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The SARS-CoV-2 leads to a worldwide COVID-19 pandemic, which has caused tremendous damage to the world. In this paper, we develop a dynamic model in vivo, fitting and estimating parameters for T lymphocytes and pro-inflammatory cytokines IL-6 in patients with mild and severe COVID-19 at Yale New Haven Hospital through the GWMCMC algorithm. Meanwhile, we also analyze the structural identifiability and practical identifiability of the model. Further, we add time-varying parameters to the model, using the least squares method to perform data fitting and parameter estimation on survivors and non-survivors of the Italian infectious disease hospital. Then analyze the similarities and differences in immune response mechanisms between the two countries. Finally, we demonstrate the existence and stability of the equilibrium state of the model and analyze the Hopf bifurcation at the positive equilibrium state by using the central manifold theory and normal form theory. This result may explain the recurrence of infection in some COVID-19 patients. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a mathematical model of pancreatic cancer, which extends the existing pancreatic cancer models with regulatory T cells (Tregs) and Treg inhibitory therapy. The model consists of tumor-immune interaction and immune suppression from Tregs. In the absence of treatments, we first characterize the system dynamics by locating equilibrium points and determining stability properties. Next, cytokine induced killer (CIK) immunotherapy is incorporated. Numerical simulations of prognostic results illustrate that the median overall survival associated with treatment can be prolonged approximately from 7 to 13 months, which is consistent with the clinical data. Furthermore, we consider cyclophosphamide (CTX) therapy as well as the combined therapy with CIK and CTX. Intensive simulation results suggest that both CTX therapy and the combined CIK/CTX therapy can reduce the number of Tregs and increase the overall survival (OS), but Tregs and tumor cells will gradually rise to equilibrium state as long as therapies are ceased. [ABSTRACT FROM AUTHOR]

This paper deals with the interaction between dispersed cancer cells and the major populations of the immune system, namely, the T helper cells, T Cytotoxic cells, B cells, and antibodies produced. The system is described by a set of five ordinary differential equations. Both local and global stability of the system has been investigated. It has been observed that under appropriate conditions this interaction is capable of controlling the growth of these cancer cells. The analytical findings are supported by numerical and computational analytical methods. [ABSTRACT FROM AUTHOR]

We present a review and comparison of several recent differential equations models of treatment of HIV-1 infection. We seek to clarify the role of the natural anti-HIV immune response and determine its effect upon optimal treatment schemes. In this paper, we consider systems in which treatment is expressed as a forcing function, as well as those in which we determine optimal treatment via control theoretic techniques. The primary goal of this study is to compare treatment schemes for systems in which a natural nonconstant immune response of the patient is considered explicitly with those that consider implicitly a constant non-specific immune response. We find that when the natural immune response can be boosted sufficiently, drug levels may not need to be as high as previously supposed. This implies that a treatment scenario in which intervals of drug treatment are alternated with some form of immune-boosting therapy may be highly beneficial in terms of reducing toxicity to the patient. Additionally, in developing countries where HIV infection is widespread and sufficient funds are not available to supply rigourous drug regimens, the implications of these models are profound, as they suggest methods of treating HIV at a minimal cost. [ABSTRACT FROM AUTHOR]

In this paper, neglecting the effects of free virus, we consider a simple model of cell-to-cell spread of HIV-1. We discuss the impact of the CD8+ cell non-cytotoxic anti-viral response (CNAR) and cytotoxic T lymphocyte (CTL) activity on infection by HIV-1. Two types of models are considered: the ordinary differential equation (ODE) model and the discrete time delay differential equation (DDE) system. The steady states of the ODE model are globally asymptotically stable respectively under two threshold criteria. In the DDE model, the global stability of the infected steady state of the ODE model becomes only ultimately stable. Moreover, at a certain interval of the time delay, the DDE model will produce Hopf bifurcation or periodic solutions. The introduction of CTL and CNAR will change the values of these steady states and induce fluctuations in the cell concentration. It also affects the critical value of the time delay and is of utility in the interpretation of typical HIV-dynamics data resulting from clinical trials. The DDE model produces sustained infective oscillations in some real parameter values, which is different from the result of the similar cell-free viral spread model with delay. [ABSTRACT FROM AUTHOR]

Immunotherapy has become a rapidly developing approach in the treatment of cancer. Cancer immunotherapy aims at promoting the immune system response to react against the tumor. In view of this, we develop a mathematical model for immune–tumor interplays with immunotherapeutic drug, and strategies for optimally administering treatment. The tumor–immune dynamics are given by a system of five coupled nonlinear ordinary differential equations which represent the interaction among tumor-specific CD4+T cells, tumor-specific CD8+T cells, tumor cells, dendritic cells and the immuno-stimulatory cytokine interleukin-2 (IL-2), extended through the addition of a control function describing the application of a dendritic cell vaccination. Dynamical behavior of the system is studied from the analytical as well as numerical points of view. The main aim is to investigate the treatment regimens which minimize the tumor cell burden and the toxicity of dendritic cell vaccination. Our numerical simulations demonstrate that the optimal treatment strategies using dendritic cell vaccination reduce the tumor cell burden and increase the cell count of CD4+T cells, CD8+T cells, dendritic cells and IL-2. The most influential parameters having significant impacts on the tumor cells are identified by employing the approach of global sensitivity analysis. [ABSTRACT FROM AUTHOR]

We consider a stochastic model for T-cell activation proposed in Refs. [1] and [2] to compare the specificity and sensitivity of two different strategies for T-cell activation that utilize the history of phosphorylation of T-cell receptor (TCR). We compare these two strategies when the temporal signals/events that are essential for progressive T-cell activation are suppressed by blockade of CD4 receptor that may have caused by disease or therapeutic effects.3–6 We show that under these conditions, a threshold-strategy which is capable of maintaining a threshold (for total number of phosphorylated TCRs by time T) for a further duration ΔT performs better in discriminating agonist peptides than a single-threshold strategy (reached by time T) leading to T-cell activation using the Wentzell-Friedlin theory for large deviations for stochastic processes.7,8 [ABSTRACT FROM AUTHOR]

We develop a mathematical model of tumor-immune interactions, including six populations (tumor cells, CD8 + T cells, natural killer (NK) cells, dendritic cells, helper T cells, cytokine interleukin-12 (IL-12)) and three potential treatments (chemotherapy, Tumor-infiltrating lymphocyte (TIL) therapy and IL-12 therapy). We characterize the dynamics of our model without treatment through stability and sensitivity analysis, which provides a broad understanding of the long-term qualitative behavior. To find the best combination of the chemo-immunotherapy regimens to eliminate tumors, we formulate an optimal control problem with path constraints of total drug dose and solve it numerically with the optimal control software Pyomo. We also simulate the scenarios of traditional treatment protocols as a comparison and find that our optimal treatment strategies have a better therapeutic effect. In addition, numerical simulation results show that IL-12 therapy is a good adjunctive therapy and has a high potential for inhibiting a large tumor in combination with other therapy. In most cases, combination therapy is more effective than a single treatment. [ABSTRACT FROM AUTHOR]

Based on experimental results of a mouse model provided in the literature, we develop a mathematical model by using system biology approach, aiming to investigate immunotherapy for 4T1 breast cancer. It is worth to mention that only 4 types of cells (tumor cells, CD8 + T cells, regular T cells (Tregs), and tumoricidal myeloid CD11b + Gr1dim cells) are quantitatively measured in experiments, which make the immunotherapy modelling more difficult since only limited system knowledge is available. To overcome the difficulty, the mathematical model is proposed by employing Evolutionary Computation to optimize the system parameters. Furthermore, with the mathematical model, analysis can be conducted to capture the inherent properties of the model, such as the number and stability of equilibria, and parameter sensitivity analysis, which disclose the nature of 4T1 breast cancer from a system biological perspective. Not limited to replication of experimental results, we further show that the mathematical model is in fact a prognostic immunotherapy model that can predict treatment outcomes of various cases; for instance, different combinations of drug delivery schedules. By virtue of computational convenience, it is relatively easy to intensively investigate most of the treatments that are impossible for animal models or clinical trials. In other words, a mathematical model based on system biology can provide meaningful reference when exploiting more effective treatment protocols. [ABSTRACT FROM AUTHOR]

A mathematical model is developed to assess humoral and cellular immune responses against Trypanosoma cruzi infection. Analysis of the model shows a unique non-trivial equilibrium, which is locally asymptotically stable, except in the case of a strong cellular response. When the proliferation of the activated CD8 T cells is increased, this equilibrium becomes unstable and a limit cycle appears. However, this behavior can be avoided by increasing the action of the humoral response. Therefore, unbalanced humoral and cellular responses can be responsible for long asymptomatic period, and the control of Trypanosoma cruzi infection is a consequence of well coordinated action of both humoral and cellular responses. [ABSTRACT FROM AUTHOR]

We consider a mathematical model that describes a viral infection with lytic and non-lytic immune responses. One of the main features of the model is that it includes a rate of linear activation of cytotoxic T lymphocytes (CTLs) immune response, a constant production rate of CTLs export from thymus, and a nonlinear attack rate for each immune effector mechanism. Stability of the infection-free equilibrium, and existence, uniqueness and stability of an immune-controlled equilibrium, are investigated. The stability results are given in terms of the basic reproductive number. We use the method of Lyapunov functions to study the global stability of the infection-free equilibrium and the immune-controlled equilibrium. We give a sufficient condition on the non-lytic-immune attack rate for the global asymptotic stability of the immune-controlled equilibrium. By theoretical analysis and numerical simulations, we show that the lytic and non-lytic activities are required to combat a viral infection. [ABSTRACT FROM AUTHOR]

Analysis of changes in viral load after initiation of treatment with potent antiretroviral agents has provided substantial insights into the dynamics of human immunodeficiency virus type 1. We built a simple mathematics model to study the effect of latent-infected resting memory CD4+ T cells during the HIV infection and highly active anti-retroviral therapy (HAART). Through analysis of eight patients who received HAART in China, we have an insight into the mechanisms of resting memory CD4+ T cells in HIV infection. Simulations show that new infections still exist in the eight patients even under the HAART. Also, because of the long half-life of resting infected memory CD4+ T cells, removal of HIV from patient could take considerably longer time or be unattainable. [ABSTRACT FROM AUTHOR]

We investigate a delayed HIV-1 infection model with general nonlinear incidence functions and two classes of target cells: CD4+ T-cells and macrophages. To account for the time lags between viruses' entry into the corresponding two types of target cells and the production of new virus particles, we incorporate four distributed intracellular delays into the model. We show that the basic reproduction number ℜ0 is the sum of the basic reproduction numbers of HIV-1 infection with CD4+ T-cells and that with macrophages; moreover, if ℜ0 is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population and macrophages; whereas if ℜ0 is larger than one, then the viral concentration maintains at some constant level. It is shown, from both our analytic and numeric results, that ignoring the contributions of macrophages to HIV-1 infection and production will underestimate both the risk of HIV-1 infection and the viral load when persisting. This highlights the important effects of multiple target cells on HIV-1 infection. [ABSTRACT FROM AUTHOR]

Current consensus postulates that the class I-antigen processing system is evolved to present microbial antigens to specific T cells. Since such cells are rare and short-lived, they require three to five days to attain fighting strength. During this critical period he innate immune system holds back the briskly multiplying pathogens. Nevertheless, a T cell response is measurable in the lymph nodes draining the infection site within 12 to 18 h. In order to explain this paradox here we suggest a new T cell model. This is based on the observation that T cells require continuous contact of the T cell receptor (TCR) with selecting self-peptide-major histocompatibility complex (MHC) molecules in the periphery for their survival. We postulate that a dynamic steady state, a so-called coupled system is formed through low affinity complementary TCR-MHC interactions between T cells and host cells. Under such condition it is sufficient to recognize what is self in order to attack what is not self. A coupled system is regulated via soluble forms of peptide-MHC and TCR molecules by the law of mass action. In a coupled system one signal is sufficient for T cell activation. The new model implies that a significant fraction of the naive polyclonal T cells are recruited into the first line of defense from the very outset of an infection, so the number of activated T cells is increased by several orders of magnitude compared to conventional models. The one-signal model also predicts that therapeutic administration of soluble agonist or antagonist T cell receptor ligands may be able to fine tune the homeostatic physiological regulatory mechanism and thus improve the treatment of some chronic diseases such as metastatic cancer, HIV/AIDS, and transplantation. [ABSTRACT FROM AUTHOR]

We present and analyze a three-species mathematical model of tumor progression and regression using prey-predator dynamics. The tumor cell population is supposed to be the prey species and the cytotoxic T-lymphocytes (CTL) and macrophages (which are killer cells of the immune system) as their predator. Using a ratio-dependent predation functional form, the ODE model is analyzed for stability criteria. A discrete time delay required for destruction of tumor cells is then incorporated into the basic model and the resulting delayed model is studied to explore the significance of various system parameters controlling the stability of the system. The active killer cells are shown to be the key population species controlling the system dynamics. A range of the delay parameter is estimated that ensures stability of different system components. Numerical simulations are performed to support analytical findings. [ABSTRACT FROM AUTHOR]