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Mycobacterium tuberculosis (TB) infection can involve all immune system components and can result in different disease outcomes. The antibiotic TB drugs require strict adherence to prevent both disease relapse and mutation of drug- and multidrug-resistant strains. To overcome the constraints of pathogen-directed therapy, host-directed therapy has attracted more attention in recent years as an adjunct therapy to enhance host immunity to fight against this intractable pathogen. The goal of this paper is to investigate in-host TB models to provide insights into therapy development. Focusing on therapy-targeting parameters, the parameter regions for different disease outcomes are identified from an established ODE model. Interestingly, the ODE model also demonstrates that the immune responses can both benefit and impede disease progression, depending on the number of bacteria engulfed and released by macrophages. We then develop two Itô SDE models, which consider the impact of demographic variations at the cellular level and environmental variations during therapies along with demographic variations. The SDE model with demographic variation suggests that stochastic fluctuations at the cellular level have significant influences on (1) the T-cell population in all parameter regions, (2) the bacterial population when parameters located in the region with multiple disease outcomes and (3) the uninfected macrophage population in the parameter region representing active disease. Further, considering environmental variations from therapies, the second SDE model suggests that disease progression can slow down if therapies (1) can have fast return rates and (2) can bring parameter values into the disease clearance regions. [ABSTRACT FROM AUTHOR]

The tumour growth paradox refers to the observation that incomplete treatment of cancers can enhance their growth. As shown here and elsewhere, the existence of cancer stem cells (CSCs) can explain this effect. CSC are less sensitive to treatments, hence any stress applied to the tumour selects for CSC, thereby increasing the fitness of the tumour. In this paper, we use a mathematical model to understand the role of CSC in the progression of cancer. Our model is a rather general system of integro-differential equations for tumour growth and tumour spread. Such a model has never been analysed, and we prove results on local and global existence of solutions, their uniqueness and their boundedness. We show numerically that this model exhibits the tumour growth paradox for all parameters tested. This effect becomes more relevant for small renewal rate of the CSC. [ABSTRACT FROM AUTHOR]