A simple model of pathogen-immune dynamics including specific and non-specific immunity

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.

Gastrointestinal helminths increase Bordetella bronchiseptica shedding and host variation in supershedding

Co-infected hosts, individuals that carry more than one infectious agent at any one time, have been suggested to facilitate pathogen transmission, including the emergence of supershedding events. However, how the host immune response mediates the interactions between co-infecting pathogens and how these affect the dynamics of shedding remains largely unclear. We used laboratory experiments and a modeling approach to examine temporal changes in the shedding of the respiratory bacterium Bordetella bronchiseptica in rabbits with one or two gastrointestinal helminth species. Experimental data showed that rabbits co-infected with one or both helminths shed significantly more B. bronchiseptica, by direct contact with an agar petri dish, than rabbits with bacteria alone. Co-infected hosts generated supershedding events of higher intensity and more frequently than hosts with no helminths. To explain this variation in shedding an infection-immune model was developed and fitted to rabbits of each group. Simulations suggested that differences in the magnitude and duration of shedding could be explained by the effect of the two helminths on the relative contribution of neutrophils and specific IgA and IgG to B. bronchiseptica neutralization in the respiratory tract. However, the interactions between infection and immune response at the scale of analysis that we used could not capture the rapid variation in the intensity of shedding of every rabbit. We suggest that fast and local changes at the level of respiratory tissue probably played a more important role. This study indicates that co-infected hosts are important source of variation in shedding, and provides a quantitative explanation into the role of helminths to the dynamics of respiratory bacterial infections.

Modelling Immune Memory Development

The cellular adaptive immune response to influenza has been analyzed through several recent mathematical models. In particular, Zarnitsyna et al. (Front Immunol 7:1-9, 2016) show how central memory CD8+ T cells reach a plateau after repeated infections, and analyze their role in the immune response to further challenges. In this paper, we further investigate the theoretical features of that model by extracting from the infection dynamics a discrete map that describes the build-up of memory cells. Furthermore, we show how the model by Zarnitsyna et al. (Front Immunol 7:1-9, 2016) can be viewed as a fast-scale approximation of a model allowing for recruitment of target epithelial cells. Finally, we analyze which components of the model are essential to understand the progressive build-up of immune memory. This is performed through the analysis of simplified versions of the model that include some components only of immune response. The analysis performed may also provide a theoretical framework for understanding the conditions under which two-dose vaccination strategies can be helpful.

Within-host mechanisms of immune regulation explain the contrasting dynamics of two helminth species in both single and dual infections

Variation in the intensity and duration of infections is often driven by variation in the network and strength of host immune responses. While many of the immune mechanisms and components are known for parasitic helminths, how these relationships change from single to multiple infections and impact helminth dynamics remains largely unclear. Here, we used laboratory data from a rabbit-helminth system and developed a within-host model of infection to investigate different scenarios of immune regulation in rabbits infected with one or two helminth species. Model selection suggests that the immunological pathways activated against Trichostrongylus retortaeformis and Graphidium strigosum are similar. However, differences in the strength of these immune signals lead to the contrasting dynamics of infections, where the first parasite is rapidly cleared and the latter persists with high intensities. In addition to the reactions identified in single infections, rabbits with both helminths also activate new pathways that asymmetrically affect the dynamics of the two species. These new signals alter the intensities but not the general trend of the infections. The type of interactions described can be expected in many other host-helminth systems. Our immune framework is flexible enough to capture different mechanisms and their complexity, and provides essential insights to the understanding of multi-helminth infections.

Stability analysis of the hiv model through incommensurate fractional-order nonlinear system

In this study, it is employed a new model of HIV infection in the form of incommensurate fractional differential equations systems involving the Caputo fractional derivative. Existence of the model's equilibrium points has been investigated. According to some special cases of the derivative-orders in the proposed model, the asymptotic stability of the infection-free equilibrium and endemic equilibrium has been proved under certain conditions. These stability conditions related to the derivative-orders depend on not only the basic reproduction rate frequently emphasized in the literature but also the newly obtained conditions in this study. Qualitative analysis results were complemented by numerical simulations in Matlab, illustrating the obtained stability result.

Stability Analysis of Mathematical Model including Pathogen-Specific Immune System Response with Fractional-Order Differential Equations

In this study, the mathematical model examined the dynamics between pathogen and specific immune system cells (memory T cells) for diseases such as chronic infection and cancer in which nonspecific immune system cells are inadequate to destroy the pathogen and has been suggested by using a system of the fractional-order differential equation with multi-orders. Qualitative analysis of the proposed model reveals the equilibrium points giving important ideas about the proliferation of the pathogen and memory T cells. According to the results of this analysis, the possible scenarios are as follows: the absence of both pathogen and memory T cells, only the existence of pathogen, and the existence of both pathogen and memory T cells. The qualitative analysis of the proposed model has expressed the persistent situations of the disease where the memory T cells either do not be able to respond to the pathogen or continue to exist with the disease-causing pathogen in the host. Results of this analysis are supported by numerical simulations. In the simulations, the time-dependent size of the tumor population under the pressure of the memory T cells was tried to be estimated.

Early treatment gains for antibiotic administration and within human host time series data

As technological improvements continue to infiltrate and impact medical practice, it has become possible to non-invasively collect dense physiological time series data from individual patients in real time. These advances continue to improve physicians' ability to detect and to treat infections early. One important benefit of early detection and treatment of nascent infections is that it leads to earlier resolution. In response to current and anticipated advances in data capture, we introduce the Early Treatment Gain (ETG) as a measure to quantify this benefit. Roughly, we define the gain to be the limiting ratio: ETG=differential change in time of resolutiondifferential change in treatment time.We study the gain using standard dynamical models and demonstrate its use with time series data from Surgical Intensive Care Unit (SICU) patients facing ventilator associated pneumonia. The main conclusion from the mathematical modelling is that the ETG is always greater than one unless there is an effective immune response, in which case the ETG can be less than one. Using real patient time series data, we observe that the formula derived for a linear model can be applied and that this produces a ETG greater than one.

BISTABILITY ANALYSIS OF AN HIV MODEL WITH IMMUNE RESPONSE

Some HIV-infected patients (the so-called post-treatment controllers) can control the virus after cessation of antiretroviral therapy. A small fraction of patients can even naturally maintain undetectable viral load without therapy (they are called elite controllers). The immune response may play an important role in viral control in these patients. In this paper, we analyze a within-host model including immune response to study the virus dynamics in HIV-infected patients. We derived two threshold values for the immune cell proliferation parameter. Below the lower immune proliferation rate, the model has a stable immune-free steady state, which predicts that patients have a high viral load. Above the higher immune proliferation rate, the model has a stable low infected steady state, which indicates that patients are under elite control. Between the two immune thresholds, the model exhibits the dynamic behavior of bistability, which suggests that patients either undergo viral rebound after treatment termination or achieve the post-treatment control. These results may explain the different virus dynamics in HIV-infected patients.

DYNAMICS OF BACTERIA-PHAGE INTERACTIONS WITH IMMUNE RESPONSE IN A CHEMOSTAT

A mathematical model of bacteria-phage interaction in the chemostat is formulated, which incorporates the host immune response with an aim to mimic phage therapy in vivo. It is shown that the host immune response induces the backward bifurcation. Thus, there exists the bistability of phage-free equilibrium with the phage-infection equilibrium. More importantly, it is found that the model exhibits the coexistence of a stable phage-infection equilibrium with a stable periodic solution. The crucial implication of these phenomena is that phage infection fails both from the smaller dose of initial injection and from the larger dose of initial injection. Thus, a proper design of phage dose is necessary for phage therapy. Further analysis indicate that the inhibition effects of bacteria and phages can induce periodic oscillations and modulated oscillation.

Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response

Resistance of developed bacteria to antibiotic treatment is a very important issue, because introduction of any new antibiotic is after a little while followed by the formation of resistant bacterial isolates in the clinic. The significant increase in clinical resistance to antibiotics is a troubling situation especially in nosocomial infections, where already defenseless patients can be unsuccessful to respond to treatment, causing even greater health issue. Nosocomial infections can be identified as those happening within 2 days of hospital acceptance, 3 days of discharge or 1 month of an operation. They influence 1 out of 10 patients admitted to hospital. Annually, this outcomes in 5000 deaths only in UK with a cost to the National Health Service of a billion pounds. Despite these problems, antibiotic therapy is still the most common method used to treat bacterial infections. On the other hand, it is often mentioned that immune system plays a major role in the progress of infections. In this context, we proposed a mathematical model defining population dynamics of both the specific immune cells produced according to the properties of bacteria by host and the bacteria exposed to multiple antibiotics synchronically, presuming that resistance is gained through mutations due to exposure to antibiotic. Qualitative analysis found out infection-free equilibrium point and other equilibrium points where resistant bacteria and immune system cells exist, only resistant bacteria exists and sensitive bacteria, resistant bacteria and immune system cells exist. As a result of this analysis, our model highlights the fact that when an individual’s immune system weakens, he/she suffers more from the bacterial infections which are believed to have been confined or terminated. Also, these results was supported by numerical simulations.

Piecewise virus-immune dynamic model with HIV-1 RNA-guided therapy

Clinical studies have used CD4 T cell counts to evaluate the safety or risk of plasma HIV-1 RNA-guided structured treatment interruptions (STIs), aimed at maintaining CD4 T cell counts above a safe level and plasma HIV-1 RNA below a certain level. However, quantifying and evaluating the impact of STIs on the control of HIV replication and on activation of the immune response remains challenging. Here we extend the virus-immune dynamic system by including a piecewise smooth function to describe the elimination of HIV viral loads and the activation of effector cells under plasma HIV-1 RNA-guided therapy, in order to quantitatively explore the STI strategies. We theoretically investigate the global dynamics of the proposed Filippov system. Our main results indicate that HIV viral loads could either go to infinity or be maintained below a certain level or stabilize at a previously given level, depending on the threshold level and initial HIV virus loads and effector cell counts. This suggests that proper combinations of threshold and initial HIV virus loads and effector cell counts, based on threshold policy, can successfully preclude exceptionally high growth of HIV virus and, in particular, maximize the controllable region.

A two-phase within-host model for immune response and its application to serological profiles of pertussis

We present a simple phenomenological within-host model describing both the interaction between a pathogen and the immune system and the waning of immunity after clearing of the pathogen. We implement the model into a Bayesian hierarchical framework to estimate its parameters for pertussis using Markov chain Monte Carlo methods. We show that the model captures some essential features of the kinetics of titers of IgG against pertussis toxin. We identify a threshold antibody level that separates a large increase in antibody level upon infection from a small increase and accordingly might be interpreted as a threshold separating clinical from subclinical infections. We contrast predictions of the model with observations reported in the literature and based on independent data and find a remarkable correspondence.

Epidemic dynamics and host immune response: a nested approach

This paper proposes an approach for building epidemiological models that incorporate the intra-host pathogen-immunity dynamics. The infected population is structured in terms of pathogen load and level of immunity, and the initial infection load may depend on the load of the individual from whom the infection is acquired. In particular, we focus on the case in which the initial inoculum is taken proportional to the load of the infectant. Possible reinfections are disregarded. Such an approach is applied to formulate an epidemic model with isolation in a closed population by introducing a specific intra-host dynamics. A numerical scheme for the solution of model equations is developed, and some numerical results illustrating the role of the initial inoculum, of the isolation threshold and of the pathogen dynamics on the epidemic evolution are presented. From the simulations the distributions of latency, infectivity, and isolation times can be also derived; however the predictions of the present models differ qualitatively from those of traditional SEIHR models with distributed latency, infectivity and isolation periods.

Mathematical Modeling of Viral Zoonoses in Wildlife

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.

A low dimensional dynamical model of the initial pulmonary innate response to infection

In order to gain a deeper understanding of the onset and progression of pulmonary infections we present and analyze a low dimensional, phenomenological model of infection and the innate immune response in the lungs. Because pulmonary innate immunity has features unique to itself, general mathematical models of the immune system may not be appropriate. The differential equations model that we propose is based on current knowledge of the biology of pulmonary innate immunity and accurately reproduces known features of the initial phase of the dynamics of pulmonary innate system as exhibited in recent experiments. Further, we propose to use the model as a starting point for interrogation with clinical data from a new noninvasive technique for sampling alveolar lining fluid.

Spatiotemporal effects of a possible chemorepulsion of tumor cells by immune system effectors

Tumor-immune system interplay is a landmark of tumor development, and chemotactic attraction of immune system cells towards a tumor is a landmark of immune control. Since tumor cells are capable of chemotactic and chemorepulsive motion, based on a number of analogies between the behavior of tumor cells and that of bacteria, and on the production of potentially chemorepulsive semaphorins by immune systems effectors, we propose here the possibility of chemorepulsion of tumor cells, elicited by chemicals produced by immune system effectors such as macrophages and cytotoxic T lymphocytes. To study the effects of this hypothesized phenomenon within the framework of the interplay of neoplasias with the innate and adaptive immune system, we appropriately extend two well-known models: the tumor-macrophage models by Owen and Sherratt [Owen and Sherratt, J. Theor. Biol., 1998] and the Matzavinos-Chaplain model of tumor-CTL interaction [Matzavinos, Chaplain and Kuznetsov, Math. Med. Biol., 2004]. Our simulations suggest that this mechanism might allow a faster expansion of tumors, and in the concluding remarks we envisage a new possible route of immunoevasion.

The role of host population heterogeneity in the evolution of virulence

I examine here the effects of host heterogeneity in the growth of immune response on the evolution and co-evolution of virulence. The analysis is based on an extension of the 'nested model' by Gilchrist and Sasaki [Modeling host-parasite coevolution, J. Theor. Biol. 218 (2002), pp. 289-308]; the criteria for host and parasite evolution, in the paradigm of adaptive dynamics, for that model are derived in generality. Host heterogeneity is assumed to be fixed at birth according to a lognormal distribution or to the presence of two discrete types. In both cases, it is found that host heterogeneity determines a dramatic decrease in pathogen virulence, since pathogens will tune to the 'weakest' hosts. Finally we clarify how contrasting results present in the literature are due to different modelling assumptions.

Bistability and bacterial infections

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.