SIR-type Model Research Articles

A Numerical Investigation of the SIR-type Model with a Susceptible R Compartment

A Numerical Investigation of the SIR-type Model with a Susceptible R Compartment

A retrospective assessment of forecasting the peak of the SARS-CoV-2 Omicron BA.1 wave in England.

We discuss the invasion of the Omicron BA.1 variant into England as a paradigm for real-time model fitting and projection. Here we use a mixture of simple SIR-type models, analysis of the early data and a more complex age-structure model fit to the outbreak to understand the dynamics. In particular, we highlight that early data shows that the invading Omicron variant had a substantial growth advantage over the resident Delta variant. However, early data does not allow us to reliably infer other key epidemiological parameters-such as generation time and severity-which influence the expected peak hospital numbers. With more complete epidemic data from January 2022 are we able to capture the true scale of the epidemic in terms of both infections and hospital admissions, driven by different infection characteristics of Omicron compared to Delta and a substantial shift in estimated precautionary behaviour during December. This work highlights the challenges of real time forecasting, in a rapidly changing environment with limited information on the variant's epidemiological characteristics.

Optimal treatment and stochastic stability on a fractional-order epidemic model incorporating media awareness

Infectious diseases have become a severe concern in our society in recent times. In this respect, conducting a qualitative investigation of infectious disease systems is very crucial. Nowadays, developing a mathematical model and its analysis for disease control has been very popular. Our present study has formulated a SIR-type epidemic model by considering a saturated incidence function with media impact. To account for the influence of memory, we have modified the model by employing a system of fractional-order differential equations of Caputo’s notion. We have determined all the system equilibrium points and discussed their stability criteria based on the basic reproduction number (R0) as the threshold parameter. It is observed that for the threshold R0<1, disease-free equilibrium becomes both locally and globally asymptotically stable. A transcritical bifurcation occurs at the threshold R0=1. Further, as the threshold R0 exceeds unity, the endemic equilibrium becomes asymptotically stable. A fractional-order optimal control problem with treatment as the control parameter is developed to reduce the impact of disease. Using Pontryagin’s Maximum Principle yields a theoretical and numerical solution to the fractional order optimal control problem. It is observed that both the media and memory effect dramatically reduces infection rates. Considering environmental variations, the stochastic stability of the endemic equilibrium point has also been studied. Sensitivity analysis is carried out to comprehend how the parameters impact the threshold value (R0). Finally, extensive numerical simulations are performed to demonstrate our theoretical examination.

On the Effects of Type II Left Censoring in Stable and Chaotic Compartmental Models for Infectious Diseases: Do Small Sample Estimates Survive Censoring?

In this paper, we discuss a selection of tools from dynamical systems and order statistics, which are most often utilized separately, and combine them into an algorithm to estimate the parameters of mathematical models for infectious diseases in the case of small sample sizes and left censoring, which is relevant in the case of rapidly evolving infectious diseases and remote populations. The proposed method relies on the analogy between survival functions and the dynamics of the susceptible compartment in SIR-type models, which are both monotone decreasing in time and are both determined by a dual variable: the hazard function in survival prediction and the number of infected people in SIR-type models. We illustrate the methodology in the case of a continuous model in the presence of noisy measurements with different distributions (Normal, Poisson, Negative Binomial) and in a discrete model, reminiscent of the Ricker map, which admits chaotic dynamics. This estimation procedure shows stable results in experiments based on a popular benchmark dataset for SIR-type models and small samples. This manuscript illustrates how classical theoretical statistical methods and dynamical systems can be merged in interesting ways to study problems ranging from more fundamental small sample situations to more complex infectious disease and survival models, with the potential that this tools can be applied in the presence of a large number of covariates and different types of censored data.

A Numerical Study of Codimension-Two Bifurcations of an SIR-Type Model for COVID-19 and Their Epidemiological Implications

We study the codimension-two bifurcations exhibited by a recently-developed SIR-type mathematical model for the spread of COVID-19, as its two main parameters —the susceptible individuals’ cautiousness level and the hospitals’ bed-occupancy rate— vary over their domains. We use AUTO to generate the model’s bifurcation diagrams near the relevant bifurcation points: two Bogdanov-Takens points and two generalised Hopf points, as well as a number of phase portraits describing the model’s orbital behaviours for various pairs of parameter values near each bifurcation point. The analysis shows that, when a backward bifurcation occurs at the basic reproduction threshold, the transition of the model’s asymptotic behaviour from endemic to disease-free takes place via an unexpectedly complex sequence of topological changes, involving the births and disappearances of not only equilibria but also limit cycles and homoclinic orbits. Epidemiologically, the analysis confirms the importance of a proper control of the values of the aforementioned parameters for a successful eradication of COVID-19. We recommend a number of strategies by which such a control may be achieved.

Well-posedness results for a new class of stochastic spatio-temporal SIR-type models driven by proportional pure-jump Lévy noise

This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semigroup theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions.

Mathematical Modeling and Georeferenced Forecasting for the COVID-19 at the State of RS, Brazil

In this contribution, we present a predictive tool developed to help in the management of the evolution of the COVID-19 pandemic situation in Rio Grande do Sul (RS) - Brazil. This tool is the result of georeferenced data analysis, mathematical modeling, and parameter calibration for the dynamics of a SIR-type model defined on a spatial structure that allows distinct subpopulations to interact, similar to the controlled distancing (A_l for l = 1, ···, 21) groupings proposed by the RS government and public health authorities. The predictive analysis, updated biweekly, provides three distinct scenarios per month (milder, average, and severe) and is made available as WebSIGs (Geographic Information System - GIS). The forecast of the average scenario for each Al group is the result of a simulation of the proposed SIRtype dynamics with calibrated parameters derived from an augmented Lagrangian maximum a posteriori estimation and data on the number of infected cases made available by the RS Health Secretariat. The milder and severe scenarios are obtained from the average scenario, with changes in the contagion rates of each Al group. When compared to the number of infections reported in each Al group, the modeling predictions for a biweekly time window (the first two weeks) were quite satisfactory, with errors ranging from 0% to 5.13%, gradually increasing over time. Therefore, we suggest a biweekly re-calibration of the parameters and corresponding forecasts as a wise strategy.

A mathematical interpretation for outbreaks of bacterial meningitis under the effect of time-dependent transmission parameters.

We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results.

On SIR-type epidemiological models and population heterogeneity effects

In this paper we elaborate on homogeneous and heterogeneous SIR-type epidemiological models. We find an unexpected correspondence between the epidemic trajectory of a transmissible disease in a homogeneous SIR-type model and radial null geodesics in the Schwarzschild spacetime. We also discuss modeling of population heterogeneity effects by considering both a one- and two-parameter gamma-distributed function for the initial susceptibility distribution, and deriving the associated herd immunity threshold. We furthermore describe how mitigation measures can be taken into account by model fitting.

PINN training using biobjective optimization: The trade-off between data loss and residual loss

Physics informed neural networks (PINNs) have proven to be an efficient tool to represent problems for which measured data are available and for which the dynamics in the data are expected to follow some physical laws. In this paper, we suggest a multiobjective perspective on the training of PINNs by treating the data loss and the residual loss as two individual objective functions in a truly biobjective optimization approach.As a showcase example, we consider COVID-19 predictions in Germany and built an extended susceptibles-infected-recovered (SIR) model with additionally considered leaky-vaccinated and hospitalized populations (SVIHR model) to model the transition rates and to predict future infections. SIR-type models are expressed by systems of ordinary differential equations (ODEs). We investigate the suitability of the generated PINN for COVID-19 predictions and compare the resulting predicted curves with those obtained by applying the method of non-standard finite differences to the system of ODEs and initial data.The approach is applicable to various systems of ODEs that define dynamical regimes. Those regimes do not need to be SIR-type models, and the corresponding underlying data sets do not have to be associated with COVID-19.

Exploring how ecological and epidemiological processes shape multi-host disease dynamics using global sensitivity analysis.

We use global sensitivity analysis (specifically, Partial Rank Correlation Coefficients) to explore the roles of ecological and epidemiological processes in shaping the temporal dynamics of a parameterized SIR-type model of two host species and an environmentally transmitted pathogen. We compute the sensitivities of disease prevalence in each host species to model parameters. Sensitivity rankings are calculated, interpreted biologically, and contrasted for cases where the pathogen is introduced into a disease-free community and cases where a second host species is introduced into an endemic single-host community. In some cases the magnitudes and dynamics of the sensitivities can be predicted only by knowing the host species' characteristics (i.e., their competitive abilities and disease competence) whereas in other cases they can be predicted by factors independent of the species' characteristics (specifically, intraspecific versus interspecific processes or a species' roles of invader versus resident). For example, when a pathogen is initially introduced into a disease-free community, disease prevalence in both hosts is more sensitive to the burst size of the first host than the second host. In comparison, disease prevalence in each host is more sensitive to its own infection rate than the infection rate of the other host species. In total, this study illustrates that global sensitivity analysis can provide useful insight into how ecological and epidemiological processes shape disease dynamics and how those effects vary across time and system conditions. Our results show that sensitivity analysis can provide quantification and direction when exploring biological hypotheses.

On the Redundancy of Birth and Death Rates in Homogeneous Epidemic SIR Models

The dynamics of fractional population sizes yi=Yi/N in homogeneous compartment models with time-dependent total population N is analyzed. Assuming constant per capita birth and death rates, the vector field Y˙i=Vi(Y) naturally projects to a vector field Fi(Y) tangent to the leaves of constant population N. A universal formula for the projected field Fi is given. In this way, in many SIR-type models with standard incidence, all demographic parameters become redundant for the dynamical system y˙i=Fi(y). They may be put to zero by shifting the remaining parameters appropriately. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to have been covered already by papers 20 years ago.

Switched NMPC for epidemiological and social-economic control objectives in SIR-type systems

Several optimal control strategies have recently been developed to minimize the infected peak prevalence or the epidemic final size. Although these two indexes are critical to assess any control policy tending to mitigate an epidemic by means of non-pharmaceutical measures, they are usually considered separately and, in general, no consensus has been reached about how to simultaneously handle them in a simple and realistic way (i.e., accounting for the limitations in the control actions, avoiding new cycles of infections or reboundings, considering side effects, etc.) Here, based on a theoretical dynamical analysis of SIR-type models, a realistic nonlinear model predictive control strategy is proposed. Apart from minimizing the epidemic final size and keeping the infected peak prevalence under an established value, the controller accounts for feedback uncertainty and different actuator constraints, such as a limited number of social distancing policies, which may remain active for a minimal and a maximal time interval. Several simulations considering different SIR-type models illustrate the benefits of the proposal.

Mathematical modeling of infectious diseases on the example of Astana city

Infectious diseases are caused by pathogens («microbes»), including viruses, bacteria, fungi and parasites, and are ranked by the World Health Organization as the second leading cause of death worldwide. These infections can cause temporary discomfort, severe tissue damage, or even death. Timely accumulation of complex information about the disease with the introduction of computer data processing into the system of the medical service will increase the level of information support for epidemiological surveillance. The paper considers the concepts and existing mathematical models of the spread of coronavirus infection COVID-19, based on SIR-type models. Variety of models of this type are most suitable for predicting, since the disease has a sufficiently long incubation period and the presence of asymptomatic carriers. A mathematical description of the SEIR model with thirteen variables is given.

Relevant mathematical modelling efforts for understanding COVID-19 dynamics: an educational challenge.

The online version contains supplementary material available at 10.1007/s11858-022-01447-2.

Uncertainty and error in SARS-CoV-2 epidemiological parameters inferred from population-level epidemic models

During the SARS-CoV-2 pandemic, epidemic models have been central to policy-making. Public health responses have been shaped by model-based projections and inferences, especially related to the impact of various non-pharmaceutical interventions. Accompanying this has been increased scrutiny over model performance, model assumptions, and the way that uncertainty is incorporated and presented. Here we consider a population-level model, focusing on how distributions representing host infectiousness and the infection-to-death times are modelled, and particularly on the impact of inferred epidemic characteristics if these distributions are mis-specified. We introduce an SIR-type model with the infected population structured by ‘infected age’, i.e. the number of days since first being infected, a formulation that enables distributions to be incorporated that are consistent with clinical data. We show that inference based on simpler models without infected age, which implicitly mis-specify these distributions, leads to substantial errors in inferred quantities relevant to policy-making, such as the reproduction number and the impact of interventions. We consider uncertainty quantification via a Bayesian approach, implementing this for both synthetic and real data focusing on UK data in the period 15 Feb–14 Jul 2020, and emphasising circumstances where it is misleading to neglect uncertainty.This manuscript was submitted as part of a theme issue on “Modelling COVID-19 and Preparedness for Future Pandemics”.

Mathematical model of COVID-19 transmission dynamics incorporating booster vaccine program and environmental contamination

COVID-19 is one of the greatest human global health challenges that causes economic meltdown of many nations. In this study, we develop an SIR-type model which captures both human-to-human and environment-to-human-to-environment transmissions that allows the recruitment of corona viruses in the environment in the midst of booster vaccine program. Theoretically, we prove some basic properties of the full model as well as investigate the existence of SARS-CoV-2-free and endemic equilibria. The SARS-CoV-2-free equilibrium for the special case, where the constant inflow of corona virus into the environment by any other means, Ω is suspended (Ω=0) is globally asymptotically stable when the effective reproduction number R0c<1 and unstable if otherwise. Whereas in the presence of free-living Corona viruses in the environment (Ω>0), the endemic equilibrium using the centre manifold theory is shown to be stable globally whenever R0c>1. The model is extended into optimal control system and analyzed analytically using Pontryagin's Maximum Principle. Results from the optimal control simulations show that strategy E for implementing the public health advocacy, booster vaccine program, treatment of isolated people and disinfecting or fumigating of surfaces and dead bodies before burial is the most effective control intervention for mitigating the spread of Corona virus. Importantly, based on the available data used, the study also revealed that if at least 70% of the constituents followed the aforementioned public health policies, then herd immunity could be achieved for COVID-19 pandemic in the community.

Modeling secondary infections with temporary immunity and disease enhancement factor: Mechanisms for complex dynamics in simple epidemiological models

Modeling insights for epidemiological scenarios characterized by chaotic dynamics have been largely unexplored. A rigorous analysis of such systems are essential for a real predictive power and a more accurate disease control decision making. Motivated by dengue fever epidemiology, we study a basic SIR–SIR type model for the host population, capturing differences between primary and secondary infections. This model is the minimalistic version to previously suggested multi-strain models for dengue fever in which deterministic chaos was found in wider parameter regions. Without strain structure of pathogens, we consider temporary immunity after a primary infection and disease enhancement in a subsequent infection to identify to which extent these biological mechanisms can generate complex behavior in simple epidemiological models.Stability analysis of the system is performed using the classical linearization theory, and the qualitative behavior of the model is investigated with a detailed bifurcation analysis. Rich dynamical structures are identified, including the Bogdanov–Takens, cusp and Bautin bifurcations which has never been described in dengue fever epidemiology. Besides the conventional transcritical bifurcation, a backward bifurcation occurs for higher disease enhancement in secondary infections, exhibiting bi-stability when biological temporary immunity period is assumed. The backward bifurcation is formalized using the center manifold theory. While the Hopf and the global homoclinic bifurcation curves were computed numerically, analytical expressions for the transcritical and tangent bifurcations are obtained. The combination of temporary immunity and disease enhancement play a significant role in the complexity of the system dynamics, with chaotic behavior observed after including seasonal forcing.

Minimizing the epidemic final size while containing the infected peak prevalence in SIR systems

Mathematical models are critical to understand the spread of pathogens in a population and evaluate the effectiveness of non-pharmaceutical interventions (NPIs). A plethora of optimal strategies has been recently developed to minimize either the infected peak prevalence (IPP) or the epidemic final size (EFS). While most of them optimize a simple cost function along a fixed finite-time horizon, no consensus has been reached about how to simultaneously handle the IPP and the EFS, while minimizing the intervention’s side effects. In this work, based on a new characterization of the dynamical behaviour of SIR-type models under control actions (including the stability of equilibrium sets in terms of herd immunity), we study how to minimize the EFS while keeping the IPP controlled at any time. A procedure is proposed to tailor NPIs by separating transient from stationary control objectives: the potential benefits of the strategy are illustrated by a detailed analysis and simulation results related to the COVID-19 pandemic.

On the Parametrization of Epidemiologic Models-Lessons from Modelling COVID-19 Epidemic.

Numerous prediction models of SARS-CoV-2 pandemic were proposed in the past. Unknown parameters of these models are often estimated based on observational data. However, lag in case-reporting, changing testing policy or incompleteness of data lead to biased estimates. Moreover, parametrization is time-dependent due to changing age-structures, emerging virus variants, non-pharmaceutical interventions, and vaccination programs. To cover these aspects, we propose a principled approach to parametrize a SIR-type epidemiologic model by embedding it as a hidden layer into an input-output non-linear dynamical system (IO-NLDS). Observable data are coupled to hidden states of the model by appropriate data models considering possible biases of the data. This includes data issues such as known delays or biases in reporting. We estimate model parameters including their time-dependence by a Bayesian knowledge synthesis process considering parameter ranges derived from external studies as prior information. We applied this approach on a specific SIR-type model and data of Germany and Saxony demonstrating good prediction performances. Our approach can estimate and compare the relative effectiveness of non-pharmaceutical interventions and provide scenarios of the future course of the epidemic under specified conditions. It can be translated to other data sets, i.e., other countries and other SIR-type models.