Mathematical models, which are based on a firm understanding of biological interactions, can provide nonintuitive insights into the dynamics of host responses to infectious agents and can suggest new avenues for experimentation. Here, a simple mathematical approach is developed to explore the relation between antiviral immune responses, virus load, and virus diversity. The model results are compared to data on cytotoxic T cell responses and viral diversity in infections with the human T cell leukemia virus (HTLV-1) and the human immunodeficiency virus (HIV-1).

http://science.sciencemag.org/content/sci/272/5258/74.full.pdf

Population dynamics of immune responses to persistent viruses.

Dedication We dedicate this book to our family members who complete us. In particular, M. Ben-chohra's dedication is to his wife, Kheira, and his children, Mohamed, Maroua, and Abdelillah; J. Henderson dedicates to his wife, Darlene, and his descendants, Kathy, Contents Preface xi 1. Preliminaries 1 1.1. Definitions and results for multivalued analysis 1 1.2. Fixed point theorems 4 1.3. Semigroups 7 1.4. Some additional lemmas and notions 9 2. Impulsive ordinary differential equations & inclusions 11 2.1. Introduction 11 2.2. Impulsive ordinary differential equations 12 2.3. Impulsive ordinary differential inclusions 24 2.4. Ordinary damped differential inclusions 49 2.5. Notes and remarks 62 3. Impulsive functional differential equations & inclusions 63 3.1. Introduction 63 3.2. Impulsive functional differential equations 63 3.3. Impulsive neutral differential equations 74 3.4. Impulsive functional differential inclusions 80 3.5. Impulsive neutral functional DIs 95 3.6. Impulsive semilinear functional DIs 107 3.7. Notes and remarks 118 4. Impulsive differential inclusions with nonlocal conditions 119 4.1. Introduction 119 4.2. Nonlocal impulsive semilinear differential inclusions 119 4.3. Existence results for impulsive functional semilinear differential inclusions with nonlocal conditions 136 4.4. Notes and remarks 145 5. Positive solutions for impulsive differential equations 147 5.1. Introduction 147 5.2. Positive solutions for impulsive functional differential equations 147 5.3. Positive solutions for impulsive boundary value problems 154 5.4. Double positive solutions for impulsive boundary value problems 159 5.5. Notes and remarks 165 viii Contents 6. Boundary value problems for impulsive differential inclusions 167 6.1. Introduction 167 6.2. First-order impulsive differential inclusions with periodic boundary conditions 167 6.3. Upper-and lower-solutions method for impulsive differential inclusions with nonlinear boundary conditions 184 6.4. Second-order boundary value problems 191 6.5. Notes and remarks 198 7. Nonresonance impulsive differential inclusions 199 7.1. Introduction 199 7.2. Nonresonance first-order impulsive functional differential inclusions with periodic boundary conditions 199 7.3. Nonresonance second-order impulsive functional differential inclusions with periodic boundary conditions 209 7.4. Nonresonance higher-order boundary value problems for impulsive functional differential inclusions 217 7.5. Notes and remarks 227 8. Impulsive differential equations & inclusions with variable times 229 8.1. Introduction 229 8.2. First-order impulsive differential equations with variable times 229 8.3. Higher-order impulsive differential equations with variable times 235 8.4. Boundary value problems for differential inclusions with variable times 241 8.5. Notes and remarks 252 9. Nondensely defined impulsive differential equations & inclusions 253 9.1. Introduction 253 9.2. Nondensely defined impulsive semilinear differential equations with nonlocal conditions 253 9.3. Nondensely defined …

Impulsive Differential Equations and Inclusions

Background/Aims: The association between postoperative infection and prolonged survival in high-grade glioma is still a matter of debate. Previously we demonstrated that the intracerebral (i.c.) injection of heat-inactivated staphylococcal epitopes (HISE) resulted in a well-defined infux of immunocompetent cells across the blood-brain barrier. The present study investigated the potential antitumoral effect of HISE-immunostimulation in an experimental glioma model. Methods: Wistar rats were intracerebrally implanted with 9L gliosarcoma cells (n=6), 9L cells mixed with HISE (n=12), or phosphate buffered saline (n=4). Tumor growth was measured by serial magnetic resonance imaging (MRI). After death due to the tumor burden, the brains were histopathologically assessed for inflammation and oncolysis. A toxicity assay was performed to quantify potential impairment of HISE on tumor cell growth in vitro. Results: Animals treated by HISE showed a significant increase in average survival and even complete regression of an already established mass in one case. Naive 9L gliosarcomas failed to recruit significant numbers of systemic immune cells. In contrast, concomitant intracerebral HISE inoculation lead to a oncolysis and a distinct peri- and intratumoral infiltration of macrophages, CD8 and CD4 co-expressing T-lymphocytes in two thirds of the tumor-bearing animals. The toxicity screening showed HISE-mediated oncolysis to be ineffective ex vivo. Conclusion: This study describes a novel approach for combatting malignant glioma using inactivated staphylococci as potent immunomodulators. Our results provide an outline for investigating the strategic potential of bacteria as emerging future therapeutics.

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Intracerebral Administration of Heat-Inactivated Staphylococcus Epidermidis Enhances Oncolysis and Prolongs Survival in a 9L Orthotopic Gliosarcoma Model

The immune system is a complex system of cells and molecules that can provide us with a basic defense against pathogenic organisms. Like the nervous system, the immune system performs pattern recognition tasks, learns, and retains a memory of the antigens that it has fought. The immune system contains more than 10{sup 7} different clones of cells that communicate via cell-cell contact and the secretion of molecules. Performing complex tasks such as learning and memory involves cooperation among large numbers of components of the immune system and hence there is interest in using methods and concepts from statistical physics. Furthermore, the immune response develops in time and the description of its time evolution is an interesting problem in dynamical systems. In this paper, the authors provide a brief introduction to the biology of the immune system and discuss a number of immunological problems in which the use of physical concepts and mathematical methods has increased our understanding. {copyright} {ital 1997} {ital The American Physical Society}

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Immunology for physicists

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

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The mathematics of cancer: integrating quantitative models

Although vaccines against measles have been routinely applied over a quarter of a century, measles is still persistent in Israel, with major epidemics roughly every 5 years. Recent serological analyses have shown that only 85% of Israelis aged 18 years have anti-measles IgG antibodies. Considering the high transmissibility of the virus and the high level of herd immunity required for disease eradication, the Israeli vaccination policy against measles is now being reevaluated. Motivated by theoretical studies of populations in perturbed environments, we examined the possibility of replacing the conventional cohort vaccination strategy by a pulse strategy--i.e., periodic vaccination of several age cohorts at the same time. Numerical studies of a deterministic age-structured model suggest that vaccination, which renders immunity to no more than 85% of the susceptible children aged 1-7 years, once every 5 years will suffice to prevent epidemics in Israel, where infection rate is highest amongst schoolchildren. The model suggests that by using such a strategy the density of susceptible individuals is always kept below the threshold above which recurrent epidemics will be maintained. Analysis of simpler, non-age-structured, models serves to clarify the basic properties of the proposed strategy. Our theoretical results indicate that the advantages and disadvantages of a pulse strategy should be seriously examined in Israel and in countries with similar patterns of measles virus transmission.

https://www.pnas.org/content/pnas/90/24/11698.full.pdf

Pulse mass measles vaccination across age cohorts

Glioblastoma (GBM), a highly aggressive (WHO grade IV) primary brain tumor, is refractory to traditional treatments, such as surgery, radiation or chemotherapy. This study aims at aiding in the design of more efficacious GBM therapies. We constructed a mathematical model for glioma and the immune system interactions, that may ensue upon direct intra-tumoral administration of ex vivo activated alloreactive cytotoxic-T-lymphocytes (aCTL). Our model encompasses considerations of the interactive dynamics of aCTL, tumor cells, major histocompatibility complex (MHC) class I and MHC class II molecules, as well as cytokines, such as TGF-β and IFN-γ, which dampen or increase the pro-inflammatory environment, respectively. Computer simulations were used for model verification and for retrieving putative treatment scenarios. The mathematical model successfully retrieved clinical trial results of efficacious aCTL immunotherapy for recurrent anaplastic oligodendroglioma and anaplastic astrocytoma (WHO grade III). It predicted that cellular adoptive immunotherapy failed in GBM because the administered dose was 20-fold lower than required for therapeutic efficacy. Model analysis suggests that GBM may be eradicated by new dose-intensive strategies, e.g., 3 × 108 aCTL every 4 days for small tumor burden, or 2 × 109 aCTL, infused every 5 days for larger tumor burden. Further analysis pinpoints crucial bio-markers relating to tumor growth rate, tumor size, and tumor sensitivity to the immune system, whose estimation enables regimen personalization. We propose that adoptive cellular immunotherapy was prematurely abandoned. It may prove efficacious for GBM, if dose intensity is augmented, as prescribed by the mathematical model. Re-initiation of clinical trials, using calculated individualized regimens for grade III–IV malignant glioma, is suggested.

Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics

Background: Therapeutic vaccination against disseminated prostate cancer (PCa) is partially effective in some PCa patients. We hypothesized that the efficacy of treatment will be enhanced by individualized vaccination regimens tailored by simple mathematical models. Methodology/Principal Findings: We developed a general mathematical model encompassing the basic interactions of a vaccine, immune system and PCa cells, and validated it by the results of a clinical trial testing an allogeneic PCa whole-cell vaccine. For model validation in the absence of any other pertinent marker, we used the clinically measured changes in prostate-specific antigen (PSA) levels as a correlate of tumor burden. Up to 26 PSA levels measured per patient were divided into each patient’s training set and his validation set. The training set, used for model personalization, contained the patient’s initial sequence of PSA levels; the validation set contained his subsequent PSA data points. Personalized models were simulated to predict changes in tumor burden and PSA levels and predictions were compared to the validation set. The model accurately predicted PSA levels over the entire measured period in 12 of the 15 vaccination-responsive patients (the coefficient of determination between the predicted and observed PSA values was R 2 =0.972). The model could not account for the inconsistent changes in PSA levels in 3 of the 15 responsive patients at the end of treatment. Each validated personalized model was simulated under many hypothetical immunotherapy protocols to suggest alternative vaccination regimens. Personalized regimens predicted to enhance the effects of therapy differed among the patients. Conclusions/Significance: Using a few initial measurements, we constructed robust patient-specific models of PCa immunotherapy, which were retrospectively validated by clinical trial results. Our results emphasize the potential value and feasibility of individualized model-suggested immunotherapy protocols.

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Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models.

A formal method is provided for predicting the effect on treatment efficacy of cell-cycle-phase-specific drugs, such as the AIDS drug zidovudine (AZT) or the cancer drug cytosine arabinoside (ara-C). Our analysis shows that the elimination of somatic cells or viruses depends not only on the drug's pharmacokinetic and pharmacodynamic properties, but also on the duration of the dosing interval per se and on the life-cycle parameters, that is, the duration of the drug-susceptible life phase, the duration of the whole life cycle, and the proliferation rate. The results support those of simplified models in showing that drug toxicity to the host may be minimized when the dosing interval is an integer multiple of the average cycle time of the host susceptible cells. This prediction has been verified in mice treated with AZT or ara-C.

A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs.

The newly characterized interleukin (IL)-21 plays a central role in the transition from innate immunity to adaptive immunity and shows substantial tumor regression in mice. IL-21 is now developed as a cancer immunotherapeutic drug, but conditions for efficacious therapy, and the conflicting immunostimulatory and immunoinhibitory influence of the cytokine, are yet to be defined. We studied the effects of IL-21 on tumor eradication in a mathematical model focusing on natural killer (NK) cell-mediated and CD8+ T-cell-mediated lysis of tumor cells. Model parameters were estimated using results in tumor-bearing mice treated with IL-21 via cytokine gene therapy (CGT), hydrodynamics-based gene delivery (HGD), or standard interval dosing (SID). Our model accurately retrieved experimental growth dynamics in the nonimmunogenic B16 melanoma and the immunogenic MethA and MCA205 fibrosarcomas, showing a strong dependence of the NK-cell/CD8+ T-cell balance on tumor immunogenicity. Moreover, in melanoma, simulations of CGT-like dosing regimens, dynamically determined according to tumor mass changes, resulted in efficient disease elimination. In contrast, in fibrosarcoma, such a strategy was not superior to that of fixed dosing regimens, HGD or SID. Our model supports clinical use of IL-21 as a potent stimulator of cellular immunity against cancer, and suggests selecting the immunotherapy strategy according to tumor immunogenicity. Nonimmunogenic tumors, but not highly immunogenic tumors, should be controlled by IL-21 dosing, which depends on tumor mass at the time of administration. This method imitates, yet amplifies, the natural anticancer immune response rather than accelerates only one of the response arms in an unbalanced manner.

https://cancerres.aacrjournals.org/content/canres/66/14/7293.full.pdf

Zvia Agur

Papers

Pulse mass measles vaccination across age cohorts

Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics

Predicting outcomes of prostate cancer immunotherapy by personalized mathematical models.

A theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs.

Cancer immunotherapy by interleukin-21: potential treatment strategies evaluated in a mathematical model.