Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

Comparison of homemade mask designs based on calculated infection risk, using actual COVID-19 infection scenarios

During pandemics such as COVID-19, shortages of approved respirators necessitate the use of alternative masks, including homemade designs. The effectiveness of the masks is often quantified in terms of the ability to filter particles. However, to formulate public policy the efficacy of the mask in reducing the risk of infection for a given population is considerably more useful than its filtration efficiency (FE). The effect of the mask on the infection profile is complicated to estimate as it depends strongly upon the behavior of the affected population. A recently introduced tool known as the dynamic-spread model is well suited for performing population-specific risk assessment. The dynamic-spread model was used to simulate the performance of a variety of mask designs (all used for source control only) in different COVID-19 scenarios. The efficacy of different masks was found to be highly scenario dependent. Switching from a cotton T-shirt of 8% FE to a 3-layer cotton-gauze-cotton mask of 44% FE resulted in a decrease in number of new infections of about 30% in the New York State scenario and 60% in the Harris County, Texas scenario. The results are valuable to policy makers for quantifying the impact upon the infection rate for different intervention strategies, e.g., investing resources to provide the community with higher-filtration masks.

A non-standard finite-difference-method for a non-autonomous epidemiological model: analysis, parameter identification and applications

In this work, we propose a new non-standard finite-difference-method for the numerical solution of the time-continuous non-autonomous susceptible-infected-recovered model. For our time-discrete numerical solution algorithm, we prove preservation of non-negativity and show that the unique time-discrete solution converges linearly towards the time-continuous unique solution. In addition to that, we introduce a parameter identification algorithm for the susceptible-infected-recovered model. Finally, we provide two numerical examples to stress our theoretical findings.

Comparison of effectiveness of enhanced infection countermeasures in different scenarios, using a dynamic-spread-function model

When formulating countermeasures to epidemics such as those generated by COVID-19, estimates of the benefits of a given intervention for a specific population are highly beneficial to policy makers. A recently introduced tool, known as the "dynamic-spread" SIR model, can perform population-specific risk assessment. Behavior is quantified by the dynamic-spread function, which includes the mechanisms of droplet reduction using facemasks and transmission control due to social distancing. The spread function is calibrated using infection data from a previous wave of the infection, or other data felt to accurately represent the population behaviors. The model then computes the rate of spread of the infection for different hypothesized interventions, over the time window for the calibration data. The dynamic-spread model was used to assess the benefit of three enhanced intervention strategies - increased mask filtration efficiency, higher mask compliance, and elevated social distancing - in four COVID-19 scenarios occurring in 2020: the first wave (i.e. until the first peak in numbers of new infections) in New York City; the first wave in New York State; the spread aboard the Diamond Princess Cruise Liner; and the peak occurring after re-opening in Harris County, Texas. Differences in the efficacy of the same intervention in the different scenarios were estimated. As an example, when the average outward filtration efficiency for facemasks worn in New York City was increased from an assumed baseline of 67% to a hypothesized 90%, the calculated peak number of new infections per day decreased by 40%. For the same baseline and hypothesized filtration efficiencies aboard the Diamond Princess Cruise liner, the calculated peak number of new infections per day decreased by about 15%. An important factor contributing to the difference between the two scenarios is the lower mask compliance (derivable from the spread function) aboard the Diamond Princess.

Optimal strategic pandemic control: human mobility and travel restriction

This paper presents a model for finding optimal pandemic control policy considering cross-region human mobility. We extend the baseline susceptible-infectious-recovered (SIR) epidemiology model by including the net human mobility from a severely-impacted region to a mildly-affected region. The strategic optimal mitigation policy combining testing and lockdown in each region is then obtained with the goal of minimizing economic cost under the constraint of limited resources. We parametrize the model using the data of the COVID-19 pandemic and show that the optimal response strategy and mitigation outcome greatly rely on the mitigation duration, available resources, and cross-region human mobility. Furthermore, we discuss the economic impact of travel restriction policies through a quantitative analysis.

Optimization strategies of human mobility during the COVID-19 pandemic: A review

The impact of the ongoing COVID-19 pandemic is being felt in all spheres of our lives - cutting across the boundaries of nation, wealth, religions or race. From the time of the first detection of infection among the public, the virus spread though almost all the countries in the world in a short period of time. With humans as the carrier of the virus, the spreading process necessarily depends on the their mobility after being infected. Not only in the primary spreading process, but also in the subsequent spreading of the mutant variants, human mobility plays a central role in the dynamics. Therefore, on one hand travel restrictions of varying degree were imposed and are still being imposed, by various countries both nationally and internationally. On the other hand, these restrictions have severe fall outs in businesses and livelihood in general. Therefore, it is an optimization process, exercised on a global scale, with multiple changing variables. Here we review the techniques and their effects on optimization or proposed optimizations of human mobility in different scales, carried out by data driven, machine learning and model approaches.

SIR model-based verification of effect of COVID-19 Contact-Confirming Application (COCOA) on reducing infectors in Japan

As of April 2021, the coronavirus disease (COVID-19) continues to spread in Japan. To overcome COVID-19, the Ministry of Health, Labor, and Welfare of the Japanese government developed and released the COVID-19 Contact-Confirming Application (COCOA) on June 19, 2020. COCOA users can know whether they have come into contact with infectors. If persons who receive a contact notification through COCOA undertake self-quarantine, the number of infectors in Japan will decrease. However, the effectiveness of COCOA in reducing the number of infectors depends on the usage rate of COCOA, the rate of fulfillment of contact condition, the rate of undergoing the reverse transcription polymerase chain reaction (RT-PCR) test, the false negative rate of the RT-PCR test, the rate of infection registration, and the self-quarantine rate. Therefore, we developed a Susceptible-Infected-Removed (SIR) model to estimate the effectiveness of COCOA. In this paper, we introduce the SIR model and report the simulation results for different scenarios that were assumed for Japan.

Poisson integral type quarantine in a stochastic SIR system

We propose a SIR system that includes a Poisson measure term to model the quarantine of infected individuals. An inequality concerning the term representing the transmission rate is given to establish the stochastic stability of the disease free equilibrium. It is further shown that if R₀ > 1 then the long-run behavior the system will reside within a neighborhood of the equilibrium in the underlying deterministic version of this system.

Feedback control problem of an SIR epidemic model based on the Hamilton-Jacobi-Bellman equation

We consider a feedback control problem of a susceptible-infective-recovered (SIR) model to design an efficient vaccination strategy for influenza outbreaks. We formulate an optimal control problem that minimizes the number of people who become infected, as well as the costs of vaccination. A feedback methodology based on the Hamilton-Jacobi-Bellman (HJB) equation is introduced to derive the control function. We describe the viscosity solution, which is an approximation solution of the HJB equation. A successive approximation method combined with the upwind finite difference method is discussed to find the viscosity solution. The numerical simulations show that feedback control can help determine the vaccine policy for any combination of susceptible individuals and infectious individuals. We also verify that feedback control can immediately reflect changes in the number of susceptible and infectious individuals.

Mathematical analysis of an SIR respiratory infection model with sex and gender disparity: special reference to influenza A

The aim of this work is to study the impact of sex and gender disparity on the overall dynamics of influenza A virus infection and to explore the direct and indirect effect of influenza A mass vaccination. To this end, a deterministic SIR model has been formulated and throughly analysed, where the equilibrium and stability analyses have been explored. The impact of sex disparity (i.e., disparity in susceptibility and in recovery rate between females and males) on the disease outcome (i.e., the basic reproduction number R₀ and the endemic prevalence of influenza in females and males) has been investigated. Mathematical and numerical analyses show that sex and gender disparities affect on the severity as well as the endemic prevalence of infection in both sexes. The analysis shows further that the efficacy of the vaccine for both sexes (e1&e2) and the response of the gender to mass-vaccination campaigns Ψ play a crucial role in influenza A containment and elimination process, where they impact significantly on the protection ratio as well as on the direct, indirect and total effect of vaccination on the burden of infection.