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Register a new userA Mathematical Model For the Spread of a Virus
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Authors
Neil R. Sheeley Jr
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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.
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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.
In late-2020, many countries around the world faced another surge in number of confirmed cases of COVID-19, including United Kingdom, Canada, Brazil, United States, etc., which resulted in a large nationwide and even worldwide wave. While there have been indications that precaution fatigue could be a key factor, no scientific evidence has been provided so far. We used a stochastic metapopulation model with a hierarchical structure and fitted the model to the positive cases in the US from the start of outbreak to the end of 2020. We incorporated non-pharmaceutical interventions (NPIs) into this model by assuming that the precaution strength grows with positive cases and studied two types of pandemic fatigue. We found that people in most states and in the whole US respond to the outbreak in a sublinear manner (with exponent k=0.5), while only three states (Massachusetts, New York and New Jersey) have linear reaction (k=1). Case fatigue (decline in peoples vigilance to positive cases) is responsible for 58% of cases, while precaution fatigue (decay of maximal fraction of vigilant group) accounts for 26% cases. If there were no pandemic fatigue (no case fatigue and no precaution fatigue), total positive cases would have reduced by 68% on average. Our study shows that pandemic fatigue is the major cause of the worsening situation of COVID-19 in United States. Reduced vigilance is responsible for most positive cases, and higher mortality rate tends to push local people to react to the outbreak faster and maintain vigilant for longer time.
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.
An epidemiological model is developed for the spread of COVID-19 in South Africa. A variant of the classical compartmental SEIR model, called the SEIQRDP model, is used. As South Africa is still in the early phases of the global COVID-19 pandemic with the confirmed infectious cases not having peaked, the SEIQRDP model is first parameterized on data for Germany, Italy, and South Korea - countries for which the number of infectious cases are well past their peaks. Good fits are achieved with reasonable predictions of where the number of COVID-19 confirmed cases, deaths, and recovered cases will end up and by when. South African data for the period from 23 March to 8 May 2020 is then used to obtain SEIQRDP model parameters. It is found that the model fits the initial disease progression well, but that the long-term predictive capability of the model is rather poor. The South African SEIQRDP model is subsequently recalculated with the basic reproduction number constrained to reported values. The resulting model fits the data well, and long-term predictions appear to be reasonable. The South African SEIQRDP model predicts that the peak in the number of confirmed infectious individuals will occur at the end of October 2020, and that the total number of deaths will range from about 10,000 to 90,000, with a nominal value of about 22,000. All of these predictions are heavily dependent on the disease control measures in place, and the adherence to these measures. These predictions are further shown to be particularly sensitive to parameters used to determine the basic reproduction number. The future aim is to use a feedback control approach together with the South African SEIQRDP model to determine the epidemiological impact of varying lockdown levels proposed by the South African Government.
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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