SIR Epidemic Model Research Articles

A Novel Approach to the Optimal Control of an SIR Epidemic Model with Vaccination

A Novel Approach to the Optimal Control of an SIR Epidemic Model with Vaccination

Effectiveness of the BNT162b2 (Pfizer-BioNTech) and the ChAdOx1 nCoV-19 (Oxford-AstraZeneca) vaccines for reducing susceptibility to infection with the Delta variant (B.1.617.2) of SARS-CoV-2

BackgroundFrom January to May 2021 the alpha variant (B.1.1.7) of SARS-CoV-2 was the most commonly detected variant in the UK. Following this, the Delta variant (B.1.617.2) then became the predominant variant. The UK COVID-19 vaccination programme started on 8th December 2020. Prior to the Delta variant, most vaccine effectiveness studies focused on the alpha variant. We therefore aimed to estimate the effectiveness of the BNT162b2 (Pfizer-BioNTech) and the ChAdOx1 nCoV-19 (Oxford-AstraZeneca) vaccines in preventing symptomatic and asymptomatic infection with respect to the Delta variant in a UK setting.MethodsWe used anonymised public health record data linked to infection data (PCR) using the Combined Intelligence for Population Health Action resource. We then constructed an SIR epidemic model to explain SARS-CoV-2 infection data across the Cheshire and Merseyside region of the UK. Vaccines were assumed to be effective after 21 days for 1 dose and 14 days for 2 doses.ResultsWe determined that the effectiveness of the Oxford-AstraZeneca vaccine in reducing susceptibility to infection is 39% (95% credible interval [34, 43]) and 64% (95% credible interval [61, 67]) for a single dose and a double dose respectively. For the Pfizer-BioNTech vaccine, the effectiveness is 20% (95% credible interval [10, 28]) and 84% (95% credible interval [82, 86]) for a single-dose and a double dose respectively.ConclusionVaccine effectiveness for reducing susceptibility to SARS-CoV-2 infection shows noticeable improvement after receiving two doses of either vaccine. Findings also suggest that a full course of the Pfizer-BioNTech provides the optimal protection against infection with the Delta variant. This reinforces the need to complete the full course programme to maximise individual protection and reduce transmission.

SIR epidemic models with spatial spread in bounded domains

SIR epidemic models with spatial spread in bounded domains

Global dynamics of a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments.

In this paper, we study a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments under Neumann boundary conditions. Firstly, the existence and boundedness of the solution of the system are addressed. Then, on the basis of the differential inclusions theory, we analysis the existence of endemic equilibrium. Furthermore, by constructing different appropriate Lyapunov functions, we investigate the global asymptotic stability of the disease free equilibrium(DFE) and the endemic equilibrium(EE), respectively. Additionally, numerical simulations are given to confirm the correctness of theorem. Finally, we give a brief conclusion and discussion in the end of the paper.

Multiple epidemic waves as the outcome of stochastic SIR epidemics with behavioral responses: a hybrid modeling approach.

In the behavioral epidemiology (BE) of infectious diseases, little theoretical effort seems to have been devoted to understand the possible effects of individuals’ behavioral responses during an epidemic outbreak in small populations. To fill this gap, here we first build general, behavior implicit, SIR epidemic models including behavioral responses and set them within the framework of nonlinear feedback control theory. Second, we provide a thorough investigation of the effects of different types of agents’ behavioral responses for the dynamics of hybrid stochastic SIR outbreak models. In the proposed model, the stochastic discrete dynamics of infection spread is combined with a continuous model describing the agents’ delayed behavioral response. The delay reflects the memory mechanisms with which individuals enact protective behavior based on past data on the epidemic course. This results in a stochastic hybrid system with time-varying transition probabilities. To simulate such system, we extend Gillespie’s classic stochastic simulation algorithm by developing analytical formulas valid for our classes of models. The algorithm is used to simulate a number of stochastic behavioral models and to classify the effects of different types of agents’ behavioral responses. In particular this work focuses on the effects of the structure of the response function and of the form of the temporal distribution of such response. Among the various results, we stress the appearance of multiple, stochastic epidemic waves triggered by the delayed behavioral response of individuals.

Optimal Control and Cost-Effectiveness Analysis in An Epidemic Model with Viral Mutation and Vaccine Intervention

This paper introduces an optimal control problem in a two-strain SIR epidemic model with viral mutation and vaccine administration. The purpose of this study was to investigate the efficacy and cost-effectiveness of two disease prevention strategies, namely restriction of community mobility to prevent disease transmission and vaccine intervention. We consider the time-dependent control case, and we use Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. We also calculate the Average Cost-Effectiveness Ratio (ACER) and the Incremental Cost-Effectiveness Ratio (ICER) to investigate the cost-effectiveness of all possible strategies of the control measures. The results of this study indicate that the most cost-effective disease control strategy is a combination of mobility restriction and vaccination.

A stochastic analysis for a triple delayed SIR epidemic model with vaccination incorporating Lévy noise

In this paper, we investigate a stochastic delayed epidemic model with double epidemic hypothesis and vaccination incorporating Lévy noise. First, we show that this model has a unique global positive solution. Furthermore, we give sufficient conditions for extinction and persistence in the mean of the two epidemics and we prove that the two diseases can coexist under some conditions. Finally, we present some examples to illustrate the analytical results by numerical simulations.

Dynamic analysis of a fractional-order SIRS model with time delay

Mathematical modeling plays a vital role in the epidemiology of infectious diseases. Policy makers can provide the effective interventions by the relevant results of the epidemic models. In this paper, we build a fractional-order SIRS epidemic model with time delay and logistic growth, and we discuss the dynamical behavior of the model, such as the local stability of the equilibria and the existence of Hopf bifurcation around the endemic equilibrium. We present the numerical simulations to verify the theoretical analysis.

Endemic bubble and multiple cusps generated by saturated treatment of an SIR model through Hopf and Bogdanov–Takens bifurcations

The current study presents complex dynamics of an SIR epidemic model that incorporates a saturated type incidence rate as well as treatment. We provide here rigorous results for asymptotic stability of equilibrium states of the proposed system. Several bifurcations including Hopf, Generalized Hopf, saddle–node, transcritical and Bogdanov–Takens are also discussed. The stability of bifurcated periodic solutions is verified with the help of first Lyapunov number. Extensive numerical simulations are performed to validate these results. In a numerical example it is observed that if the saturation factor increases slowly, then the unique endemic equilibrium state is asymptotically stable for a certain range. The further increase in the value of saturation parameter, the endemic equilibrium state loses its stability and periodic solutions appear through Hopf bifurcation. It is also observed that the increase in saturation parameter beyond Hopf bifurcation threshold, results in regaining the stability of the endemic equilibrium state, which forms an interesting dynamical phenomenon in the bifurcation diagram named as an endemic bubble. It is pointed out that in the case of two endemic equilibrium states, one of these two is always saddle, whereas, the other one becomes unstable through Hopf bifurcation. In this scenario, the periodic solution is initially stable and it becomes unstable through generalized Hopf bifurcation. In numerical example for Bogdanov–Takens bifurcation two pairs of feasible bifurcation thresholds exist for the same set of parameters value. The bifurcation diagrams and equilibrium surfaces are also plotted to observe the combined effects of medication and saturation parameters.

A spatiotemporal SIR epidemic model two-dimensional with problem of optimal control

In this work, we are interested in studying a spatiotemporal two-dimensional SIR epidemic model, in the form of a system of partial differential equations (PDE). A distribution of a vaccine in the form of a control variable is considered to force immunity. The purpose is to characterize a control that minimizes the number of susceptible, infected individuals and the costs associated with vaccination over a nite space and time domain. In addition, the existence of the solution of the state system and the optimal control is proved. The characterization of the control is given in terms of state function and adjoint function. The numerical resolution of the state system shows the effectiveness of our control strategy.

A stochastic Sir epidemic evolution model with non-concave force of infection: Mathematical modeling and analysis

In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in [Formula: see text]th moment for any [Formula: see text]. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations.

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps.

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and Lacute{e}vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold lambda by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.

Asymptotic profiles of a diffusive SIRS epidemic model with standard incidence mechanism and a logistic source

Asymptotic profiles of a diffusive SIRS epidemic model with standard incidence mechanism and a logistic source

Fractional order SIR epidemic model with Beddington-De Angelis incidence and Holling type II treatment rate for COVID-19.

In this paper, an attempt has been made to study and investigate a non-linear, non-integer SIR epidemic model for COVID-19 by incorporating Beddington–De Angelis incidence rate and Holling type II saturated cure rate. Beddington–De Angelis incidence rate has been chosen to observe the effects of measure of inhibition taken by both: susceptible and infective. This includes measure of inhibition taken by susceptibles as wearing proper mask, personal hygiene and maintaining social distance and the measure of inhibition taken by infectives may be quarantine or any other available treatment facility. Holling type II treatment rate has been considered for the present model for its ability to capture the effects of available limited treatment facilities in case of Covid 19. To include the neglected effect of memory property in integer order system, Caputo form of non-integer derivative has been considered, which exists in most biological systems. It has been observed that the model is well posed i.e., the solution with a positive initial value is reviewed for non-negativity and boundedness. Basic reproduction number R_{0} is determined by next generation matrix method. Routh Hurwitz criteria has been used to determine the presence and stability of equilibrium points and then stability analyses have been conducted. It has been observed that the disease-free equilibrium Q^{d} is stable for R_{0} < 1 i.e., there will be no infection in the population and the system tends towards the disease-free equilibrium Q^{d} and for R_{0} > 1, it becomes unstable, and the system will tend towards endemic equilibrium Q^{e}. Further, global stability analysis is carried out for both the equilibria using R_{0}. Lastly numerical simulations to assess the effects of various parameters on the dynamics of disease has been performed.

Agent-based modelling approach for internet rumour propagation and its empirical study

This paper analyses the current research status of internet rumour propagation at home and abroad. Based on the SIR epidemic model, considering the two methods of spreading rumours and dispelling rumours, the dynamics of the spread rate and the mechanism for individuals to forget information, an expanded SPNR rumour spreading model was proposed. Then, the agent method is used to simulate and analyse, which provides a basis for the designation of rumour control strategy. Finally, the SPNR model is empirically studied using two methods, dataset verification and case verification, to verify the applicability of model in the process of rumour propagation. The research results show that: 1) the final scale of the spread of rumours can be reduced through a continuous stream of hot event; 2) the influence of the rumours can be increased by enhancing the right of the rumours' discourse, increasing the number of rumours, etc., thereby effectively reducing the number of rumours and reducing the spread of rumours; 3) in this model, small-world networks and scale-free networks have no effect on the spread of rumours.

Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

&lt;p style='text-indent:20px;'&gt;In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ I $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) exceeds a certain level, the incidence rate is a decreasing function with respect to &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ I $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ I $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \widetilde{I_0} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ ( = \frac{b}{d}) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for the infective level &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ I_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; at which the health care system reaches its capacity such that:&lt;b&gt;(i)&lt;/b&gt; When &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the transmission dynamics of the model is determined by the basic reproduction number &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;: &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ R_0 = 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; separates disease persistence from disease eradication. &lt;b&gt;(ii)&lt;/b&gt; When &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ I_0 &amp;lt; \widetilde{I_0} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.&lt;/p&gt;

Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect.

In this paper, an SIR model with a strong Allee effect and density-dependent transmission is proposed, and its characteristic dynamics are investigated. The elementary mathematical characteristic of the model is studied, including positivity, boundedness and the existence of equilibrium. The local asymptotic stability of the equilibrium points is analyzed using linear stability analysis. Our results indicate that the asymptotic dynamics of the model are not only determined using the basic reproduction number ${R_0}$. If ${R_0} < 1$, there are three disease-free equilibrium points, and a disease-free equilibrium is always stable. At the same time, the conditions for other disease-free equilibrium points to be bistable were determined. If ${R_0} > 1$ and in certain conditions, either an endemic equilibrium emerges and is locally asymptotically stable, or the endemic equilibrium becomes unstable. What must be emphasized is that there is a locally asymptotically stable limit cycle when the latter happens. The Hopf bifurcation of the model is also discussed using topological normal forms. The stable limit cycle can be interpreted in a biological significance as a recurrence of the disease. Numerical simulations are used to verify the theoretical analysis. Taking into account both density-dependent transmission of infectious diseases and the Allee effect, the dynamic behavior becomes more interesting than when considering only one of them in the model. The Allee effect makes the SIR epidemic model bistable, which also makes the disappearance of diseases possible, since the disease-free equilibrium in the model is locally asymptotically stable. At the same time, persistent oscillations due to the synergistic effect of density-dependent transmission and the Allee effect may explain the recurrence and disappearance of disease.

Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

&lt;p style='text-indent:20px;'&gt;The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.&lt;/p&gt;

The Dynamic Reproduction Index: Accurate Determination from Incidence and Application for an Early Warning System

wo methods of calculating the reproduction index from daily new infection data (incidence) are considered, one by using the generation time Gt as a shift ( GR ), and an incidence-based method directly derived from the differential equation system of an SIR epidemic dynamics model ( IR ). While the former (which s commonly used) is shown to be at variance with the true reproduction index, we find that the latter provides a sensitive detection device for intervention effects and other events affecting the epidemic, making it well-suited for diagnostic purposes in policy making. Furthermore, we introduce a similar quantity, calc IR , which can be calculated directly from GR . It shows largely the same behaviour as IR , with less fine structure. However, it is accurate in particular in the vicinity of = 1R , where accuracy is important for the correct prediction of epidemic dynamics. We introduce an entirely new, self-consistent method to derive, an improved corr IR which is both accurate and contains the details of the epidemic spreading dynamics. Hence we obtain R accurately from incidence data alone. Moreover, plotting R versus incidence reveals the orbital structure of epidemic waves, whose fine structure features clearly correlate with public events and interventions, thus providing a sensitive diagnostic tool for policy making. It is demonstrated that the widespread use of only incidence as a diagnostic tool is clearly inappropriate

A vertically transmitted epidemic model with two state-dependent pulse controls.

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the Poincare´map, an analogue of Poincare´ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.