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Register a new userMulti-year optimization of malaria intervention: a mathematical model
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Malaria is a mosquito-borne, lethal disease that affects millions and kills hundreds of thousands of people each year. In this paper, we develop a model for allocating malaria interventions across geographic regions and time, subject to budget constraints, with the aim of minimizing the number of person-days of malaria infection. The model considers a range of several conditions: climatic characteristics, treatment efficacy, distribution costs, and treatment coverage. We couple an expanded susceptible-infected-recovered (SIR) compartment model for the disease dynamics with an integer linear programming (ILP) model for selecting the disease interventions. Our model produces an intervention plan for all regions, identifying which combination of interventions, with which level of coverage, to use in each region and year in a five-year planning horizon. Simulations using the model yield high-level, qualitative insights on optimal intervention policies: The optimal policy is different when considering a five-year time horizon than when considering only a single year, due to the effects that interventions have on the disease transmission dynamics. The vaccine intervention is rarely selected, except if its assumed cost is significantly lower than that predicted in the literature. Increasing the available budget causes the number of person-days of malaria infection to decrease linearly up to a point, after which the benefit of increased budget starts to taper. The optimal policy is highly dependent on assumptions about mosquito density, selecting different interventions for wet climates with high density than for dry climates with low density, and the interventions are found to be less effective at controlling malaria in the wet climates when attainable intervention coverage is 60% or lower. However, when intervention coverage of 80% is attainable, then malaria prevalence drops quickly.
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We present a new mathematical model to explicitly capture the effects that the three restriction measures: the lockdown date and duration, social distancing and masks, and, schools and border closing, have in controlling the spread of COVID-19 infections $i(r, t)$. Before restrictions were introduced, the random spread of infections as described by the SEIR model grew exponentially. The addition of control measures introduces a mixing of order and disorder in the systems evolution which fall under a different mathematical class of models that can eventually lead to critical phenomena. A generic analytical solution is hard to obtain. We use machine learning to solve the new equations for $i(r,t)$, the infections $i$ in any region $r$ at time $t$ and derive predictions for the spread of infections over time as a function of the strength of the specific measure taken and their duration. The machine is trained in all of the COVID-19 published data for each region, county, state, and country in the world. It utilizes optimization to learn the best-fit values of the models parameters from past data in each region in the world, and it updates the predicted infections curves for any future restrictions that may be added or relaxed anywhere. We hope this interdisciplinary effort, a new mathematical model that predicts the impact of each measure in slowing down infection spread combined with the solving power of machine learning, is a useful tool in the fight against the current pandemic and potentially future ones.
Disease transmission is studied through disciplines like epidemiology, applied mathematics, and statistics. Mathematical simulation models for transmission have implications in solving public and personal health challenges. The SIR model uses a compartmental approach including dynamic and nonlinear behavior of transmission through three factors: susceptible, infected, and removed (recovered and deceased) individuals. Using the Lambert W Function, we propose a framework to study solutions of the SIR model. This demonstrates the applications of COVID-19 transmission data to model the spread of a real-world disease. Different models of disease including the SIR, SIRm and SEIR model are compared with respect to their ability to predict disease spread. Physical distancing impacts and personal protection equipment use will be discussed in relevance to the COVID-19 spread.
Single species population models and discrete stochastic gene frequency models are two standards of mathematical biology important for the evolution of populations. An agent based model is presented which reproduces these models and then explores where these models agree and disagree under relaxed specifications. For the population models, the requirement of homogeneous mixing prevents prediction of extinctions due to local resource depletion. These models also suggest equilibrium based on attainment of constant population levels though underlying population characteristics may be nowhere close to equilibrium. The discrete stochastic gene frequency models assume well mixed populations at constant levels. The models predictions for non-constant populations in strongly oscillating and chaotic regimes are surprisingly good, only diverging from the ABM at the most chaotic levels.
This paper describes a mathematical model for the spread of a virus through an isolated population of a given size. The model uses three, color-coded components, called molecules (red for infected and still contagious; green for infected, but no longer contagious; and blue for uninfected). In retrospect, the model turns out to be a digital analogue for the well-known SIR model of Kermac and McKendrick (1927). In our RGB model, the number of accumulated infections goes through three phases, beginning at a very low level, then changing to a transition ramp of rapid growth, and ending in a plateau of final values. Consequently, the differential change or growth rate begins at 0, rises to a peak corresponding to the maximum slope of the transition ramp, and then falls back to 0. The properties of these time variations, including the slope, duration, and height of the transition ramp, and the width and height of the infection rate, depend on a single parameter - the time that a red molecule is contagious divided by the average time between collisions of the molecules. Various temporal milestones, including the starting time of the transition ramp, the time that the accumulating number of infections obtains its maximum slope, and the location of the peak of the infection rate depend on the size of the population in addition to the contagious lifetime ratio. Explicit formulas for these quantities are derived and summarized. Finally, Appendix E has been added to describe the effect of vaccinations.
The goal of this note is to present a simple mathematical model with two parameters for the number of deaths due to the corona (COVID-19) virus. The model only requires basic knowledge in differential calculus, and can also be understood by pupils attending secondary school. The model can easily be implemented on a computer, and we will illustrate it on the basis of case studies for different countries.
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