Figure 2 - uploaded by Julien Arino
Content may be subject to copyright.
Source publication
Delay differential equation models of disease transmission arise due to the assumption of constant sojourn times in certain epidemiological states. The notions of sojourn times and survival functions are briefly presented and illustrated by a simple SIS model with a general form for the survival function in the infective state. A step function surv...
Contexts in source publication
Context 1
... model, which is similar to that in [2], has the transfer diagram shown in Figure 2. The number of individuals in the susceptible, infective and vaccinated states are given by S(t), I(t), V (t), respectively. ...
Context 2
... the initial susceptible and infective proportions be S(0) > 0, I(0) > 0 and let V 0 (t) be the proportion of individuals who are initially in the vaccinated state and for whom the vaccine is still effective at time t. With the above assumptions, the following integro-differential system describes the model depicted in Figure 2. ...
Similar publications
Landing-gear noise is predicted by conducting a Detached-Eddy Simulation of the turbulent region, and extrapolating the sound to the far field using the Ffowcs-Williams-Hawkings (FWH) equation. We use the Rudimentary Landing Gear (RLG), as well as two simpler configurations. Our past results conflicted with expectations based on Curle's approximati...
Citations
... It is common to use differential equations to construct these models. The most well-known models are based on ordinary differential equations (ODEs), partial differential equations (PDEs), and delay differential equations (DDEs) [2][3][4][5]. The SIR (susceptible-infected-recovered) model is by far the most commonly used mathematical model based on differential equations [1]. ...
... In this way, the stability of these steady states is shown. The Neumann boundary conditions are given in (2) and indicate that ...
... there is only one endemic state E e (S e , I e , A e , R e ) of system (1) with the conditions given in (2), which is ...
The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible (S), infected (I), asymptomatic (A), and recovered (R). An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic (Ee) and disease-free (E0) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R0, which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior.
... With an average latent period of 1/α, the progression to an infectious state in E q B depends on the amount of time spent in E A before traveling to P B and the duration of quarantine. Let e(t, a) be the density of E q B (t) (Alexander et al. 2008;Arino and Van Den Driessche 2006;Robertson et al. 2018) at time t with respect to a, representing the time since entering quarantine. Thus, ...
Travel restrictions, while delaying the spread of an emerging disease from the source, could inflict substantial socioeconomic burden. Travel-related policies, such as quarantine and testing of travelers, may be considered as alternative strategies to mitigate the negative impact of travel bans. We developed a meta-population, delay-differential model to evaluate a strategy that combines testing of travelers prior to departure from the source of infection with quarantine and testing at exit from quarantine in the destination population. Our results, based on early parameter estimates of SARS-CoV-2 infection, indicate that testing travelers at exit from quarantine is more effective in delaying case importation than testing them before departure or upon arrival. We show that a 1-day quarantine with an exit test could outperform a longer, 3-day quarantine without testing in delaying the outbreak peak. Rapid, large-scale testing capacities with short turnaround times provide important means of detecting infectious cases and reducing case importation, while shortening quarantine duration for travelers at destination.
... A wide range of literary studies have been published on COVID-19 models, including review papers, special issues, and books [9][10][11]; however, deterministic modeling of COVID-19 and vaccination is very limited [6]. In mathematical epidemic models, time delays are investigated to better understand and characterize the dynamics of infectious diseases; for example, see [12][13][14][15]. Further, time delays may lead to periodic solutions via the Hopf bifurcation; see, e.g., Reference [16] and references therein. ...
In this study, we provide a fractional-order mathematical model that considers the effect of vaccination on COVID-19 spread dynamics. The model accounts for the latent period of intervention strategies by incorporating a time delay τ. A basic reproduction number, R0, is determined for the model, and prerequisites for endemic equilibrium are discussed. The model’s endemic equilibrium point also exhibits local asymptotic stability (under certain conditions), and a Hopf bifurcation condition is established. Different scenarios of vaccination efficacy are simulated. As a result of the vaccination efforts, the number of deaths and those affected have decreased. COVID-19 may not be effectively controlled by vaccination alone. To control infections, several non-pharmacological interventions are necessary. Based on numerical simulations and fitting to real observations, the theoretical results are proven to be effective.
... The introduction of time delays to mathematical epidemic models has been studied in order to better understand and describe the transmission dynamics of infectious diseases; see, e.g., [14,[19][20][21]. Moreover, time delays may have an important effect on the stability of the equilibrium points, leading, for example, to periodic solutions by Hopf bifurcation; see, e.g., [22] and references cited therein. ...
... if R 0 satisfies relations (14) and (15) with condition (19), then we have ...
We analyze mathematical models for COVID-19 with discrete time delays and vaccination. Sufficient conditions for the local stability of the endemic and disease-free equilibrium points are proved for any positive time delay. The stability results are illustrated through numerical simulations performed in MATLAB.
... The introduction of time delays to mathematical epidemic models has been studied in order to better understand and describe the transmission dynamics of infectious diseases; see, e.g., [14,[19][20][21]. Moreover, time delays may have an important effect on the stability of the equilibrium points, leading, for example, to periodic solutions by Hopf bifurcation; see, e.g., [22] and references cited therein. ...
... if R 0 satisfies relations (14) and (15) with condition (19), then we have ...
We analyze mathematical models for COVID-19 with discrete time delays and vaccination. Sufficient conditions for the local stability of the endemic and disease-free equilibrium points are proved for any positive time delay. The stability results are illustrated through numerical simulations performed in MATLAB.
... [29,30,39,43], and population dynamics or epidemics in mathematical biology, e.g. [4,8,35,37,38,42] (see [28,34] for further applications). ...
In recent years we provided numerical methods based on pseudospectral collocation for computing the Floquet multipliers of different types of delay equations, with the goal of studying the stability of their periodic solutions. The latest work of the series concerns the extension of these methods to a piecewise approach, in order to take the properties of numerically computed solutions into account. In this chapter we describe the MATLAB implementation of this method and provide practical usage examples.
... We can equivalently express Equation 1 as a renewal equation (Arino & Driessche, 2006), namely In Equation 2, I(u) gives the density of infected individuals at some previous time u, u < t. The function e − (t−u) defines the pathogen survival function in the environment, assuming that the pathogen decays at a constant rate . ...
The ongoing explosion of fine‐resolution movement data in animal systems provides a unique opportunity to empirically quantify spatial, temporal and individual variation in transmission risk and improve our ability to forecast disease outbreaks. However, we lack a generalizable model that can leverage movement data to quantify transmission risk and how it affects pathogen invasion and persistence on heterogeneous landscapes. We developed a flexible model ‘Movement‐driven modelling of spatio‐temporal infection risk’ (MoveSTIR) that leverages diverse data on animal movement to derive metrics of direct and indirect contact by decomposing transmission into constituent processes of contact formation and duration and pathogen deposition and acquisition. We use MoveSTIR to demonstrate that ignoring fine‐scale animal movements on actual landscapes can mis‐characterize transmission risk and epidemiological dynamics. MoveSTIR unifies previous work on epidemiological contact networks and can address applied and theoretical questions at the nexus of movement and disease ecology. The ongoing explosion of fine‐resolution movement data in animal systems provides a unique opportunity to empirically quantify spatio‐temporal variation in transmission risk and improve our ability to forecast disease outbreaks. We develop a novel, flexible model ‘Movement‐driven modeling of spatio‐temporal infection risk’ (MoveSTIR) that leverages diverse data on animal movement to quantify potential transmission risk. Using MoveSTIR, we demonstrate that ignoring fine‐scale animal movements can underestimate transmission risk and mis‐characterize epidemiological dynamics.
... These DDE compartment models simulate a biologically appropriate constant incubation period. In contrast, the exposed compartment in the SEIR and SEIRS models implies an exponential distribution of incubation time (Huang et al. 2010;Arino and van den Driessche 2006). The delay from infection to infectious is typically included with the force of infection term as I(t − )S(t) or I(t)S(t − ). ...
... The delay from infection to infectious is typically included with the force of infection term as I(t − )S(t) or I(t)S(t − ). Hethcote et al. introduced alternative delay models (Arino and van den Driessche 2006;Hethcote and Tudor 1980;Hethcote et al. 1981aHethcote et al. , b, 1989Hethcote and van den Driessche 1995). ...
The susceptible-transmissible-removed (STR) model is a deterministic compartment model, based on the susceptible-infected-removed (SIR) prototype. The STR replaces 2 SIR assumptions. SIR assumes that the emigration rate (due to death or recovery) is directly proportional to the infected compartment’s size. The STR replaces this assumption with the biologically appropriate assumption that the emigration rate is the same as the immigration rate one infected period ago. This results in a unique delay differential equation epidemic model with the delay equal to the infected period. Hamer’s mass action law for epidemiology is modified to resemble its chemistry precursor—the law of mass action. Constructing the model for an isolated population that exists on a surface bounded by the extent of the population’s movements permits compartment density to replace compartment size. The STR reduces to a SIR model in a timescale that negates the delay—the transmissible timescale. This establishes that the SIR model applies to an isolated population in the disease’s transmissible timescale. Cyclical social interactions will define a rhythmic timescale. It is demonstrated that the geometric mean maps transmissible timescale properties to their rhythmic timescale equivalents. This mapping defines the hybrid incidence (HI). The model validation demonstrates that the HI-STR can be constructed directly from the disease’s transmission dynamics. The basic reproduction number (R0) is an epidemic impact property. The HI-STR model predicts that R0∝ρnB where ρn is the population density, and B is the ratio of time increments in the transmissible- and rhythmic timescales. The model is validated by experimentally verifying the relationship. R0’s dependence on ρn is demonstrated for droplet-spread SARS in Asian cities, aerosol-spread measles in Europe and non-airborne Ebola in Africa.
... The introduction of time delays to mathematical epidemic models has been studied in order to better understand and describe the transmission dynamics of infectious diseases; see, e.g., [14,[19][20][21]. Moreover, time delays may have an important effect on the stability of the equilibrium points, leading, for example, to periodic solutions by Hopf bifurcation; see, e.g., [22] and references cited therein. ...
... if R 0 satisfies relations (14) and (15) with condition (19), then we have ...
... These DDE compartment models simulate a biologically appropriate constant incubation period. In contrast, the exposed compartment in the SEIR and SEIRS models implies an exponential distribution of incubation time [37,39]. The delay from infection to infectious is typically included with the force of infection term as βI(t − τ )S(t) or βI(t)S(t − τ ). ...
... The delay from infection to infectious is typically included with the force of infection term as βI(t − τ )S(t) or βI(t)S(t − τ ). Hethcote et al introduced alternative delay models [39][40][41][42][43][44]. ...
... Conjecturing isolation at day 3 of the prodrome, ∆τ is 3 days; δt = 1 day and B = 3. Substituting the latter into (37), for measles: Table 4 documents the experimental R 0 for 5 countries at 8 historical periods [130]. Figure 2 demonstrates the linear relationship predicted by (39). increased R 2 is likely due to the several methods used by several investigators to calculate R 0 for measles. ...
The susceptible-transmissible-removed (STR) model is a deterministic compartment model, based on the susceptible-infected-removed (SIR) prototype. The STR replaces 2 SIR assumptions. SIR assumes that the emigration rate (due to death or recovery) is directly proportional to the infected compartment's size. The STR replaces this assumption with the biologically appropriate assumption that the emigration rate is the same as the immigration rate one infected period ago. This results in a unique delay differential equation epidemic model with the delay equal to the infected period. Hamer's mass action law for epidemiology is modified to resemble its chemistry precursor -- the law of mass action. Constructing the model for an isolated population that exists on a surface bounded by the extent of the population's movements permits compartment density to replace compartment size. The STR reduces to a SIR model in a timescale that negates the delay -- the transmissible timescale. This establishes that the SIR model applies to an isolated population in the disease's transmissible timescale. Cyclical social interactions will define a rhythmic timescale. It is demonstrated that the geometric mean maps transmissible timescale properties to their rhythmic timescale equivalents. This mapping defines the hybrid incidence (HI). The model validation demonstrates that the HI-STR can be constructed directly from the disease's transmission dynamics. The basic reproduction number (R0) is an epidemic impact property. The HI-STR model predicts that R0 \propto \sqrt[B]{pn} where pn is the population density, and B is the ratio of time increments in the transmissible- and rhythmic timescales. The model is validated by experimentally verifying the relationship. R0's dependence on pn is demonstrated for droplet-spread SARS in Asian cities, aerosol-spread measles in Europe and non-airborne Ebola in Africa.
