Pulse vaccination in a modified SIR model: global dynamics, bifurcations and seasonality

We analyze a periodically-forced dynamical system inspired by the SIR model with pulse vaccination. We fully characterize its dynamics according to the proportion $p$ of vaccinated individuals and the time $T$ between doses. If the basic reproduction number is less than 1 (i.e. $\mathcal{R}_p<1$), then we obtain precise conditions for the existence and global stability of a disease-free $T$-periodic solution. Otherwise, if $\mathcal{R}_p>1$, then a globally stable $T$-periodic solution emerges with positive coordinates. We draw a bifurcation diagram $(T,p)$ and we describe the associated bifurcations. We also find analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. In a realistic context, low vaccination coverage and intense seasonality may result in unpredictable dynamics. Previous experiments have suggested chaos in periodically-forced biological impulsive models, but no analytic proof has been given

On a Discrete SEIR Epidemic Model with Two-Doses Delayed Feedback Vaccination Control on the Susceptible

A new discrete susceptible-exposed-infectious-recovered (SEIR) epidemic model is presented subject to a feedback vaccination effort involving two doses. Both vaccination doses, which are subject to a non-necessarily identical effectiveness, are administrated by respecting a certain mutual delay interval, and their immunity effect is registered after a certain delay since the second dose. The delays and the efficacies of the doses are parameters, which can be fixed in the model for each concrete experimentation. The disease-free equilibrium point is characterized as well as its stability properties, while it is seen that no endemic equilibrium point exists. The exposed subpopulation is supposed to be infective eventually, under a distinct transmission rate of that of the infectious subpopulation. Some simulation examples are presented by using disease parameterizations of the COVID-19 pandemic under vaccination efforts requiring two doses.This research was funded by MCIU/AEI/FEDER, UE, grant number RTI2018-094336-B-I00; Spanish Institute of Health Carlos III, grant number COV 20/01213 and Basque Government, grant number IT1207-19. The APC was funded by MCIU/AEI/FEDER, UE

On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.The work has been funded by Grant RTI2018-094336-B-I00 from MCIU/AEI/FEDER, UE; by Grant IT1207-19, by the Basque Government and by Grant COV 20/01213 from Spanish Institute of Health Carlos III

Global dynamics of a state-dependent feedback control system

The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order-k (k ≥ 3) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control

A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications

[EN] This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.Spanish Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Generalitat Valenciana, Grant/Award Number: APOSTD/2019/128; Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCortés, J.; El-Labany, S.; Navarro-Quiles, A.; Selim, MM.; Slama, H. (2020). A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications. Mathematical Methods in the Applied Sciences. 43(14):8204-8222. https://doi.org/10.1002/mma.6482S820482224314Hamra, G., MacLehose, R., & Richardson, D. (2013). Markov Chain Monte Carlo: an introduction for epidemiologists. International Journal of Epidemiology, 42(2), 627-634. doi:10.1093/ije/dyt043Becker, N. (1981). A General Chain Binomial Model for Infectious Diseases. Biometrics, 37(2), 251. doi:10.2307/2530415Allen, L. J. S. (2010). An Introduction to Stochastic Processes with Applications to Biology. doi:10.1201/b12537Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599-653. doi:10.1137/s0036144500371907Brauer, F., & Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. doi:10.1007/978-1-4757-3516-1Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Some results about randomized binary Markov chains: theory, computing and applications. International Journal of Computer Mathematics, 97(1-2), 141-156. doi:10.1080/00207160.2018.1440290Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Slama, H., Hussein, A., El-Bedwhey, N. A., & Selim, M. M. (2019). An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique. Applied Mathematics and Computation, 361, 144-156. doi:10.1016/j.amc.2019.05.019Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Van den Driessche, P. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3), 288-303. doi:10.1016/j.idm.2017.06.002Heffernan, J. ., Smith, R. ., & Wahl, L. . (2005). Perspectives on the basic reproductive ratio. Journal of The Royal Society Interface, 2(4), 281-293. doi:10.1098/rsif.2005.0042Khalil, K. M., Abdel-Aziz, M., Nazmy, T. T., & Salem, A.-B. M. (2012). An Agent-Based Modeling for Pandemic Influenza in Egypt. Intelligent Systems Reference Library, 205-218. doi:10.1007/978-3-642-25755-1_1

On the equivalence of the integral and differential Bellman equations in impulse control problems

When solving optimal impulse control problems, one can use the dynamic programming approach in two different ways: at each time moment, one can make the decision whether to apply a particular type of impulse, leading to the instantaneous change of the state, or apply no impulses at all; or, otherwise, one can plan an impulse after a certain interval, so that the length of that interval is to be optimized along with the type of that impulse. The first method leads to the differential Bellman equation, while the second method leads to the integral Bellman equation. The target of the current article is to prove the equivalence of those Bellman equations. Firstly, we prove that, for the simple deterministic optimal stopping problem, the equations in the integral and differential form are equivalent under very mild conditions. Here, the impulse means that the uncontrolled process is stopped, i.e., sent to the so called cemetery. After that, the obtained result immediately implies the similar equivalence of the Bellman equations for other models of optimal impulse control. Those include abstract dynamical systems, controlled ordinary differential equations, piece-wise deterministic Markov processes and continuous-time Markov decision processes.publishe

Optimal Lockdown Policies driven by Socioeconomic Costs

none4sìIn this research paper we modify a classical SIR model to better adapt to the dynamics of COVID-19, that is we propose the heterogeneous SQAIRD model where COVID-19 spreads over a population of economic agents, namely: the elderly, adults and young people. We then compute and simulate an optimal control problem faced by a Government, where its objective is to minimize the costs generated by the pandemics using as control a compulsory quarantine measure (that is, a lockdown). We first analyze the problem from a theoretical perspective, claiming that different lockdown policies (total lockdown, no lockdown or partial lockdown) may justified by different cost (concave or convex) structures of the economies. We then focus on a particular cost structure (convex costs) and we simulate a targeted optimal policy vs. a uniform optimal policy, by dividing the whole population in three demographic groups (young, adults and old). We also simulate the dynamic of the pandemic with no policy implemented. Simulations highlighted the fact that: a) a policy of lockdown is always better than the laissez faire policy, because it limits the costs that the pandemic generates in an uncontrolled situation; b) a targeted policy based on age of the individuals outperforms a uniform policy in terms of costs that it generates, being a targeted policy less costly and equally effective in the control of the pandemic.openElena Gubar, Laura Policardo, Edgar J. Sanchez Carrera, Vladislav TaynitskiyGubar, Elena; Policardo, Laura; Sanchez Carrera, Edgar J.; Taynitskiy, Vladisla

Three Essays on Individuals’ Vulnerability to Security Attacks in Online Social Networks: Factors and Behaviors

With increasing reliance on the Internet, the use of online social networks (OSNs) for communication has grown rapidly. OSN platforms are used to share information and communicate with friends and family. However, these platforms can pose serious security threats to users. In spite of the extent of such security threats and resulting damages, little is known about factors associated with individuals’ vulnerability to online security attacks. We address this gap in the following three essays. Essay 1 draws on a synthesis of the epidemic theory in infectious disease epidemiology with the social capital theory to conceptualize factors that contribute to an individual’s role in security threat propagation in OSN. To test the model, we collected data and created a network of hacked individuals over three months from Twitter. The final hacked network consists of over 8000 individual users. Using this data set, we derived individual’s factors measuring threat propagation efficacy and threat vulnerability. The dependent variables were defined based on the concept of epidemic theory in disease propagation. The independent variables are measured based on the social capital theory. We use the regression method for data analysis. The results of this study uncover factors that have significant impact on threat propagation efficacy and threat vulnerability. We discuss the novel theoretical and managerial contributions of this work. Essay 2 explores the role of individuals’ interests in their threat vulnerability in OSNs. In OSNs, individuals follow social pages and post contents that can easily reveal their topics of interest. Prior studies show high exposure of individuals to topics of interest can decrease individuals’ ability to evaluate the risks associated with their interests. This gives attackers a chance to target people based on what they are interested in. However, interest-based vulnerability is not just a risk factor for individuals themselves. Research has reported that similar interests lead to friendship and individuals share similar interests with their friends. This similarity can increase trust among friends and makes individuals more vulnerable to security threat coming from their friends’ behaviors. Despite the potential importance of interest in the propagation of online security attacks online, the literature on this topic is scarce. To address this gap, we capture individuals’ interests in OSN and identify the association between individuals’ interests and their vulnerability to online security threats. The theoretical foundation of this work is a synthesis of dual-system theory and the theory of homophily. Communities of interest in OSN were detected using a known algorithm. We test our model using the data set and social network of hacked individuals from Essay 1. We used this network to collect additional data about individuals’ interests in OSN. The results determine communities of interests which were associated with individuals’ online threat vulnerability. Moreover, our findings reveal that similarities of interest among individuals and their friends play a role in individuals’ threat vulnerability in OSN. We discuss the novel theoretical and empirical contributions of this work. Essay 3 examines the role addiction to OSNs plays in individuals’ security perceptions and behaviors. Despite the prevalence of problematic use of OSNs and the possibility of addiction to these platforms, little is known about the functionalities of brain systems of users who suffer from OSN addiction and their online security perception and behaviors. In addressing these gaps, we have developed the Online addiction & security behaviors (OASB) theory by synthesizing dual-system theory and extended protection motivation theory (PMT). We collected data through an online survey. The results indicate that OSN addiction is rooted in the individual’s brain systems. For the OSN addicted, there is a strong cognitive-emotional preoccupation with using OSN. Our findings also reveal the positive and significant impact of OSN addiction on perceived susceptibility to and severity of online security threats. Moreover, our results show the negative association between OSN addiction and perceived self-efficacy. We discuss the theoretical and practical implications of this work