Identifiability in Epidemic Models with Prior Immunity and Under-Reporting

We treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.

The emergence of highly pathogenic H5N1 avian influenza in dairy cattle herds across the United States has caused multiple mild human infections. There is an urgent need to understand the risk of spillover into humans. Here, we show that pre-existing immunity from the 2009 H1N1 pandemic influenza virus provided protection from mortality and severe clinical disease to ferrets intranasally infected with bovine H5N1. H1N1 immune ferrets exhibited a differential tissue tropism with little bovine H5N1 viral dissemination to organs outside the respiratory tract and significantly less H5N1 virus found in nasal secretions and the respiratory tract. Additionally, ferrets with H1N1 prior immunity produced antibodies that cross-reacted with H5N1 neuraminidase protein. Taken together, these results suggest that mild disease in humans may be linked to prior immunity to human seasonal influenza viruses.

Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a nonlinear dynamic system and that transitions between different attractors are driven by slow changes in birth rates and vaccination levels. The same analysis can explain the main features of chickenpox dynamics, but fails for rubella and whooping cough. We show that an additional (perturbative) analysis of the model, together with knowledge of the population size in question, can account for all the observed incidence patterns by predicting how stochastically sustained transient dynamics should be manifested in these systems.

Structural and practical identifiability analysis of outbreak models

This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution.

Description Tuberculosis (TB) lung pathology comes in a range of lesion types including necrotic granulomas defined by a necrotic core, and lesions of alveolitis where lung integrity is intact. Granulomas develop in response to primary Mycobacterium tuberculosis (Mtb) infection, whereas alveolitis is associated with prior Mtb immunity. To investigate the mechanisms of Mtb lesion formation we use a model of concomitant Mtb infection (coMtb) in C3HeB/FeJ mice which form necrotizing granulomas in primary infections and alveolitis with coMtb. Features of coMtb include: 1) an early and robust CD4 T cell response that is necessary and sufficient to prevent necrotizing granulomas and 2) prevention of a dysregulated neutrophil response that drives granuloma necrosis. TB lesions in coMtb are also associated with non-hematopoietic indoleamine 2,3-dioxygenase (IDO-1), a critical enzyme for tryptophan metabolism into immunomodulatory kynurenines. Importantly, we find non-hematopoietic IDO-1 within TB lesions is dependent on CD4-derived IFN-γ, leading to the hypothesis that IFN-γ induces non-hematopoietic IDO-1 which prevents the establishment of a dysregulated neutrophil response and promotes lesions of alveolitis. We determined non-hematopoietic IFN-γ signaling and IDO-1 expression was protective in prior immunity and prevented necrotizing granulomas. These results highlight a context dependent role for IDO-1 in Mtb infection and a novel role for non-hematopoietic cells in TB lesion formation. Topic Categories Microbial, Parasitic, and Fungal Immunology (MPF)

The recent outbreak of COVID-19 underlined the need for a fast and trustworthy methodology to identify the features of a pandemic, whose early identification is of help for designing non-pharmaceutical interventions (including lockdown and social distancing) to limit the progression of the disease. A common approach in this context is the parameter identification from deterministic epidemic models, which, unfortunately, cannot take into account the inherent randomness of the epidemic phenomenon, especially in the initial stage; on the other hand, the use of raw data within the framework of a stochastic model is not straightforward. This note investigates the stochastic approach applied to a basic SIR (Susceptible, Infected, Recovered) epidemic model to enhance information from raw data generated in silico. The stochastic model consists of a Continuous-Time Markov Model, describing the epidemic outbreak in terms of stochastic discrete infection and recovery events in a given region, and where independent random paths are associated to different provinces of the same region, which are assumed to share the same set of model parameters. The estimation procedure is based on the building of a loss function that symmetrically weighs first-order and second-order moments, differently from the standard approach that considers a highly asymmetrical choice, exploiting only first-order moments. Instead, we opt for an innovative symmetrical identification approach which exploits both moments. The new approach is specifically proposed to enhance the statistical information content of the raw epidemiological data.

In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in [Formula: see text]th moment for any [Formula: see text]. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations.

Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching

Petri nets are an increasingly used modeling framework for the spread of disease across populations or within an individual. For example, the Susceptible-Infectious-Recovered (SIR) compartment model is foundational for population epidemiological modeling and has been implemented in several prior Petri net studies. While the SIR model is typically expressed as Ordinary Differential Equations (ODEs), with continuous time and variables, Petri nets operate as discrete event simulations with deterministic or stochastic timings. We present the first systematic study of the numerical convergence of two distinct Petri net implementations of the SIRS compartment model relative to the standard ODE. In particular, we introduce a novel deterministic implementation of the SIRS model using variable transition weights in the GPenSIM package and stochastic Petri net models using Spike. We show how rescaling and rounding procedures are critical for the numerical convergence of Petri net SIR models relative to the ODEs, and we achieve a relative root mean squared error of less than 1% compared to ODE simulations for biologically relevant parameter ranges. Our findings confirm that both stochastic and deterministic discrete time Petri nets are valid for modeling SIR-type dynamics with appropriate numerical procedures, laying the foundations for larger-scale use of Petri net models.

Stochastic modeling predicts various outcomes from stochasticity in the data, parameters and dynamical system. Stochastic models are deemed more appropriate than deterministic models accounting in terms of essential and practical information about a system. The objective of the current investigation is to address the issue above through the development of a novel deep neural network referred to as a stochastic epidemiology-informed neural network. This network learns knowledge about the parameters and dynamics of a stochastic epidemic vaccine model. Our analysis centers on examining the nonlinear incidence rate of the model from the perspective of the combined effects of vaccination and stochasticity. Based on empirical evidence, stochastic models offer a more comprehensive understanding than deterministic models, mainly when we use error metrics. The findings of our study indicate that a decrease in randomness and an increase in vaccination rates are associated with a better prediction of nonlinear incidence rates. Adopting a nonlinear incidence rate enables a more comprehensive representation of the complexities of transmitting diseases. The computational analysis of the proposed method, focusing on sensitivity analysis and overfitting analysis, shows that the proposed method is efficient. Our research aims to guide policymakers on the effects of stochasticity in epidemic models, thereby aiding the development of effective vaccination and mitigation policies. Several case studies have been conducted on nonlinear incidence rates using data from Tennessee, USA.

Influenza A viruses in swine have considerable genetic diversity and continue to pose a pandemic threat to humans due to a potential lack of population level immunity. Here we describe a pipeline to characterize and triage influenza viruses for their pandemic risk and examine the pandemic potential of two widespread swine origin viruses. Our analysis reveals that a panel of human sera collected from healthy adults in 2020 has no cross-reactive neutralizing antibodies against a α-H1 clade strain (α-swH1N2) but do against a γ-H1 clade strain. The α-swH1N2 virus replicates efficiently in human airway cultures and exhibits phenotypic signatures similar to the human H1N1 pandemic strain from 2009 (H1N1pdm09). Furthermore, α-swH1N2 is capable of efficient airborne transmission to both naïve ferrets and ferrets with prior seasonal influenza immunity. Ferrets with H1N1pdm09 pre-existing immunity show reduced α-swH1N2 viral shedding and less severe disease signs. Despite this, H1N1pdm09-immune ferrets that became infected via the air can still onward transmit α-swH1N2 with an efficiency of 50%. These results indicate that this α-swH1N2 strain has a higher pandemic potential, but a moderate level of impact since there is reduced replication fitness and pathology in animals with prior immunity.

Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.

Mathematical modeling has proved to be useful in predicting the spread of infectious diseases and assessing the dynamical behavior of contagious diseases, including COVID-19. Various models aid in forecasting COVID-19 spread, such as SEIR (Susceptible – Exposed – Infected – Recovered), SIR (Susceptible – Infected – Recovered), SIRD (Susceptible – Infected – Recovered – Death), and SIRVD (Susceptible – Infected – Recovered – Vaccinated – Death). With recent technological advancements, forecasting of COVID-19 can also be done using machine learning techniques such as SVM (support vector machine), decision tree, random forest, and linear regression. This chapter delves into the various mathematical models and provides simulations using Python and machine learning techniques for COVID-19. These simulations provide essential insights into the spread of infectious diseases and evaluate which machine learning algorithm performs better using evaluation metrics.

Deterministic compartment models (CMs) and stochastic models, including stochastic CMs and agent-based models, are widely utilized in epidemic modeling. However, the relationship between CMs and their corresponding stochastic models is not well understood. The present study aimed to address this gap by conducting a comparative study using the susceptible, exposed, infectious, and recovered (SEIR) model and its extended CMs from the coronavirus disease 2019 modeling literature. We demonstrated the equivalence of the numerical solution of CMs using the Euler scheme and their stochastic counterparts through theoretical analysis and simulations. Based on this equivalence, we proposed an efficient model calibration method that could replicate the exact solution of CMs in the corresponding stochastic models through parameter adjustment. The advancement in calibration techniques enhanced the accuracy of stochastic modeling in capturing the dynamics of epidemics. However, it should be noted that discrete-time stochastic models cannot perfectly reproduce the exact solution of continuous-time CMs. Additionally, we proposed a new stochastic compartment and agent mixed model as an alternative to agent-based models for large-scale population simulations with a limited number of agents. This model offered a balance between computational efficiency and accuracy. The results of this research contributed to the comparison and unification of deterministic CMs and stochastic models in epidemic modeling. Furthermore, the results had implications for the development of hybrid models that integrated the strengths of both frameworks. Overall, the present study has provided valuable epidemic modeling techniques and their practical applications for understanding and controlling the spread of infectious diseases.