Backward bifurcations, turning points and rich dynamics in simple disease models

Effects of Vitamin D Supplementation and Degradation on the Innate Immune System Response: Insights on SARS-CoV-2

Vitamin D has been proven to be a strong stimulator of mechanisms associated with the elimination of pathogens. Because of its recognized effectiveness against viral infections, during SARS-CoV-2 infection, the effects of Vitamin D supplementation have been the object of debate. This study aims to contribute to this debate by the means of a qualitative phenomenological mathematical model in which the role of Vitamin D and its interactions with the innate immune system are explicitly considered. We show that Vitamin D influx and degradation can be considered as possible control parameters for the disease evaluation and recovery. By varying Vitamin D influx, three dynamical scenarios have been found with different modalities of recovery from the disease. Inside each scenario, Vitamin D degradation has been related to different degrees of severity in disease development. Interestingly, the emergence of hysteretic phenomenologies when Vitamin D influx is too low can be related to the onset of Long-COVID syndrome, confirming clinical evidence from recent studies on the topic.

Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics

In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich the model introduced in (Gaurav et al., 2019) with the mechanism of self-information that accounts for information about the product performance basing on consumers’ experience on the recent past. We obtain conditions for which the model exhibits a forward or a backward phenomenology and evaluate the impact of self-information on both these scenarios. Our analysis suggests that, even if hysteretic dynamics in referral campaigns is intimately linked to the mechanism of referrals, an adequate level of self-information and a fairly high level of customer-satisfaction can act as strategic tools to manage hysteresis and allow the campaign to spread in more controllable conditions.

Models of cytokine dynamics in the inflammatory response of viral zoonotic infectious diseases

Inflammatory responses to an infection from a zoonotic pathogen, such as avian influenza viruses, hantaviruses and some coronaviruses, are distinctly different in their natural reservoir versus human host. While not as well studied in the natural reservoirs, the pro-inflammatory response and viral replication appear controlled and show no obvious pathology. In contrast, infection in humans results in an initial high viral load marked by an aggressive pro-inflammatory response known as a cytokine storm. The key difference in the course of the infection between the reservoir and human host is the inflammatory response. In this investigation, we apply a simple two-component differential equation model for pro-inflammatory and anti-inflammatory responses and a detailed mathematical analysis to identify specific regions in parameter space for single stable endemic equilibrium, bistability or periodic solutions. The extensions of the deterministic model to two stochastic models account for variability in responses seen at the cell (local) or tissue (global) levels. Numerical solutions of the stochastic models exhibit outcomes that are typical of a chronic infection in the natural reservoir or a cytokine storm in human infection. In the chronic infection, occasional flare-ups between high and low responses occur when model parameters are in a region of bistability or periodic solutions. The cytokine storm with a vigorous pro-inflammatory response and less vigorous anti-inflammatory response occurs in the parameter region for a single stable endemic equilibrium with a strong pro-inflammatory response. The results of the model analyses and the simulations are interpreted in terms of the functional role of the cytokines and the inflammatory responses seen in infection of the natural reservoir or of the human host.

On the Z-type control of backward bifurcations in epidemic models

We investigate how the Z-type dynamic approach can be applied to control backward bifurcation phenomena in epidemic models. Because of its rich phenomenology, that includes stationary or oscillatory subcritical persistence of the disease, we consider the SIR model introduced by Zhou & Fan in [Nonlinear Analysis: Real World Applications, 13(1), 312-324, 2012] and apply the Z-control approach in the specific case of indirect control of the infective population. We derive the associated Z-controlled model both when the desired Z-controlled equilibrium is an endemic equilibrium with a very low number of infectives and when the Z-controlled equilibrium is a disease-free equilibrium. We investigate the properties of these Z-controlled models from the point of view of the dynamical system theory and elucidate the key role of the design parameter λ. Numerical investigations on the model also highlight the impacts of the Z-control method on the backward scenario and on a variety of dynamical regimes emerging from it.

The effect of sexual transmission on Zika virus dynamics

Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated. The effect of the incidence function used to model sexual transmission from infectious males to susceptible females is investigated. It is proved that for such functions that are sublinear, if the basic reproduction [Formula: see text], then the disease dies out and [Formula: see text] is a sharp threshold. Moreover, under certain conditions on model parameters and assuming mass action incidence for sexual transmission, it is proved that if [Formula: see text], there exists a unique endemic equilibrium that is globally asymptotically stable. However, under nonlinear incidence, it is shown that for certain functions backward bifurcation and Hopf bifurcation may occur, giving rise to subthreshold equilibria and periodic solutions, respectively. Numerical simulations for various parameter values are displayed to illustrate these behaviours.

Mate Limitation in Fungal Plant Parasites Can Lead to Cyclic Epidemics in Perennial Host Populations

Fungal plant parasites represent a growing concern for biodiversity and food security. Most ascomycete species are capable of producing different types of infectious spores both asexually and sexually. Yet the contributions of both types of spores to epidemiological dynamics have still to been fully researched. Here we studied the effect of mate limitation in parasites which perform both sexual and asexual reproduction in the same host. Since mate limitation implies positive density dependence at low population density, we modeled the dynamics of such species with both density-dependent (sexual) and density-independent (asexual) transmission rates. A first simple SIR model incorporating these two types of transmission from the infected compartment, suggested that combining sexual and asexual spore production can generate persistently cyclic epidemics in a significant part of the parameter space. It was then confirmed that cyclic persistence could occur in realistic situations by parameterizing a more detailed model fitting the biology of the Black Sigatoka disease of banana, for which literature data are available. We discuss the implications of these results for research on and management of Sigatoka diseases of banana.