Modeling precaution, immunity loss and dispersal on disease dynamics: a two-patch SIRS model

<p style='text-indent:20px;'>An SIR epidemic model with age structure in the infected class is investigated. The model is transformed into a non-densely defined abstract Cauchy problem. The positivity and boundedness of solutions, basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal R_0 $\end{document}</tex-math></inline-formula> and the existence of equilibria are established. The linearized system and characteristic equation at an equilibrium from the corresponding abstract Cauchy problem are obtained. When <inline-formula><tex-math id="M2">\begin{document}$ \mathcal R_0\leq 1 $\end{document}</tex-math></inline-formula>, local and global stability of the disease-free equilibrium is proved, and hence the disease will be deracinated. For the model with the latent period described by infection age, when <inline-formula><tex-math id="M3">\begin{document}$ \mathcal R_0>1 $\end{document}</tex-math></inline-formula> and Assumptions 5.1 and 5.2 are satisfied, local stability of the endemic equilibrium and the existence of Hopf bifurcation are established. This shows that the disease with the latent period of infection age has complex dynamical behavior at the endemic equilibrium. Finally, numerical examples are presented to verify the theoretical results.</p>

We propose a stage-structured model of childhood infectious disease transmission dynamics, with the population demographics dynamics governed by a certain family and population planning strategy giving rise to nonlinear feedback delayed effects on the reproduction ageing and rate. We first describe the long-term aging-profile of the population by describing the pattern and stability of equilibrium of the demographic model. We also investigate the disease transmission dynamics, using the epidemic model when the population reaches the positive equilibrium (limiting equation). We establish conditions for the existence, uniqueness and global stability of the disease endemic equilibrium. We then prove the global stability of the endemic equilibrium for the original epidemic model with varying population demographics. The global stability of the endemic equilibrium allows us to examine the effects of reproduction ageing and rate, under different family planning strategies, on the childhood infectious disease transmission dynamics. We also examine demographic distribution, diseases reproductive number, infant disease rate and age distribution of disease, and as such, the work can be potentially used to inform targeted age group for optimal vaccine booster programs.

Mean-field dispersal induced synchrony and stability in an epidemic model under patchy environment

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node's degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coefficient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.

Stability and Hopf bifurcation for age-structured SVIR epidemic model with different compartment ages and two delays effects

<abstract><p>Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.</p></abstract>

Testing–culling is one of the important prevention and control measures considered in the study of animal infectious diseases. However, the process of finding infected animals (animal testing) is still not well studied through the kinetic model. In this paper, based on the characteristics of animal testing, a time-delayed model on brucellosis transmission is established. Under the general hypothesis of biological significance, the existence and stability of equilibria are first investigated. The results find that the global stability of equilibria depends on the basic reproduction number [Formula: see text] without the information delay: if [Formula: see text], the disease dies out; if [Formula: see text], the endemic equilibrium exists and the disease persists. Next, the impact of information delay on the dynamics of the model is analyzed and Hopf bifurcation is found in the established model when the information delay is greater than a critical value. Finally, the theoretical results are then further explained through numerical analysis and the significance of these results for the development of risk management measures is elaborated.

Global analysis of some epidemic models with general contact rate and constant immigration

A model for the dynamic transmission of SEITR pandemics is proposed, incorporating awareness and a mass action incidence function. The stability of both endemic and disease-free equilibrium points was examined, revealing that the disease-free equilibrium is locally and globally asymptotically stable when the Basic Reproduction Number is less than 1, while the endemic equilibrium is stable when . Additionally, it's shown that the Basic Reproduction Number (without awareness) is greater than (with awareness), this means that when , then . A Lyapunov function is constructed to demonstrate the global stability of both disease-free equilibrium when and the endemic equilibrium when . Numerical simulations illustrate the impact of awareness on disease spread, highlighting the importance of awareness in informing and educating individuals about infection risks during outbreaks, thereby reducing the health burden of the epidemic by lowering its peak.

This work considers an epidemic model with saturated incidence rate and saturated isolation and treatment functions. The incidence function (as well as the treatment and isolation functions) is of the Holling type II form, where, in the cases of treatment and isolation, these functions describes the effect of delayed treatment and isolation with large number of infected individuals in a population especially in situations where isolation facilities are available but fewer medical personnel. It is shown that the disease-free equilibrium (DFE) is locally-asymptotically stable whenever the effective reproduction number is less than unity. However, it is shown that the global-asymptotic stability of the disease-free equilibrium was largely dictated by the delay in treatment (of non-isolated infected individuals) and isolation of infected individuals; when such delay effect is weak, then the DFE is globally asymptotically stable when the effective reproduction number is less than unity. When the delay effect is strong, it is shown that there is the possibility of the existence of the backward bifurcation phenomenon whereby the DFE will co-exists with two endemic equilibria, when the effective reproduction number is less than unity. The backward bifurcation phenomenon existed when the delays occurred either singly or jointly. Mathematical analysis provides threshold conditions that allows for the global stability of the endemic equilibrium whenever the associated reproduction number is greater than unity. The results suggests that with timely isolation of infected individuals (for treatment under isolation) and treatment of non-isolated infectious cases, the ultimate goal of disease eradication is possible.

In this paper, we formulate the epidemic model that describes the dynamics of the spread of infectious transmission in the host population. This epidemic model combines three classes of infectious individuals with different infectivity and the nonlinear incidence rate. We investigated the basic reproductions number. We found that this model has two equilibrium points, one of them is free-equilibrium point and the other is endemic equilibrium point. By analyzing the existence and stability of the equilibria, we observed that the disease- free equilibrium is globally asymptotically stable when the basic reproduction number R 0 is less than or equal unity, that is the disease dies out. While the endemic equilibrium point is locally asymptotically stable when the reproduction number is more than unity. The local and global stability for all possible equilibria are carried out with the help of Lyapunov function and LaSalle’s invariant principle. The global stability of the endemic equilibrium is discussed.

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

Iris Yellow Spot Disease (IYSD), caused by Iris Yellow Spot Virus (IYSV) is a destructive and fastspreading virus disease of onion plants worldwide. It is mainly transmitted by an insect vector called thrips tabaci in a persistent and propagative manner and as such, there is a significant latent time after acquisition of the virus by the vector and an incubation time is needed for the appearance of disease symptoms on plants. In this paper, we formulate and analyze a non-linear mathematical model to explore the dynamics of IYSV disease in onion plants using system of delay differential equations by incorporating incubation and latent periods as time delays factors. The delays are introduced by adding an exposed population for onion plants representing the plants that are infected but not yet infective and by taking into account that there is fraction of the newly exposed onion plants that do not die during incubation period before becoming infective. It is assumed that the onion plant grows logistically in the farm so that the total onion plant population is taken as variable. The local stability of the disease-free equilibrium in the presence of delays is investigated using Descartes’s rule of signs. We establish the sufficient conditions for the stability of the endemic equilibrium in presence of delays and we investigate the occurrence of Hopf bifurcation when certain conditions are satisfied by considering the two delays as bifurcation parameters. We compute the critical values of the delays which preserve the local asymptotic stability of the endemic equilibrium and the model shows an oscillatory behavior beyond these critical values. Finally, numerical simulations are performed and displayed graphically to support the analytical results, and the eco-epidemiological implications of the key outcomes are briefly discussed.

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

This paper introduces differential susceptible e-epidemic model SSiiIIII (susceptible class-1 for virus (S1) susceptible class-2 for worms (S2) -susceptible class-3 for Trojan horse (S3) – infectious (I) – recovered (R)) for the transmission of malicious codes in a computer network. We derive the formula for reproduction number (R0) to study the spread of malicious codes in computer network. We show that the Infectious free equilibrium is globally asymptotically stable and endemic equilibrium is locally asymptotically sable when reproduction number is less than one. Also an analysis has been made on the effect of antivirus software in the infectious nodes. Numerical methods are employed to solve and simulate the system of equations developed. Index Terms — Computer network; Worms; Virus; Trojan horse; Epidemic Model; Reproduction number; Global stability