N1 infection in domestic bird populations. Culling is one of the most widely used control measures and has proved effective for isolated outbreaks. However, the socio-economic impacts of mass culling, in the face of a disease which has become endemic in many regions of the world, can affect the implementation and success of culling as a control measure. We use mathematical modeling to understand the dynamics of avian influenza under different culling approaches. We incorporate culling into an SI model by considering the per capita culling rates to be general functions of the number of infected birds. Complex dynamics of the system, such as backward bifurcation and forward hysteresis, along with bi-stability, are detected and analyzed for two distinct culling scenarios. In these cases, employing other control measures temporarily can drastically change the dynamics of the solutions to a more favorable outcome for disease control.