Analysis of a mathematical model for tuberculosis: What could be done to increase case detection

This paper presents qualitative and quantitative study of a TB mathematical model to test results from a survey carried out in Benin City, Nigeria. The purpose of the survey was to determine factors that could enhance the case detection rate of tuberculosis. Results from the survey identified four key factors that must be combined for an effective control of TB and increase the case detection rate: effective awareness programme, active cough identification, associated cost factor for treatment of identified cases and effective treatment. The overall effect of these factors on the basic reproduction number under treatment, R_T, of the TB model was considered. In all, a serious concentration on tuberculosis awareness programmes and active cough identification as a marker for someone having TB was shown to significantly reduce the value of the reproduction number, hereby reducing the severity of the disease in the presence of treatment.     

Bacteriological conversion in twenty urinary tuberculosis patients treated with ofloxacin, rifampin and isoniazid: a 10-year follow-up study

Twenty patients with urinary tuberculosis were treated with ofloxacin (200 mg/day, 6 months), rifampin (600 mg/day, 3 months) and isoniazid (300 mg/day, 3 months) between 1989 and 1990. All patients were new cases, diagnosed by observation and/or isolation of Mycobacterium tuberculosis in one of the three morning urine samples. Bacteriological culture conversion (negativization) was assessed as a clinical guide of efficacy, comparing it, as the only parameter, against a control group (150 patients) with urinary tuberculosis who received conventional therapy. Bacteriological follow-up studies were performed in both groups monthly for 6 months, then again 6 months later and then every year for 10 years after completion of treatment. In the 20 patients, the initial culture was positive with over 100 colonies per culture (>50%); the smear was positive in 45% of the patients (most were 2+). All strains were susceptible to rifampin, isoniazid and ofloxacin. Two patients discontinued treatment. Beginning with the first month of treatment, the bacteriological conversion was 100%, 89.5% and 100% in the remaining controls. In the control group, which received conventional treatment, the conversion was: 90%, 87%, 93% and 100% in the remaining controls. Treatment with ofloxacin resulted in a bacteriological conversion similar to that following conventional treatment (p>0.05, Fisher's exact test). After 10 years of patient follow-up, we conclude that ofloxacin, in combination with rifampin and isoniazid (both for 3 months only is effective against M. tuberculosis, providing satisfactory bacteriological and clinical efficacy. Electronic Publication     

A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea

We have developed a dynamic model for tuberculosis (TB) transmission in South Korea using a SEIR model with the time-dependent parameters. South Korea ranked the highest TB incidence among members of the Organization for Economic Cooperation and Development (OECD) in 2005 yr. The observed data from the Korea Center for Disease Control and Prevention (KCDC) shows a certain rise of active-TB incidence individuals after 2001 yr. Because of this sudden jump, we have considered two different periods for best fitting the model: prior to 2001 yr and posterior to 2001 yr. The least-squares fitting has been used for estimating model parameters to the observed data of active-TB incidence. Our model agrees well with the observed data. In this work, we also propose optimal treatment strategies of TB model in South Korea for the future. We have considered three control mechanisms representing distancing, case finding and case holding efforts. Optimal control programs have been proposed in various scenarios, in order to minimize the number of exposed and infectious individuals and the cost of implementing the control treatment.     

Mathematical modeling of the hypothalamic-pituitary-adrenal system activity

Mathematical modeling has proven to be valuable in understanding of the complex biological systems dynamics. In the present report we have developed an initial model of the hypothalamic-pituitary-adrenal system self-regulatory activity. A four-dimensional non-linear differential equation model of the hormone secretion was formulated and used to analyze plasma cortisol levels in humans. The aim of this work was to explore in greater detail the role of this system in normal, homeostatic, conditions, since it is the first and unavoidable step in further understanding of the role of this complex neuroendocrine system in pathophysiological conditions. Neither the underlying mechanisms nor the physiological significance of this system are fully understood yet.     

Mathematical modeling of human granulopoiesis: the possible importance of regulated apoptosis

Steady state human granulopoiesis was modeled by a convection-reaction differential equation of the Rubinow type for the bone marrow granulocyte precursors and an ordinary differential equation for the blood granulocytes. Measured values reported from several laboratories were used as sources for the model proliferation, maturation, and mobilization rates. Due to the large variability in the measured input data, four alternative models were constructed initially, each one with a specific combination of proliferation rate and maturation rate. They were all able to produce output values for the bone marrow neutrophil count and turnover rate close to accepted data, but neither of them could reproduce good values for the differential fractions of the neutrophil precursor stages. The model output was especially sensitive to changes in transit time in the mitotic relative to the postmitotic precursor compartments. When the net proliferation rate was modeled to optimize the bone marrow differential fractions according to published data, the total bone marrow neutrophil count would not fit with published data. However, a composite model optimizing differential fractions, bone marrow neutrophil count, and turnover rate yielded plausible output values and a reduced proliferation rate in the myelocyte stage. This result opens for a possibly substantial apoptosis rate at the myelocyte stage in accordance with results from earlier investigators. However, the result was based on a special choice of precursor transit times, taken from the literature. More precise data concerning granulocyte precursor cycle times, transit times, and differential fractions would radically improve the model's ability to clarify the role of apoptosis during granulocyte production and storage.     

Mathematical modeling of thrombus growth in mesenteric vessels

Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe the realistic features of the phenomenon. The main directions of the generalization of Richardson’s model are: (1) the dependence of platelet activation time on the distance from the injured vessel wall; (2) the non-homogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall; (3) the adequate choice of the phenomenological function describing the dependence of blood velocity on the thrombus size. The generalization of the model corresponds to the main experimental results and theoretical considerations concerning thrombus formation obtained in recent years. The extended model permits to achieve qualitative agreement between model and experimental data.     

Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions

One-third of the world population (approximately 2 billion individuals) is currently infected with Mycobacterium tuberculosis, the vast majority harboring a latent infection. As the risk of reactivation is around 10% in a lifetime, it follows that 200 million of these will eventually develop active pulmonary disease. Only therapeutic or post-exposure interventions can tame this vast reservoir of infection. Treatment of latent infections can reduce the risk of reactivation, and there is accumulating evidence that combination with post-exposure vaccines can reduce the risk of reinfection. Here we develop mathematical models to explore the potential of these post-exposure interventions to control tuberculosis on a global scale. Intensive programs targeting recent infections appear generally effective, but the benefit is potentially greater in intermediate prevalence scenarios. Extending these strategies to longer-term persistent infections appears more beneficial where prevalence is low. Finally, we consider that susceptibility to reinfection is altered by therapy, and explore its epidemiological consequences. When we assume that therapy reduces susceptibility to subsequent reinfection, catastrophic dynamics are observed. Thus, a bipolar outcome is obtained, where either small or large reductions in prevalence levels result, depending on the rate of detection and treatment of latent infections. By contrast, increased susceptibility after therapy may induce an increase in disease prevalence and does not lead to catastrophic dynamics. These potential outcomes are silent unless a widespread intervention is implemented.     

Mathematical modeling of humoral immune response suppression by passively administered antibodies in mice

Although passively administered antibodies are known to suppress the humoral immune response, the mechanism is not fully understood. Here, we developed a mathematical model to better understand the suppression phenomena in mice. Using this model, we tested the generally accepted but difficult to prove "epitope masking hypothesis." To simulate the hypothesis and clearly observe masking of epitopes, we modeled epitope-antibody and epitope-B-cell receptor interactions at the epitope level. To validate this model, we simulated the effect of the antibody affinity and quantity as well as the timing of administration on the suppression, and we compared the results with experimental observations reported in the literature. We then developed a simulation to determine whether the epitope-masking hypothesis alone can explain known immune suppression phenomena, especially the conflicting results on F(ab')2 fragment-induced suppression, which has been shown to be no suppression, or similar to or up to 1000-fold weaker than the suppression by intact antibody. We found that suppression was caused by a synergistic effect of both epitope masking and rapid antigen clearance. Although the latter hypothesis has lost support because FcgammaRI/III mutant mice show antibody-mediated suppression, our simulations predict that, even in FcgammaRI/III mutant mice, the immune response can be suppressed according to the antibody affinity. Our model also effectively reproduced the conflicting results obtained using F(ab')2 fragments. Thus, in contrast to the idea that the F(ab')2 results prove the FcgammaRIIb involvement in suppression, our mathematical model suggests that the epitope-masking hypothesis together with rapid antigen clearance explains the conflicting results.     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy

Ovarian cancer has long been one of the most common forms of cancer in women. The main treatment for ovarian cancer comprises a combination of surgery and chemotherapy. In an effort to improve treatment strategies, a variety of mathematical models have been developed in the literature. In this paper, we consider a simple mathematical model that incorporates tumor growth as well as the effects of chemotherapeutic and surgical treatments in ovarian cancer. We consider several growth models and combine them with different cell-kill hypotheses. Surgery is assumed to eliminate a fixed fraction of tumor cells instantaneously. We discuss how different models predict the optimal sequencing of chemotherapeutic and surgical treatments. This work has been carried out in the context of ovarian cancer; however, the results may also be useful for other kind of cancers.     

Low red cell production may protect against severe anemia during a malaria infection—Insights from modeling

The malaria parasite causes lysis of red blood cells, resulting in anemia, a major cause of mortality and morbidity. Intuitively, one would expect the production of red blood cells to increase in order to compensate for this loss. However, it has been observed that this response is weaker than would be expected. Furthermore, iron supplementation for iron deficient children in malaria endemic regions can paradoxically adversely affect the clinical outcome of malaria infection. A possible explanation may lie in the preference that some malaria parasites show for infecting immature red blood cells (reticulocytes). In the presence of a parasite preference for immature red cells, a rise in red cell production can ‘fuel the fire’ of infection by increasing the availability of the parasite's preferred target cell.We present a mathematical model of red blood cell production and infection in order to explore this hypothesis. We assess the effect of varying the reticulocyte replacement rate and preference of the parasite for reticulocytes on four key outcome measures assessing anemia and parasitemia.For a given level of parasite preference for reticulocytes we uncover an optimal erythropoietic response which minimizes disease severity. Increasing red blood cell production much above this optimum confers no benefit to the patient, and in fact can increase the degree of anemia and parasitemia. These conclusions are consistent with epidemiological studies demonstrating that both iron deficiency and anemia are protective against severe malaria, whilst iron supplementation in malaria endemic regions is with an increased number of malaria related adverse effects. Thus, suppression of red blood cell production, rather than being an unfortunate side effect of inflammation, may be a host protective effect against severe malarial anemia.     

气雾攻击法感染结核病小鼠模型的建立及其综合评价

目的 模拟自然感染方式建立结核病小鼠模型,并对其病理变化进行综合评价.方法 通过气雾攻击方式将结核分枝杆菌H37Rv接种至C57BL/6J小鼠体内.在感染后的4周、6周、8周对小鼠进行micro-CT活体动态扫描,无菌分离肺脏和脾脏,肉眼观察病变情况,活菌菌落计数,组织病理检测(HE和抗酸染色).结果 肉眼观察和micro-CT扫描发现,不同时间小鼠肺部感染情况逐渐加重,至感染后第8周时病变弥漫至整个肺部;HE染色肺组织出现弥漫性肉芽肿样实变;抗酸染色可见结核分枝杆菌.结论 通过大体病变、病理、影像、菌落计数几个方面对建立的小鼠模型进行综合分析,证明利用气雾攻击法感染的结核病小鼠模型建立成功;该模型在形成病变时与结核患者的情况存在一定差异,对其完善的综合评价有助于在相关研究中对该小鼠模型的合理应用.     

Mathematical analysis of a basic model for epidermal wound healing

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.     

Mathematical analysis of multisolute renal flow in a single nephron model of the kidney

A single nephron model, which includes the Bowman's space, Cortical interstitium, and Pelvis as well-stirred baths, is investigated. A boundary value problem, which allows for pelvic reflux, is established for the fluid-multisolute flow in the nephron. The implicit function theorem is used to establish the existence and uniqueness of a solution of the boundary value problem for the case of small permeability coefficients and transport rates.Dedicated to Professor Emertius L. P. Burton, Auburn University, for the research guidance he gave this author thirty years agoSupported in part by NIH Grant 7-ROl-DK 38817     

Lifelong dynamics of human CD4CD25 regulatory T cells: Insights from in vivo data and mathematical modeling

Despite their limited proliferation capacity, regulatory T cells (T_regs) constitute a population maintained over the entire lifetime of a human organism. The means by which T_regs sustain a stable pool in vivo are controversial. Using a mathematical model, we address this issue by evaluating several biological scenarios of the origins and the proliferation capacity of two subsets of T_regs: precursor CD4⁺CD25⁺CD45RO⁻ and mature CD4⁺CD25⁺CD45RO⁺ cells. The lifelong dynamics of T_regs are described by a set of ordinary differential equations, driven by a stochastic process representing the major immune reactions involving these cells. The model dynamics are validated using data from human donors of different ages. Analysis of the data led to the identification of two properties of the dynamics: (1) the equilibrium in the CD4⁺CD25⁺FoxP3⁺T_regs population is maintained over both precursor and mature T_regs pools together, and (2) the ratio between precursor and mature T_regs is inverted in the early years of adulthood. Then, using the model, we identified three biologically relevant scenarios that have the above properties: (1) the unique source of mature T_regs is the antigen-driven differentiation of precursors that acquire the mature profile in the periphery and the proliferation of T_regs is essential for the development and the maintenance of the pool; there exist other sources of mature T_regs, such as (2) a homeostatic density-dependent regulation or (3) thymus- or effector-derived T_regs, and in both cases, antigen-induced proliferation is not necessary for the development of a stable pool of T_regs. This is the first time that a mathematical model built to describe the in vivo dynamics of regulatory T cells is validated using human data. The application of this model provides an invaluable tool in estimating the amount of regulatory T cells as a function of time in the blood of patients that received a solid organ transplant or are suffering from an autoimmune disease.     

Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.     

Mathematical model for the estimation of hemodynamic and oxygenation variables by tissue spectroscopy

This article presents a quasistatic, compartmental model of tissue-level hemodynamics and oxygenation that leads to a set of formulas, which is suitable to calculate important physiological variables from the mean tissue concentration and saturation of hemoglobin, measured by tissue spectroscopy. Dimensioned quantities are represented relative to their baseline value in the equations (relative value = perturbed/baseline). All model parameters are non-dimensional. The model is based and extends on a number of previous works: previous models of similar aim and scope are consolidated, and every critical assumptions and approximations are treated explicitly; extensions include for example the incorporation of the Fahraeus-effect and the separate estimation of the volume changes of the arterial and the venous compartments. The information content of spectroscopic data alone is shown to be valuable, but limited: the relative venous volume, the oxygen extraction fraction and the relative cellulovascular coupling (defined as the ratio of blood flow and oxygen consumption) can be calculated from these data, if the alterations in arterial blood volume are negligible. The number of variables estimated by the derived formulas can be increased if local blood flow is measured simultaneously: in this case, the relative arterial and venous volume and resistance, the oxygen extraction fraction, and the relative oxygen consumption can be determined. Given that this model considers arterial blood pressure, saturation and hematocrit as its inputs, when measured, the model becomes applicable in such conditions as hyper- or hypotension, hypoxic hypoxia, hemodilution and hemorrhage, where these variables do change. The estimation of the changes in arterial resistance can be applied to estimate the extent of an autoregulatory response.