The Law of Mass Action in Epidemiology: II

Infection risk in cable cars and other enclosed spaces

As virus-laden aerosols can accumulate and remain suspended for hours in insufficiently ventilated enclosed spaces, indoor environments can heavily contribute to the spreading of airborne infections. In the COVID-19 pandemics, the role possibly played by cable cars has attracted media attention following several outbreaks in ski resort. To assess the real risk of infection, we experimentally characterize the natural ventilation in cable cars and develop a general stochastic model of infection in an arbitrary indoor space that accounts for the epidemiological situation, the virological parameters, and the indoor characteristics (ventilation rate and occupant number density). As a results of the high air exchange rate (we measured up to 180 air changes per hour) and the relatively short duration of the journey, the infection probability in cable cars traveling with open windows is remarkably lower than in other enclosed spaces such as aircraft cabins, train cars, offices, classrooms, and dining rooms. Accounting for the typical duration of the stay, the probability of infection during a cable-car ride is lower by two to three orders of magnitude than in the other examples considered (the highest risk being estimated in case of a private gathering in a poorly ventilated room). For most practical purposes, the infection probability can be approximated by the inhaled viral dose, which provides an upper bound and allows a simple comparison between different indoor situations once the air exchange rate and the occupant number density are known. Our approach and findings are applicable to any indoor space in which the viral transmission is predominately airborne and the air is well mixed.

An epidemic model for non-first-order transmission kinetics

Compartmental models in epidemiology characterize the spread of an infectious disease by formulating ordinary differential equations to quantify the rate of disease progression through subpopulations defined by the Susceptible-Infectious-Removed (SIR) scheme. The classic rate law central to the SIR compartmental models assumes that the rate of transmission is first order regarding the infectious agent. The current study demonstrates that this assumption does not always hold and provides a theoretical rationale for a more general rate law, inspired by mixed-order chemical reaction kinetics, leading to a modified mathematical model for non-first-order kinetics. Using observed data from 127 countries during the initial phase of the COVID-19 pandemic, we demonstrated that the modified epidemic model is more realistic than the classic, first-order-kinetics based model. We discuss two coefficients associated with the modified epidemic model: transmission rate constant k and transmission reaction order n. While k finds utility in evaluating the effectiveness of control measures due to its responsiveness to external factors, n is more closely related to the intrinsic properties of the epidemic agent, including reproductive ability. The rate law for the modified compartmental SIR model is generally applicable to mixed-kinetics disease transmission with heterogeneous transmission mechanisms. By analyzing early-stage epidemic data, this modified epidemic model may be instrumental in providing timely insight into a new epidemic and developing control measures at the beginning of an outbreak.

Mathematical modeling of dengue epidemic: control methods and vaccination strategies

Dengue is, in terms of death and economic cost, one of the most important infectious diseases in the world. So, its mathematical modeling can be a valuable tool to help us to understand the dynamics of the disease and to infer about its spreading by the proposition of control methods. In this paper, control strategies, which aim to eliminate the Aedes aegypti mosquito, as well as proposals for the vaccination campaign are evaluated. In our mathematical model, the mechanical control is accomplished through the environmental support capacity affected by a discrete function that represents the removal of breedings. Chemical control is carried out using insecticide and larvicide. The efficiency of vaccination is studied through the transfer of a fraction of individuals, proportional to the vaccination rate, from the susceptible to the recovered compartments. Our major find is that the dengue fever epidemic is only eradicated with the use of an immunizing vaccine because control measures, directed against its vector, are not enough to halt the disease spreading. Even when the infected mosquitoes are eliminated from the system, the susceptible ones are still present, and infected humans cause dengue fever to reappear in the human population.

Dynamics of an SVEIRS Epidemic Model with Vaccination and Saturated Incidence Rate

Measles and influenza are two major diseases-caused an epidemic in India. Therefore, in this paper, a SVEIRS epidemic mathematical model for measles and influenza is proposed and analyzed, where pre and post vaccinations are considered as control strategies with waning natural, vaccine-induced immunity and saturation incidence rate. The dissection of the proposed model is conferred in terms of the associated reproduction number R v , which is determined by the next-generation approach and obtained that ifR v ≤ 1 , the disease-free equilibrium exists and it is locally as well as globally asymptotically stable. Further forR v > 1 , a unique endemic equilibrium exists and it is also locally as well as globally asymptotically stable under certain conditions, which shows the prevalence and persistence of the disease in the population.

Dynamics of SVEIS epidemic model with distinct incidence

In this paper, we study the global dynamics of a SVEIS epidemic model with distinct incidence for exposed and infectives. The model is analyzed for stability and bifurcation behavior. To account for the realistic phenomenon of non-homogeneous mixing, the effect of diffusion on different population subclasses is considered. The diffusive model is analyzed using matrix stability theory and conditions for Turing bifurcation are derived. Numerical simulations support our analytical results on the dynamic behavior of the model.

An SEIV Epidemic Model for Childhood Diseases with Partial Permanent Immunity

An SEIV epidemic model for childhood disease with partial permanent immunity is studied. The basic reproduction number R 0 has been worked out. The local and global asymptotical stability analysis of the equilibria are performed, respectively. Furthermore, if we take the treated rate τ as the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium when τ passes through a critical value. Finally, some numerical simulations are given to support our analytic results.

DYNAMICS OF A DELAYED EPIDEMIOLOGICAL MODEL WITH NONLINEAR INCIDENCE: THE ROLE OF INFECTED INCIDENCE FRACTION

We present and analyze an epidemiological model containing Susceptible (S(t)) and Infected (I(t)) populations. The incidence rate is assumed to be nonlinear in the infected fraction (Ip(t)) as well as the susceptible fraction (Sq(t)). The dynamical behavior of the system is investigated from the point of view of stability and bifurcation. To model the recovery time of infected populations, a recovery delay, both in distributed and discrete form is introduced. In all the cases, it is shown that the infected incidence fraction p plays a vital role in controlling the dynamical behavior of the system. Numerical simulations are performed to justify the analytical findings.

Dose-dependent infection rates of parasites produce the Allee effect in epidemiology

In many epidemiological models of microparasitic infections it is assumed that the infection process is governed by the mass-action principle, i.e. that the infection rate per host and per parasite is a constant. Furthermore, the parasite-induced host mortality (parasite virulence) and the reproduction rate of the parasite are often assumed to be independent of the infecting parasite dose. However, there is empirical evidence against those three assumptions: the infection rate per host is often found to be a sigmoidal rather than a linear function of the parasite dose to which it is exposed; and the lifespan of infected hosts as well as the reproduction rate of the parasite are often negatively correlated with the parasite dose. Here, we incorporate dose dependences into the standard modelling framework for microparasitic infections, and draw conclusions on the resulting dynamics. Our model displays an Allee effect that is characterized by an invasion threshold for the parasite. Furthermore, in contrast to standard epidemiological models a parasite strain needs to have a basic reproductive rate that is substantially greater than 1 to establish an infection. Thus, the conditions for successful invasion of the parasite are more restrictive than in mass-action infection models. The analysis further suggests that negative correlations of the parasite dose with host lifespan and the parasite reproduction rate helps the parasite to overcome the invasion constraints of the Allee-type dynamics.

Dynamical behavior of epidemiological models with nonlinear incidence rates

Epidemiological models with nonlinear incidence rates lambda IpSq show a much wider range of dynamical behaviors than do those with bilinear incidence rates lambda IS. These behaviors are determined mainly by p and lambda, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models

When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.