A mathematical model for the Andean Tiwanaku civilization collapse: climate variations

Classical mathematical models for prediction of response to chemotherapy and immunotherapy

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

Mathematical oncology uses quantitative models for prediction of tumor growth and treatment response. The theoretical foundation of mathematical oncology is provided by six classical mathematical models: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. These models have been introduced decades ago, have been used in thousands of scientific articles and are part of textbooks and curricula in mathematical oncology. However, these models have not been systematically tested in clinical data from actual patients. In this study, we have collected quantitative tumor volume measurements from thousands of patients in five large clinical trials of cancer immunotherapy. We use this dataset to systematically investigate how accurately mathematical models can describe tumor growth, showing that there are pronounced differences between models. In addition, we show that two of these models can predict tumor response to immunotherapy and chemotherapy at later time points when trained on early tumor growth dynamics. Thus, our article closes a conceptual gap in the literature and at the same time provides a simple tool to predict response to chemotherapy and immunotherapy on the level of individual patients.

Interconnections Accelerate Collapse in a Socio-Ecological Metapopulation

Over-exploitation of natural resources can have profound effects on both ecosystems and their resident human populations. Simple theoretical models of the dynamics of a population of human harvesters and the abundance of a natural resource being harvested have been studied previously, but relatively few models consider the effect of metapopulation structure (i.e., a population distributed across discrete patches). Here we analyze a socio-ecological metapopulation model based on an existing single-population model used to study persistence and collapse in human populations. Resources grow logistically on each patch. Each population harvests resources on its own patch to support population growth, but can also harvest resources from other patches when their own patch resources become scarce. We show that when populations are allowed to harvest resources from other patches, the peak population size is higher, but subsequent population collapse is significantly accelerated and across a broader parameter regime. As the number of patches in the metapopulation increases, collapse is more sudden, more severe, and occurs sooner. These effects persist under scenarios of asymmetry and inequality between patches. Our model makes simplifying assumptions in order to facilitate insight and understanding of model dynamics. However, the robustness of the model prediction suggests that more sophisticated models should be developed to ascertain the impact of metapopulation structure on socio-ecological sustainability.

Innovation and the growth of human population

Biodiversity is sustained by and is essential to the services that ecosystems provide. Different species would use these services in different ways, or adaptive strategies, which are sustained in time by continuous innovations. Using this framework, we postulate a model for a biological species (Homo sapiens) in a finite world where innovations, aimed at increasing the flux of ecosystem services (a measure of habitat quality), increase with population size, and have positive effects on the generation of new innovations (positive feedback) as well as costs in terms of negatively affecting the provision of ecosystem services. We applied this model to human populations, where technological innovations are driven by cumulative cultural evolution. Our model shows that depending on the net impact of a technology on the provision of ecosystem services (θ), and the strength of technological feedback (ξ), different regimes can result. Among them, the human population can fill the entire planet while maximizing their well-being, but not exhaust ecosystem services. However, this outcome requires positive or green technologies that increase the provision of ecosystem services with few negative externalities or environmental costs, and that have a strong positive feedback in generating new technologies of the same kind. If the feedback is small, then the technological stock can collapse together with the human population. Scenarios where technological innovations generate net negative impacts may be associated with a limited technological stock as well as a limited human population at equilibrium and the potential for collapse. The only way to fill the planet with humans under this scenario of negative technologies is by reducing the technological stock to a minimum. Otherwise, the only feasible equilibrium is associated with population collapse. Our model points out that technological innovations per se may not help humans to grow and dominate the planet. Instead, different possibilities unfold for our future depending on their impact on the environment and on further innovation.This article is part of the themed issue 'Process and pattern in innovations from cells to societies'.

Climatic volatility, agricultural uncertainty, and the formation, consolidation and breakdown of preindustrial agrarian states

The episodic formation, consolidation and breakdown of preindustrial states occurred in multiple contexts worldwide during the last 5000 years and are contingent upon interacting endogenous economic, demographic and political mechanisms. In some instances, there is support for climate change stimulating integration or inducing sociopolitical fragmentation in these complex systems. Here, we build upon this paradigm and introduce the hypothesis that stable climatic conditions favour the formation of agrarian states, while persistently volatile climatic conditions can contribute to the episodic collapse of these complex societies. It is generally recognized that agrarian economies underwrite preindustrial state-level societies. In these contexts, the economic uncertainty associated with highly volatile climatic regimes makes it difficult for individuals or institutions to determine the costs and benefits of one agricultural strategy over another. We argue that this fosters sociopolitical instability and decentralization. As a first test of this hypothesis, we examine the historical dynamics of state formation and decline in the Mexican and Andean highlands within the last 2000 years. The available data in these regions are consistent with the hypothesis that the formation and consolidation of regional polities and empires is favoured in stable climatic regimes and that political decentralization can be stimulated and perpetuated by highly volatile climatic conditions.

Can government be self-organized? A mathematical model of the collective social organization of ancient Teotihuacan, central Mexico

Teotihuacan was the first urban civilization of Mesoamerica and one of the largest of the ancient world. Following a tradition in archaeology to equate social complexity with centralized hierarchy, it is widely believed that the city’s origin and growth was controlled by a lineage of powerful individuals. However, much data is indicative of a government of co-rulers, and artistic traditions expressed an egalitarian ideology. Yet this alternative keeps being marginalized because the problems of collective action make it difficult to conceive how such a coalition could have functioned in principle. We therefore devised a mathematical model of the city’s hypothetical network of representatives as a formal proof of concept that widespread cooperation was realizable in a fully distributed manner. In the model, decisions become self-organized into globally optimal configurations even though local representatives behave and modify their relations in a rational and selfish manner. This self-optimization crucially depends on occasional communal interruptions of normal activity, and it is impeded when sections of the network are too independent. We relate these insights to theories about community-wide rituals at Teotihuacan and the city’s eventual disintegration.