Waning immunity and reinfection are critical features of many infectious diseases, but epidemiological models often fail to capture the intricate interaction between an individual's history of immunity and their current infection status; when they do, the approach is usually overly simplistic. We develop a novel dual-age structured model that simultaneously tracks immunity age (time since the last recovery from infection) and infection age (time since infection) to analyze epidemic dynamics under conditions of waning immunity and reinfection. The model is formulated as a system of age-structured partial differential equations that describe susceptible and infected populations stratified by both immunity and infection ages. We derive basic reproduction numbers associated with the model and numerically solve the system using a second-order Runge-Kutta scheme along the characteristic lines. We further extend the model to explore vaccination interventions, specifically targeting individuals according to their immunity age. Numerical results reveal that higher contact rates produce larger amplitude oscillations with longer interepidemic periods. The relationship between initial infectio
As pathogens spread in a population of hosts, immunity is built up and the pool of susceptible individuals is depleted. This generates selective pressure, to which many human RNA viruses, such as influenza virus or SARS-CoV-2, respond with rapid antigenic evolution and frequent emergence of immune evasive variants. However, the host's immune systems adapt and older immune responses wane, such that escape variants only enjoy a growth advantage for a limited time. If variant growth dynamics and reshaping of host-immunity operate on comparable time scales, viral adaptation is determined by eco-evolutionary interactions that are not captured by models of rapid evolution in a fixed environment. Here, we use a Susceptible/Infected model to describe the interaction between an evolving viral population in a dynamic but immunologically diverse host population. We show that depending on strain cross-immunity, heterogeneity of the host population, and durability of immune responses, escape variants initially grow exponentially, but lose their growth advantage before reaching high frequencies. Their subsequent dynamics follows an anomalous random walk determined by future escape variants and r
Infectious disease time series are often used to estimate a pathogen's basic reproduction number, $R_0$. However, fits of epidemic models to time series conflate pathogen transmissibility with pre-existing population immunity, so only the *effective* reproduction number, $R_{eff}$, can be inferred. This composite parameter is the product of the underlying $R_0$ and the pre-epidemic susceptible fraction, $x^-$. We show that a conservation law associated with epidemic momentum---prevalence weighted by potential to infect---makes it possible to disentangle transmissibility from prior immunity and to infer $R_0$ and $x^-$ separately from a single epidemic time series. We test the methodology using stochastic epidemic simulations, and illustrate the approach with a reappraisal of influenza transmissibility during the 1918 pandemic, estimating rather than assuming the degree of prior population immunity. For the autumn wave in Philadelphia, USA, we find $R_0\approx2.7$ and $x^-\approx0.8$, implying that about 20\% of the population was already immune before that wave, plausibly as a result of infection during the spring 1918 herald wave.
Diffusion processes pervade numerous areas of AI, abstractly modeling the dynamics of exchanging, oftentimes volatile, information in networks. A central question is how long the information remains in the network, known as survival time. For the commonly studied SIS process, the expected survival time is at least super-polynomial in the network size already on star graphs, for a wide range of parameters. In contrast, the expected survival time of the SIRS process, which introduces temporary immunity, is always at most polynomial on stars and only known to be super-polynomial for far denser networks, such as expanders. However, this result relies on featuring full temporary immunity, which is not always present in actual processes. We introduce the cSIRS process, which incorporates gradually declining immunity such that the expected immunity at each point in time is identical to that of the SIRS process. We study the survival time of the cSIRS process rigorously on star graphs and expanders and show that its expected survival time is very similar to that of the SIS process, which features no immunity. This suggests that featuring gradually declining immunity is almost as having non
Herd immunity is a critical concept in epidemiology, describing a threshold at which a sufficient proportion of a population is immune, either through infection or vaccination, thereby preventing sustained transmission of a pathogen. In the classic Susceptible-Infectious-Recovered (SIR) model, which has been widely used to study infectious disease dynamics, the achievement of herd immunity depends on key parameters, including the transmission rate ($β$) and the recovery rate ($γ$), where $γ$ represents the inverse of the mean infectious period. While the transmission rate has received substantial attention, recent studies have underscored the significant role of $γ$ in determining the timing and sustainability of herd immunity. Additionally, it is becoming increasingly evident that assuming $γ$ as a constant parameter might oversimplify the dynamics, as variations in recovery times can reflect diverse biological, social, and healthcare-related factors. In this paper, we investigate how heterogeneity in the recovery rate affects herd immunity. We show empirically that the mean of the recovery rate is not a reliable metric for determining the achievement of herd immunity. Furthermore
Antimicrobial resistance (AMR) is escalating and outpacing current antibiotic development. Thus, discovering antibiotics effective against emerging pathogens is becoming increasingly critical. However, existing approaches cannot rapidly identify effective molecules against novel pathogens or emerging drug-resistant strains. Here, we introduce ApexOracle, an artificial intelligence (AI) model that both predicts the antibacterial potency of existing compounds and designs de novo molecules active against strains it has never encountered. Departing from models that rely solely on molecular features, ApexOracle incorporates pathogen-specific context through the integration of molecular features captured via a foundational discrete diffusion language model and a dual-embedding framework that combines genomic- and literature-derived strain representations. Across diverse bacterial species and chemical modalities, ApexOracle consistently outperformed state-of-the-art approaches in activity prediction and demonstrated reliable transferability to novel pathogens with little or no antimicrobial data. Its unified representation-generation architecture further enables the in silico creation of
SIRS epidemic models assume that individual immunity (from infection and vaccination) wanes in one big leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that's become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and where booster vaccines are given to increase immunity. This paper considers an epidemic model allowing for such gradual waning of immunity (either linear or exponential waning) thereby extending SIRS epidemics, and also incorporates vaccination. The two versions for gradual waning of immunity are compared with the classic SIRS epidemic, where the three models are calibrated by having the same \emph{average cumulative immunity}. All models are shown to have identical basic reproduction number $R_0$. However, if no prevention is put in place, the exponential waning model has highest prevalence and the classic SIRS model has lowest. Similarly, the amount of vaccine supply needed to reach and maintain herd immunity is highest for the model with exponential decay of immunity and lowest for the classic SIRS model. consequently, if truth lies close t
When a fraction of a population becomes immune to an infectious disease, the population-wide infection risk decreases nonlinearly due to collective protection, known as herd immunity. Some studies based on mean-field models suggest that natural infection in a heterogeneous population may induce herd immunity more efficiently than homogeneous immunization. However, we theoretically show that this is not necessarily the case when the population is modeled as a network instead of using the mean-field approach. We identify two competing mechanisms driving disease-induced herd immunity in networks: the biased distribution of immunity toward socially active individuals enhances herd immunity, while the topological localization of immune individuals weakens it. The effect of localization is stronger in networks embedded in a low-dimensional space, which can make disease-induced immunity less effective than random immunization. Our results highlight the role of networks in shaping herd immunity and call for a careful examination of model predictions that inform public health policies.
This work is devoted to the study of the probability of immunity, i.e. the effect occurs whether exposed or not. We derive necessary and sufficient conditions for non-immunity and $ε$-bounded immunity, i.e. the probability of immunity is zero and $ε$-bounded, respectively. The former allows us to estimate the probability of benefit (i.e., the effect occurs if and only if exposed) from a randomized controlled trial, and the latter allows us to produce bounds of the probability of benefit that are tighter than the existing ones. We also introduce the concept of indirect immunity (i.e., through a mediator) and repeat our previous analysis for it. Finally, we propose a method for sensitivity analysis of the probability of immunity under unmeasured confounding.
Epidemic spreading over populations networks has been an important subject of research for several decades, and especially during the Covid-19 pandemic. Most epidemic outbreaks are likely to create multiple mutations during their spreading over the population. In this paper, we study the evolution of a pathogen which can mutate continuously during the epidemic spreading. We consider pathogens whose mutating parameter is the mortality mean-time, and study the evolution of this parameter over the spreading process. We use analytical methods to compute the dynamic equation of the epidemic and the conditions for it to spread. We also use numerical simulations to study the pathogen flow in this case, and to understand the mutation phenomena. We show that the natural selection leads to less violent pathogens becoming predominant in the population. We discuss a wide range of network structures and show how different effects are manifested in each case. We also applied our theory in the context of the Covid-19 pandemic, using relevant epidemiological data collected for this outbreak. We provided explanations for the variants spreading processes observed throughout this pandemic.
Understanding social immunity mechanisms in ant colonies remains crucial to comprehending the evolution of defense strategies in eusocial organisms. This study assumes the absence of the role of memory in the ants' defense strategy, considering that they can make a new exploration of collective cleaning behaviors. We investigate how worker interactions and previous behaviors influence the evolution of social immunity strategies in response to the presence of pathogens and vulnerable colony members. In this context, with the application of Markov transition models it was possible to describe changes in cleaning behavior over time influenced by the presence of vulnerable members and treatment with an entomopathogenic fungus. We found a significant effect on ant behavior when exposed to \it{Metarhizium anisopliae} when a fungus garden fragment and one larva were present. Our findings confirm a link between prophylactic cleaning and the presence of vulnerable members. Remarkably, distinct behaviors and transition times vary among treatments, revealing workers' adaptability to threats. Allogrooming shows adaptive changes when exposed to pathogens, potentially affecting pathogen transmis
We analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individual waning parameters are unobserved, thus only allowing vaccination schemes to depend on time since last vaccination. The optimal vaccination policies for both the informed and uniformed heterogeneous situation are derived and compared with the homogeneous waning model (meaning all individuals have the same immunity waning rate), as well as to the classic SIRS model where immunity at the individual level drops from complete immunity to complete susceptibility in one leap. It is shown that the classic SIRS model requires least vaccines, followed by the SIRS with homogeneous gradual waning, follo
Deepfakes pose an evolving threat to cybersecurity, which calls for the development of automated countermeasures. While considerable forensic research has been devoted to the detection and localisation of deepfakes, solutions for reversing fake to real are yet to be developed. In this study, we introduce cyber vaccination for conferring immunity to deepfakes. Analogous to biological vaccination that injects antigens to induce immunity prior to infection by an actual pathogen, cyber vaccination simulates deepfakes and performs adversarial training to build a defensive immune system. Aiming at building up attack-agnostic immunity with limited computational resources, we propose to simulate various deepfakes with one single overpowered attack: face masking. The proposed immune system consists of a vaccinator for inducing immunity and a neutraliser for recovering facial content. Experimental evaluations demonstrate effective immunity to face replacement, face reenactment and various types of corruptions.
We adapt the article of Forien, Pang, Pardoux and Zotsa: Arxiv preprint Arxiv2210.04667(2022), on epidemic models with varying infectivity and waning immunity, to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.
Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.
Infectious diseases caused by pathogens including viruses, bacteria, fungi, and parasites are ranked as the second leading cause of death worldwide by the World Health Organization. Despite tremendous improvements in global public health since 1950, a number of challenges remain to either prevent or eradicate infectious diseases. Many pathogens can cause acute infections that are effectively cleared by the host immunity, but a subcategory of these pathogens called "intracellular pathogens" can establish persistent and sometimes lifelong infections. Several of these intracellular pathogens manage to evade the host immune monitoring and cause disease by replicating inside the host cells. These pathogens have evolved diverse immune escape strategies and overcome immune responses by residing and multiplying inside host immune cells, primarily macrophages. While these intracellular pathogens that cause persistent infections are phylogenetically diverse and engage in diverse immune evasion and persistence strategies, they share common pathogen type-specific mechanisms during host-pathogen interaction inside host cells. Likewise, the host immune system is also equipped with a diverse range of effector functions to fight against the establishment of pathogen persistence and subsequent host damage. This article provides an overview of the immune effector functions used by the host to counter pathogens and various persistence strategies used by intracellular pathogens to counter host immunity, which enables their extended period of colonization in the host. The improved understanding of persistent intracellular pathogen-derived infections will contribute to develop improved disease diagnostics, therapeutics, and prophylactics.
SIRS models capture transmission dynamics of infectious diseases for which immunity is not lifelong. Extending these models by a W compartment for individuals with waning immunity, the boosting of the immune system upon repeated exposure may be incorporated. Previous analyses assumed identical waning rates from R to W and from W to S. This implicitly assumes equal length for the period of full immunity and of waned immunity. We relax this restriction, and allow an asymmetric partitioning of the total immune period. Stability switches of the endemic equilibrium are investigated with a combination of analytic and numerical tools. Then, continuation methods are applied to track bifurcations along the equilibrium branch. We find rich dynamics: Hopf bifurcations, endemic double bubbles, and regions of bistability. Our results highlight that the length of the period in which waning immunity can be boosted is a crucial parameter significantly influencing long term epidemiological dynamics.
A persistent public health challenge is finding immunization schemes that are effective in combating highly mutable pathogens such as HIV and influenza viruses. To address this, we analyze a simplified model of affinity maturation, the Darwinian evolutionary process B cells undergo during immunization. The vaccination protocol dictates selection forces that steer affinity maturation to generate antibodies. We focus on determining the optimal selection forces exerted by a generic time-dependent vaccination protocol to maximize production of broadly neutralizing antibodies (bnAbs) that can protect against a broad spectrum of pathogen strains. The model lends itself to a path integral representation and operator approximations within a mean-field limit, providing guiding principles for optimizing time-dependent vaccine-induced selection forces to enhance bnAb generation. We compare our analytical mean-field results with the outcomes of stochastic simulations and discuss their similarities and differences.
Using a simple economic model in which social-distancing reduces contagion, we study the implications of waning immunity for the epidemiological dynamics and social activity. If immunity wanes, we find that COVID-19 likely becomes endemic and that social-distancing is here to stay until the discovery of a vaccine or cure. But waning immunity does not necessarily change optimal actions on the onset of the pandemic. Decentralized equilibria are virtually independent of waning immunity until close to peak infections. For centralized equilibria, the relevance of waning immunity decreases in the probability of finding a vaccine or cure, the costs of infection (e.g., infection-fatality rate), and the presence of other NPIs that lower contagion (e.g., quarantining and mask use). In simulations calibrated to July 2020, our model suggests that waning immunity is virtually unimportant for centralized equilibria until at least 2021. This provides vital time for individuals and policymakers to learn about immunity against SARS-CoV-2 before it becomes critical.
We analyse a model that describes the propagation of many pathogens within and between many species. A branching process approximation is used to compute the probability of disease outbreaks. Special cases of aquatic environments with two host species and one or two pathogens are considered both analytically and computationally.