, r ( 1 ) ) = ( s ( 0 ) , n − r ( 0 ) − s ( 0 ) , r ( 0 ) ) , u t < u < 1 , (1),(1),(1))=(s(0),n-r(0)-s(0),r(0)),\quad u_<u<1,} where u t satisfies i ( u t ) = 0 (u_)=0} . by the transcendental equation for r ∞ above, it follows that u t = e − ( r ∞ − r ( 0 ) ) / ρ ( = s ∞ / s ( 0 ) =e^-r(0))/\rho }(=s_/s(0)} , if s ( 0 ) ≠ 0 ) and i ∞ = 0 =0} . an equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by miller yields s ( t ) = s ( 0 ) e − ξ ( t ) i ( t ) = n − s ( t ) − r ( t ) r ( t ) = r ( 0 ) + ρ ξ ( t ) ξ ( t ) = β n ∫ 0 t i ( t ∗ ) d t ∗ s(t)&=s(0)e^\\i(t)&=n-s(t)-r(t)\\r(t)&=r(0)+\rho \xi (t)\\\xi (t)&=\int _^i(t^)\,dt^\end here ξ ( t ) can be interpreted as the expected number of transmissions an individual has received by time t . the two solutions are related by e − ξ ( t ) = u =u} . effectively the same result can be found in the original work by kermack and mckendrick. these solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to i . the equations may thus be divided through by i , and the time rescaled so that the differential operator on the left-hand side becomes simply d / d τ , where d τ = i d t , i.e. τ = ∫ i d t . the differential equations are now all linear, and the third equation, of the form d r / d τ = const., shows that τ and r (and ξ above) are simply linearly related. a highly accurate analytic approximant of the sir model as well as exact analytic expressions for the final values s ∞ , i ∞ , and r ∞ were provided by kröger and schlickeiser, so that there is no need to perform a numerical integration to solve the sir model (a simplified example practice on covid-19 numerical simulation using microsoft excel can be found here ), to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the sir model. the approximant involves the lambert w function which is part of all basic data visualization software such as microsoft excel, matlab, and mathematica. while kendall considered the so-called all-time sir model where the initial conditions s ( 0 ) , i ( 0 ) , and r ( 0 ) are coupled through the above relations, kermack and mckendrick proposed to study the more general semi-time case, for which s ( 0 ) and i ( 0 ) are both arbitrary. this latter version, denoted as semi-time sir model, makes predictions only for future times t > 0 . an analytic approximant and exact expressions for the final values are available for the semi-time sir model as well. numerical solutions to the sir model with approximations numerical solutions to the sir model can be found in the literature. an example is using the model to analyze covid-19 spreading data. three reproduction numbers can be pulled out from the data analyzed with numerical approximation, the basic reproduction number: r 0 = β 0 γ 0 = the real-time reproduction number: r t = β t γ t = and the real-time effective reproduction number: r e = β t s γ t n =s}n}}} r 0 represents the speed of reproduction rate at the beginning of the spreading when all populations are assumed susceptible, e.g. if β 0 = 0.4 d a y − 1 =0.4day^ and γ 0 = 0.2 d a y − 1 =0.2day^ meaning one infectious person on average infects 0.4 susceptible people per day and recovers in 1/0.2=5 days. thus when this person recovered, there are two people still infectious directly got from this person and r 0 = 2 =2} , i.e. the number of infectious people doubled in one cycle of 5 days. the data simulated by the model with r 0 = 2 =2} or real data fitted will yield a doubling of the number of infectious people faster than 5 days because the two infected people are infecting people. from the sir model, we can tell that β is determined by the nature of the disease and also a function of the interactive frequency between the infectious person i with the susceptible people s and also the intensity/duration of the interaction like how close they interact for how long and whether or not they both wear masks, thus, it changes over time when the average behavior of the carriers and susceptible people changes. the model use s i to represent these factors but it indeed is referenced to the initial stage when no action is taken to prevent the spread and all population is susceptible, thus all changes are absorbed by the change of β . γ is usually more stable over time assuming when the infectious person shows symptoms, she/he will seek medical attention or be self-isolated. so if we find r t changes, most probably the behaviors of people in the community have changed from their normal patterns before the outbreak, or the disease has mutated to a new form. costive massive detection and isolation of susceptible close contacts have effects on reducing 1 / γ but whose efficiencies are under debate. this debate is largely on the uncertainty of the number of days reduced from after infectious or detectable whichever comes first to before a symptom shows up for an infected susceptible person. if the person is infectious after symptoms show up, or detection only works for a person with symptoms, then these prevention methods are not necessary, and self-isolation and/or medical attention is the best way to cut the 1 / γ values. the typical onset of the covid-19 infectious period is in the order of one day from the symptoms showing up, making massive detection with typical frequency in a few days useless. r t does not tell us whether or not the spreading will speed up or slow down in the latter stages when the fraction of susceptible people in the community has dropped significantly after recovery or vaccination. r e corrects this dilution effect by multiplying the fraction of the susceptible population over the total population. it corrects the effective/transmissible interaction between an infectious person and the rest of the community when many of the interaction is immune in the middle to late stages of the disease spreading. thus, when r e > 1 >1} , we will see an exponential-like outbreak; when r e = 1 =1} , a steady state reached and no number of infectious people changes over time; and when r e < 1 <1} , the disease decays and fades away over time. using the differential equations of the sir model and converting them to numerical discrete forms, one can set up the recursive equations and calculate the s, i, and r populations with any given initial conditions but accumulate errors over a long calculation time from the reference point. sometimes a convergence test is needed to estimate the errors. given a set of initial conditions and the disease-spreading data, one can also fit the data with the sir model and pull out the three reproduction numbers when the errors are usually negligible due to the short time step from the reference point. any point of the time can be used as the initial condition to predict the future after it using this numerical model with assumption of time-evolved parameters such as population, r t , and γ . however, away from this reference point, errors will accumulate over time thus convergence test is needed to find an optimal time step for more accurate results. among these three reproduction numbers, r 0 is very useful to judge the control pressure, e.g., a large value meaning the disease will spread very fast and is very difficult to control. r t is most useful in predicting future trends, for example, if we know the social interactions have reduced 50% frequently from that before the outbreak and the interaction intensities among people are the same, then we can set r t = 0.5 r 0 =0.5r_ . if social distancing and masks add another 50% cut in infection efficiency, we can set r t = 0.25 r 0 =0.25r_ . r e will perfectly correlate with the waves of the spreading and whenever r e > 1 >1} , the spreading accelerates, and when r e < 1 <1} , the spreading slows down thus useful to set a prediction on the short term trends. also, it can be used to directly calculate the threshold population of vaccination/immunization for the herd immunity stage by setting r t = r 0 =r_ . the sir model with vital dynamics and constant population consider a population characterized by a death rate μ and birth rate λ , and where a communicable disease is spreading. the model with mass-action transmission is: d s d t = λ − μ s − β i s n d i d t = β i s n − γ i − μ i d r d t = γ i − μ r &=\lambda -\mu s-\\&=-\gamma i-\mu i\\&=\gamma i-\mu r\end for which the disease-free equilibrium (dfe) is: ( s ( t ) , i ( t ) , r ( t ) ) = ( λ μ , 0 , 0 ) . ,0,0\right).} in this case, we can derive a basic reproduction number: r 0 = β μ + γ , =,} which has threshold properties. in fact, independently from biologically meaningful initial values, one can show that: r 0 ≤ 1 ⇒ lim t → ∞ ( s ( t ) , i ( t ) , r ( t ) ) = dfe = ( λ μ , 0 , 0 ) \leq 1\rightarrow \lim _(s(t),i(t),r(t))==\left(,0,0\right)} r 0 > 1 , i ( 0 ) > 0 ⇒ lim t → ∞ ( s ( t ) , i ( t ) , r ( t ) ) = ee = ( γ + μ β , μ β ( r 0 − 1 ) , γ β ( r 0 − 1 ) ) . >1,i(0)>0\rightarrow \lim _(s(t),i(t),r(t))==\left(,\left(r_-1\right),\left(r_-1\right)\right).} the point ee is called the endemic equilibrium (the disease is not totally eradicated and remains in the population). with heuristic arguments, one may show that r 0 may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population. the sir model diagram of the sir model with initial values s ( 0 ) = 997 , i ( 0 ) = 3 , r ( 0 ) = 0 , and rates for infection β = 0.4 and for recovery γ = 0.04 animation of the sir model with initial values s ( 0 ) = 997 , i ( 0 ) = 3 , r ( 0 ) = 0 , and rate of recovery γ = 0.04 . the animation shows the effect of reducing the rate of infection from β = 0.5 to β = 0.12 . if there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by appropriate measures such as social distancing. in 1927, w. o. kermack and a. g. mckendrick created a model in which they considered a fixed population with only three compartments: susceptible, s ( t