H2/ Establishing Definitive Points in a Mathematical Model to Determine Percentage Points For Herd Immunity
Herd Immunity
Establishing Definitive Points in a Mathematical Model Equation to Determine When Herd Immunity Exists Utilizing Inflection Points
EP= End Point= Herd Immunity = R=(1-pC)(1-pI) Ro
When discussing what herd immunity is, one of the preliminary factors is to state how to quantify relevant elements in the situation described in a mathematical model. In the previous post, the equation for establishing herd immunity is outlined such that R=(1-pC)(1-pI)R0. From this point an observation cyclical graphs will be used to determine the specific parameters in terms of inflection points regarding establishing and tracking the course of a virus spread within a given population to reach its end point (EP). Reaching EP is the end point at which herd immunity can be declared within a given population, as numbers start to decline and herd immunity transpires throughout, leading to a change in the classification of the viral infections, and towards an end of a previously identified pandemic.
A review of the previous equation for herd immunity is outlined as the following. For R0 to transpire, in a given situation regarding the spread of a virus across a specific population, in a geographic locale during a specific time frame. R (that is, the average number of persons infected by a case) dropping below 1 in the absence of interventions. In a population in which individuals mix homogeneously and are equally susceptible and contagious. R = (1 − pC)(1 − pI)R0 (equation 1)* where pC is the relative reduction in transmission rates due to non-pharmaceutical interventions pI is the proportion of immune individuals. R0 is the reproduction number in the absence of control measures in a fully susceptible population. R0 may vary across populations and over time, depending on the nature and number of contacts among individuals and potentially environmental factors. In the absence of control measures. (pC = 0) the condition for herd immunity. (R < 1, where R = (1 − pI)R0) is therefore achieved when the proportion of immune individuals reaches pI = 1 – 1/R0. For SARS-CoV-2 most estimates of R0 are in the range 2.5–4, with no clear geographical pattern.
To obtain specific percentages and fractions at the end point (EP=0) of COVID, regarding different groups within a population becoming infected in a pandemic. An additional set of equations must be established and solved. The evolution of the pandemic, can be documented using a graph showing various cycles and inflection points. In the case of COVID, the understanding is that individuals infected, are asymptomatic for an average of 3 days, followed by an infectious period of an average of 4 days, hence outlining an average course of the viral cycle in a particular case scenario.
To determine what the specific percentage rates within a given population to achieve herd immunity are, several elements must be considered. The natural course of viral infections have occurred throughout most of history, and have shown that within the second cycle of a virus, rates will start to decline.
In COVID-19, which has an estimated infection fatality ratio of 0.3–1.3%, the cost of reaching herd immunity through natural infection has been clearly quite high, especially due to the lack of an early establishment of patient management strategies by identifying preexisting conditions upon initial registration of incoming COVID cases.
Early estimates of attaining a maximum herd immunity threshold of 50% as a general calculated percentage point in the United States, could result in the approximation of 100,000–450,000 cases and 500,000–2,100,000 fatalities. Far past that point at the current time with over 65 million cases world wide, and a confirmed 14 million positive cases in the US, this point has clearly passed. The positive outlook for this would be that past a certain inflection point then the population can assert that it is in the process of attaining herd immunity.
