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Ischaemic Hepatitis (IH) or Hypoxic Hepatitis (HH) also known as centrilobular liver cell necrosis is an acute liver injury characterized by a rapid increase in serum aminotransferase. The liver injury typically results from another underlying medical conditions like cardiac failure, respiratory failure and septic shock in which the liver becomes damaged due to deprivation of either blood or oxygen. IH is a potentially lethal condition which is often preventable if diagnosed properly. {\em Unfortunately, mechanism that causes IH are often not well understood, making it difficult to diagnose or accurately quantify the patterns of related biomarkers}. In most cases, currently the only way to determine a case of IH (i.e., to diagnose it) is to rule out all other possible conditions for other liver injuries. A better understanding of the liver9s response to IH is necessary to aid in its diagnosis, measurement and improve outcomes. {\bf The goal of this study, is to identify mechanisms that can alter a few associated biomarkers for reducing density of damaged hepatocytes, and thus reduce chances of IH}. To this end, we develop a mathematical model capturing dynamics of hepatocytes in the liver through the rise and fall of associated liver enzymes aspartate transaminase (AST), alanine transaminase (ALT) and lactate dehydrogenase (LDH) related to condition of IH. %It studies liver cell death in response to IH caused by chronic heart failure and cardiovascular shock. The model analysis provides a novel approach to predict the level of biomarkers given variations in the systemic oxygen in the body. Using IH patient data in US, novel model parameters are described and then estimated for the first time to capture real time dynamics of hepatocytes in the presence and absence of IH condition. Different scenarios of patient conditions were also analyzed and validated using empirical information. This study and its results may allow physicians to estimate the extent of liver damage in a IH patient based on their enzyme levels and receive faster treatment on real time basis.

Prions are proteins that cause fatal neurodegenerative diseases. The misfolded conformation adopted by prions can be transmitted to other normally folded proteins. Therapeutics to stop prion proliferation have been studied experimentally; however, it is not clear how the combination of different types of treatments can decrease the growth rate of prions in the brain. In this article, we combine the implementation of pharmacological chaperones and interferons to develop a novel model using a non-linear system of ordinary differential equations and study the quantitative effects of these two treatments on the growth rate of prions. This study aims to identify how the two treatments affect prion proliferation, both individually and in tandem. We analyze the model, and qualitative global results on the disease-free and disease equilibria are proved analytically. Numerical simulations, using parameter values from in vivo experiments that provide a pharmaceutically important demonstration of the effects of these two treatments, are presented here. This mathematical model can be used to identify and optimize the best combination of the treatments within their safe ranges.

Standard heuristic mathematical models of population dynamics are often constructed using ordinary differential equations (ODEs). These deterministic models yield pre-dictable results which allow researchers to make informed recommendations on public policy. A common immigration, natural death, and fission ODE model is derived from a quantum mechanics view. This macroscopic ODE predicts that there is only one stable equilibrium point . We therefore presume that as t → ∞, the expected value should be . The quantum framework presented here yields the same standard ODE model, however with very unexpected quantum results, namely . The obvious questions are: why isn’t , why are the probabilities ≈ 0.37, and where is the missing probability of 0.26? The answer lies in quantum tunneling of probabilities. The goal of this paper is to study these tunneling effects that give specific predictions of the uncertainty in the population at the macroscopic level. These quantum effects open the possibility of searching for “black–swan” events. In other words, using the more sophisticated quantum approach, we may be able to make quantitative statements about rare events that have significant ramifications to the dynamical system.

Prion diseases are lethal neurodegenerative disorders such as mad cow disease in bovines, chronic wasting disease in cervids, and Creutzfeldt-Jakob disease in humans. They are caused when the prion protein PrPC misfolds into PrPSc, which is capable of inducing further misfolding in healthy PrPC proteins. Recent in vivo experiments show that pharmacological chaperones can temporarily prevent this conversion by binding to PrPC molecules, and thus constitute a possible treatment. A second strategic approach uses interferons to decrease the concentration of PrPSc. In order to study the quantitative effects of these treatments on prion proliferation, we develop a model using a non-linear system of ordinary differential equations. By evaluating their efficacy and potency, we find that interferons act at lower doses and achieve greater prion decay rates. However, there are benefits in combining them with pharmacological chaperones in a two-fold therapy. This research is crucial to guide future prion experiments and inform potential treatment protocols.

Infection of Herpes Simplex Virus type 2 (HSV-2) is a lifelong sexually transmitted disease. According to the Center for Disease Control and Prevention (CDC), 11.9% of the United States (U.S.) population was infected with HSV-2 in 2015-2016. The HSV-2 pathogen establishes latent infections in neural cells and can reactivate causing lesions later in life, a strategy that increases pathogenicity and allows the virus to evade the immune system. HSV-2 infections are currently treated by Acyclovir only in the non-constitutional stage, marked by genital skin lesions and ulcers. However, patients in the constitutional stage expressing mild and common (with other diseases) symptoms, such as fever, itching and painful urination, remain difficult to detect and are untreated. In this study, we develop and analyze a mathematical model to study the transmission and control of HSV-2 among the U.S. population between the ages of 15-49 when there are options to treat individuals in different stages of their pathogenicity. In particular, the goals of this work are to study the effect on HSV-2 transmission dynamics and to evaluate and compare the cost-effectiveness of treating HSV-2 infections in both constitutional and non-constitutional stages (new strategy) against the current conventional treatment protocol for treating patients in the non-constitutional stage (current strategy). Our results distinguish model parameter regimes where each of the two treatment strategies can optimize the available resources and consequently gives the long-term reduced cost associated with each treatment and incidence. Moreover, we estimated that the public health cost of HSV-2 with the proposed most cost-effective treatment strategy would increase by approximately 1.63% in 4 years of implementation. However, in the same duration, early treatment via the new strategy will reduce HSV-2 incidence by 42.76% yearly and the reproduction number will decrease to 0.84 from its current estimate of 2.5. Thus, the proposed new strategy will be significantly cost-effective in controlling the transmission of HSV-2 if the strategy is properly implemented.

Cutaneous leishmaniasis (CL) is a neglected tropical disease transmitted by species of Phlebotominae sand flies. CL is responsible for more than 1000 reported cases per year in Ecuador. Vector collection studies in Ecuador suggest that there is a strong association between the ecological diversity of an ecosystem, the presence of potential alternative or reservoir hosts and the abundance of sand fly species. Data collected from a coastal community in Ecuador showed that Leishmania parasites may be circulating in diverse hosts, including mammalian and potentially avian species, and these hosts may serve as potential hosts for the parasite. There has been limited reporting of CL cases in Ecuador because the disease is non-fatal and its surveillance system is passive. Hence, the actual incidence of CL is unknown. In this study, an epidemic model was developed and analysed to understand the complexity of CL transmission dynamics with potential non-human hosts in the coastal ecosystem and to estimate critical epidemiological quantities for Ecuador. The model is fitted to the 2010 CL outbreak in the town of Valle Hermoso in the Santo Domingo de los Tsachilas province of Ecuador and parameters such as CL transmission rates in different types of hosts (primary and alternative), and levels of case reporting in the town are estimated. The results suggest that the current surveillance in this region fails to capture 38% (with 95% CI (29%, 47%)) of the actual number of cases under the assumption that alternative hosts are dead-end hosts and that the mean CL reproduction number in the town is 3.9. This means that on the average 3.9 new human CL cases were generated by a single infectious human in the town during the initial period of the 2010 outbreak. Moreover, major outbreaks of CL in Ecuador in coastal settings are unavoidable until reporting through the surveillance system is improved and alternative hosts are managed properly. The estimated infection transmission probabilities from alternative hosts to sand flies, and sand flies to alternative hosts are 27% and 32%, respectively. The analysis highlights that vector control and alternative host management are two effective programmes for Ecuador but need to be implemented concurrently to avoid future major outbreaks.

All mathematical models are approximate and their usefulness depends on our understanding the uncertainty inherent in the predictions. Uncertainties can affect the reliability of the results at every stage of computation; they may grow or even shrink as the solution of the model evolves. Often these inherent uncertainties cannot be made arbitrarily small by a more complex model or additional computation and we must understand how the uncertainty in the model parameters, the initial conditions, and the model itself, lead to uncertainties in the model predictions. This chapter is an introductory survey of sensitivity analysis and illustrates how to define the derivative of the model solution as a function of the model input and determine the relative importance of the model parameters on the model predictions.