The first linear model (LM) proposed contains three free parameters. These are characterized by the rate constant of β for the virus converting healthy people into sick people, a rate constant γ₂ at which sick people leave the infectious pool either through recovery with bestowed immunity or death into the “recovered pool”, and a third rate parameter γ₁, which will allow for the possibility of infected people returning to the susceptible pool, i.e., without immunity. The latter provides a sliding parameter that may be adjusted to allow for SIR towards SIS conditions. The units of the rate constants are specified in days⁻¹. The following set of differential equations describes the linearized version of the SIR/SIS conditions of the top row in Figure 1:

d f d t = − β f + γ 1 g (1a)

d g d t = β f + γ 1 g − γ 2 g (1b)

d m d t = γ 2 g (1c)

where f is the susceptible fraction of the population, g is the infectious fraction of the population and m is the recovered fraction of the population where the vanishing sum of Equations (1a)-(1c) yields the normalization condition,

f + g + m = f 0 + g 0 + m 0 = 1 (2)

where f 0 , g 0 , and m 0 are the starting fractions of each pool at t = 0. With apologies to Kermack and McKendrick [1], who used x, y and z for the actual pool numbers, not fractions, and modern adaptations of KM models which use S, I and R, for fractional populations once properly normalized [2] [3], our notation of f, g and m allow us to reserve their capital counterparts F, G and M for their respective Laplace transforms. Equations (1a)-(1c) are what we refer to as linearized versions of the non-linear equations provided by Kermack and McKendrick, albeit without the γ₁ term [1]. Despite the simplification which allows for analytic solutions for f, g and m (vide infra), the essentials of three of the SIR, SIS and SI may be gleaned from them which will also apply to the non-linear KM versions discussed below. In the SIR model, the parameter γ₁ is set to 0 which has the following implication. Once an individual passes from the S pool to the I pool, there is no going back. The assumption is that this individual will end up in the R pool, having either recovered with immunity or having died, in either case escaping reinfection. In the SIS model, γ₂ is set to 0, effectively implying that there is no recovered pool R and individuals get sick and then return to the healthy but susceptible fraction. With SIS, a steady state level of the infected fraction is achieved. The SI model sets both γ₁ and γ₂ to 0 and is a limit of the SIS model in which everyone ultimately becomes infected. The SI model is the simplest of the three and interestingly the model selected by Mohazzabi et al. to fit CDC COVID-19 case data over the course of a year within the United States (3) and throughout the world [4]. Regardless, the Laplace transform of the system of Equations (1a)-(1c) leads to the following algebraic relations among F, G and M, with α being the Laplace transform variable:

α F − f 0 = − β F + γ 1 G (3a)

α G − g 0 = β F + γ 1 G − γ 2 G (3b)

α M − m 0 = γ 2 G (3c)

where F = ∫ f   e − α t d t , etc., as integrated from t = 0 to ∞ with the tacit assumption that terms like f   e − α t at t = ∞ vanish. Solving for G from Equations (3a)-(3b) leads to:

G = ( α + β ) g 0 + β f 0 α 2 + α ( β + γ ) + β γ 2 = ( α + β ) g 0 + β f 0 ( α − Ψ + ) ( α − Ψ − ) (4)

with γ ≡ γ 1 + γ 2 . The second equality in Equation (4) introduces the roots of the quadratic defining the denominator of the first equality which are:

Ψ ± = [ − ( β + γ ) ± ( β + γ ) 2 − 4 β γ 2 ] / 2 . (5)

A similar derivation for F from Equations (3a)-(3b) yields,

F = f 0 ( α − Ψ + ) ( α − Ψ − ) + γ 1 ( α + β ) g 0 + γ 1 β f 0 ( α + β ) ( α − Ψ + ) ( α − Ψ − ) . (6)

To extract f and g from F and G, the following inverse Laplace transform rule [5] which utilizes the residues (Res) of F and G at their poles is invoked:

g = ∑ Res ( G ( α i ) ) e α i t (7a)

f = ∑ Res ( F ( α i ) ) e α i t (7b)

where the sum is over two poles for G and three poles for F. Proceeding by inspection, we find that g and f consist of two and three exponential functions of t, respectively, which read:

g = [ ( ( Ψ + + β ) g 0 + β f 0 ) e Ψ + t − ( ( Ψ − + β ) g 0 + β f 0 ) e Ψ − t ] / ( Ψ + − Ψ − ) (8a)

f = f 0 e − β t + γ 1 β f 0 e − β t ( β + Ψ + ) ( β + Ψ − ) + ( γ 1 ( Ψ + + β ) g 0 + γ 1 β f 0 ) e Ψ + t ( Ψ + + β ) ( Ψ + − Ψ − )     + ( γ 1 ( Ψ − + β ) g 0 + γ 1 β f 0 ) e Ψ − t ( Ψ − + β ) ( Ψ − − Ψ + ) . (8b)

One could also solve for m via Equation (3c) and Equation (4) using the inverse Laplace transform but there is no need as the normalization condition of Equation (2) may be used once f and g are specified. A further set of differential equations which will underpin a linearized version of the SIRS model considers a flux from the susceptible fraction S to the infected fraction I and then to the recovered fraction R but then also incorporates the possibility of “immunity” within the recovered fraction to “wane” by introducing a flux from the recovered back to the susceptible fraction, as shown in the lower row of Figure 1. The differential equations governing this situation read:

d f d t = − β f + γ 1 m (9a)

d g d t = β f − γ 2 g (9b)

d m d t = γ 2 g − γ 1 m . (9c)

Again, the Laplace transform approach may be gainfully applied to this system with the details left to the reader. One obtains for f, g and m the following time courses:

f = γ 1 γ 2 β γ 1 + β γ 2 + γ 1 γ 2 + ( f 0 ( β + Ψ + ) ( β + Ψ − ) − γ 1 γ 2 ) e − β t ( β + Ψ + ) ( β + Ψ − )     + ( m 0 γ 1 ( β + Ψ + ) ( γ 2 + Ψ + ) + g 0 γ 1 γ 2 ( β + Ψ + ) + γ 1 γ 2 β f 0 ) e Ψ + t Ψ + ( Ψ + + β ) ( Ψ + − Ψ − )     + ( m 0 γ 1 ( Ψ − + β ) ( Ψ − + γ 2 ) + g 0 γ 1 γ 2 ( Ψ − + β ) + γ 1 γ 2 β f 0 ) e Ψ − t Ψ − ( Ψ − + β ) ( Ψ − − Ψ + ) (10a)

m = β γ 2 β γ 1 + β γ 2 + γ 1 γ 2 + ( m 0 ( Ψ + + β ) ( Ψ + + γ 2 ) + g 0 γ 2 ( Ψ + + β ) + f 0 β γ 2 ) e Ψ + t Ψ + ( Ψ + − Ψ − )     + ( m 0 ( Ψ − + β ) ( Ψ − + γ 2 ) + g 0 γ 2 ( Ψ − + β ) + f 0 β γ 2 ) e Ψ − t Ψ − ( Ψ − − Ψ + ) (10b)

g = 1 − f − m (10c)

with

Ψ ± = [ − ( β + γ 1 + γ 2 ) ± ( β + γ 1 + γ 2 ) 2 − 4 ( β γ 1 + β γ 2 + γ 1 γ 2 ) ] / 2 . (11)

For non-vanishing rates of β, γ₁ and γ₂, the susceptible, infected, and recovered fractions achieve steady state values as t approaches ∞ which are given by:

f s s = γ 1 γ 2 β γ 1 + β γ 2 + γ 1 γ 2 (12a)

g s s = β γ 1 β γ 1 + β γ 2 + γ 1 γ 2 (12b)

m s s = β γ 2 β γ 1 + β γ 2 + γ 1 γ 2 . (12c)

Utilizing Equations (8a)-(8b), we first examine how decreasing the β parameter, the transmissibility rate, through such factors as social distancing, the use of personal protective gear, business closings, limited capacity recreational events, etc., can flatten the curve within the context of pure SIR conditions, e.g., γ 1 = 0 in Equations (1a)-(1c). The γ₂ parameter, in contrast and if given no effective treatment, is largely based on the biology of the virus and the host which may be somewhat beyond our control, though the advent of vaccines and more nuanced control of sick patients does allow some control over the γ₂ parameter and allows us to demonstrate how increasing this parameter via effective medical practices can also flatten the curve within the context of pure SIR conditions.

Figure 2(a) shows the effects of decreasing β (social distancing, etc.) on “flattening the curve” when β is decreased for a fixed γ₂. The solid curves in the figure

represent f, the healthy but susceptible fraction of the population. The dashed curves represent g, the sick fraction of the population. The curves were generated with a fixed γ₂ value of 0.072 days⁻¹ and β values of 0.096 days⁻¹ (black curves), 0.048 days⁻¹ (blue curves) and 0.024 days⁻¹ (red curves). In this case, the strategy to strongly advise or otherwise mandate various degrees of social distancing to reduce the rate at which the virus spreads clearly worked. With the highest β value, perhaps representing the case of no social distancing, one sees the highest peak height of the g curve, a worst-case scenario with the potential to surpass various hospital capacities while the next two curves with lower β values show a successive lowering of the peak heights, hopefully flattened sufficiently so as to not overwhelm various hospital capacities. It is worth noting however that “flattening the curve” via increasing levels of social distancing also results in more infected people in the later stages of the pandemic. For example, Figure 2(a) indicates that at 70 days from t = 0 approximately 10% of the population remains infected in the case of the highest social distancing while only ~2% remains infected in the absence of social distancing. In fact, one finds identical areas beneath each of the three g curves in Figure 2(a) for this model as proven by calculating the area under g curves over all time from 0 to ∞. For SIR conditions ( γ 1 = 0 ) we have:

∫ g ( t ) d t = f 0 β γ 2 − β ∫ ​ ( e − β t − e − γ 2 t ) d t + g 0 ∫ e − γ 2 t d t = f 0 + g 0 γ 2 (13)

thus, proving identical areas under the g curves for fixed γ₂ and areas which are independent of β. This result has some implications regarding “flattening the curve”. Namely, within the confines of this model and with no way to affect γ₂, then regardless of the degree of social distancing, masking, etc., invoked to “flatten the curve”, the same number of people will ultimately get sick, reach ICU units and/or ventilators, and die or recover.

Figure 2(b) shows the effects of potential pharmaceutical interventions when social distancing is fixed by using a β of 0.072 days⁻¹ with the susceptible fraction f plotted as the solid black curve. The blue, red and black dashed curves plotted the infected fraction g for increasing levels of the effectiveness of potential pharmaceutical interventions and were generated using γ₂ values of 0.024, 0.048 and 0.096 days⁻¹, respectively. Figure 2(b) shows that effective interventions can alleviate the burden of disease as quantified by the area under each of the three curves, the red and black dashed curves showing half and one-fourth of the area of the blue curve, in accordance with their respective γ₂ values (see Equation (13)).

The effects on the curves when the immunity bestowed upon the recovered fraction is allowed to wane, as portrayed by the lower row of Figure 1 and characterized via Equations (9)-(12) are provided in Figure 2(c) which shows that as one allows a flux from the recovered to the susceptible pool with a non-vanishing γ₁ the steady state levels of the infected fraction rise from 0, in the pure SIR condition, to levels commensurate with Equation (12b) while the susceptible fraction also rises to non-vanishing levels (Equation (12a)) at long times. This “unflattening” of the curves are shown as solid and dashed lines in Figure 2(c) for the susceptible fraction f and the infected fraction g, respectively, for fixed β and γ₂ values of 0.076 days⁻¹ and 0.096 days⁻¹, respectively, as γ₁ value ranges from 0 to 0.07 days⁻¹. This range of values for γ₁ has been gleaned from some recent literature regarding various measurements of the length of immunity from COVID-19 following recovery [6] [7] [8]. The lifting of the steady state values from 0 to non-zero values for the infected and susceptible pools heralds the transition from an epidemic situation to an endemic situation.

The linear models proposed above successfully achieved the goal of generating curves amenable to flattening, curves which resemble those reported in the media. Furthermore, the curves so generated behaved quite sensibly with respect to alterations in the parameters such as β and γ₂ as indicative of the effects of social distancing type interventions and pharmacological type interventions, respectively. We defer a discussion of the γ₁ parameter in Equations (3a)-(3c) which allows for the possibility of a return of an individual from the I pool to the S pool for potential reinfection until after a discussion of the non-linear KM type models. There are a number of differences between the linear models and the KM models more commonly deployed for epidemiological considerations, differences we now discuss in the following exposition of the KM models.

The article which Kermack and McKendrick published [1] shortly after the end of the “Great Influenza” of the last century [9] laid the foundations for much of the mathematical modeling associated with epidemics since. The following system of equations, normalized [2], slightly modified to include an I to S rate γ₁, and cast in modern terminology, represent our KM starting points:

d S d t = − β S I + γ 1 I (14a)

d I d t = β S I − γ 1 I − γ 2 I (14b)

d R d t = γ 2 I (14c)

where the normalization relation:

S + I + R = 1 (15)

follows from the vanishing sum of Equations (14a)-(14c). As Breda et al. [10] have recently and somewhat caustically remarked regarding the 1927 KM paper that “…even experienced experts believe that the paper is just about (this) system…” while pointing out that Kermack and McKendrick introduced this system of equations only as a special case of their more general formulation. Not being such experts, we happily consider these equations, noting that the most notable difference between the linear models and those of Equations (14a)-(14c) is the recognition that in the latter, the rate at which people are removed from the S fraction to the I fraction is not simply proportional to the S fraction but also to the infectious fraction I. The more people who are sick, the more likely a healthy person will encounter a sick person and so, quite reasonably, arises the ± βIS terms in Equations (14a) and (14b). Though this assumption has been challenged, particularly at high infection fractions [11], the equations posed have formed a very important backdrop from which to study not just disease spread but also some interesting mathematics [10] - [17]. Despite statements to the contrary [13], Equations (14a)-(14c) are non-linear so any hope that the Laplace transform approach to their solution might be of value is quickly vanquished, though this also means that notationally we no longer need to reserve capital letters for the Laplace transform counterparts of f, g and m and directly utilize the S, I and R representations for the three fractions. Rather than attempting to solve these equations as written, let’s first look at two simplified versions of them which are commonly deployed [3] [4] [10] [11] [12] [14] [15] [16] [17], the SI and SIS models, both of which have useful analytic solutions but do not lead to curves with “peaks” to flatten.

The SI model involves considering only the susceptible fraction S and the infectious fraction I with no loss from the latter to a recovered fraction. This is done by setting γ 1 = γ 2 = 0 in Equations (14a)-(14c) to obtain:

d S d t = − β S I (16a)

d I d t = β S I (16b)

with S + I = 1 . To solve this set of equations, this normalization condition is used to substitute for S into Equation (16b) as in:

d I d t = β ( 1 − I ) I (17a)

or rearranging

d I I ( 1 − I ) = β d t (17b)

Now since d d l ( ln ( I 1 − I ) ) = 1 I ( 1 − I ) direct integration of the left side from I₀ to I and the right side from 0 to t may be performed to obtain, after some tedious rearrangement, the following expression for I:

I = I 0 e β t 1 − I 0 ( 1 − e β t ) . (18)

Thus, we see that the SI model has an analytic solution which yields an infectious fraction I which is a monotonically increasing function of t, as shown by the black curve in Figure 3(a) which assumed a β value of 0.1 days⁻¹ and an initially infected fraction, I₀, of 0.01. For any value of the single parameter β, the infectious fraction tends towards unity given enough time. The model simply implies that sooner or later all the susceptible fraction becomes infected, though there is no curve to flatten. The SI model effectively contains no further information as the only other variable of interest S, is simply 1 − I. There is one interesting feature of the SI model which is shared by the other KM models discussed below. This is the feature that if I 0 = 0 , as in no sick people, the infectious fraction I will remain 0 and there will be no spread of disease with time. This is a very sensible feature indeed for an infectious phenomenon and one not shared by the linear model where the increase of the infected fraction (g in Equation (1b)) occurs from the start no matter what. Turning to the slightly more complicated SIS model, the governing differential equations read:

d S d t = − β S I + γ I (19a)

d I d t = β S I − γ I (19b)

In this model, the rate γ represents the flux of sick people back to the susceptible pool, recovery without immunity. We still have the normalization condition for the two fractions S + I = 1 so that Equation (19b) may be written as:

d I d t = β ( 1 − I ) I − γ I (20)

After Shabbir et al. (16) we seek the solution to this equation by making the substitution I = 1 y and with d d t ( 1 y ) being ( − 1 y 2 ) d y / d t , Equation (20) becomes:

d y d t = y ( γ − β ) + β (21a)

or equivalently

d y y ( γ − β ) + β = d t (21b)

Integrating the left and right sides from y₀ to y and 0 to t, respectively, may be readily accomplished by noting that d d y ( ln ( β + y ( γ − β ) ) ) = γ − β β + y ( γ − β ) .

Performing the requisite integrations and rearranging leads to the solution for y as:

y = ( β + ( γ − β ) y 0 ) e ( γ − β ) t − β γ − β (22)

whose inverse finally yields the sought after I as:

I = I 0 ( γ − β ) e − ( γ − β ) t γ − β + β I 0 ( 1 − e − ( γ − β ) t ) ≡ I 0 e − ( γ − β ) t 1 + Ψ I 0 ( 1 − e − ( γ − β ) t ) (23)

where we have defined Ψ ≡ β γ − β . It is easily shown that Equation (23) of the SIS model reduces to Equation (18) of the SI model when γ is set to 0. The primary difference between SI and SIS is that the endpoint of the infectious fraction is not unity but rather some fraction less than unity. That is, given enough time not all the population becomes infected, as in SI, but rather the steady state achieved involves some smaller portion of the full population. Figure 3(a) shows SIS model simulations for the same β and I₀ used for the SI model, β = 0.1 days⁻¹, I₀ = 0.01, but with γ values of 0.025, 0.05 and 0.075 days⁻¹, respectively. Figure 3(b) shows what happens when the γ values exceed the β value. Namely, the infectious fraction dies out as shown using the β of 0.1 days⁻¹ and γ values of 0.1 (black curve), 0.12 (red curve), 0.14 (blue curve) and 0.16 (magenta curve) days⁻¹, respectively.

As mentioned briefly above Mohazzabi et al. used the SI model to fit available CDC measurements of the 5-day rolling average of individuals infected with COVID-19 over the course of a year within the United States [3] and also for similar data reported on a worldwide basis [4]. The SI model is, of course, just the SIS model with γ in Equations (19a) and (19b) set to 0. We have applied the SI and the SIS model to the same CDC data for the United States used by Mohazzabi et al. The results are shown in Figures 4(a)-(c). Figure 4(a) shows the actual data (black curve) while the red and blue curves fit the data as performed with the SI and SIS models, respectively.

To convert the fractional population of infected people I to the numbers of infected people, multiplication by the estimated total population for people living in the USA was performed assuming this population was 3.31 × 10⁸ individuals, the same estimate used by Mohazzabi et al. [3]. Also, as in that work, the total number of infected individuals at t = 0, n₀, was incorporated as a free parameter in addition to the free parameters β and γ. Our fits yielded the parameters shown

in Table 1, with the parameters for the SI model very similar to those reported by Mohazzabi et al. for this model. The model fitting for SI model was based on Equation (18) and the fitting coefficients were n₀ and β. For the SIS model, the estimated n₀ from SI model fit was used as an input and β and γ were estimated by fitting the data to Equation (23).

Using the SIS model with one additional free parameter a slightly higher β is returned and of course a non-vanishing γ, but with equivalent correlation coefficients R² of 0.993 for both models fits. Visualization of this equivalence is provided in Figure 4(b) which shows the residuals from both fits. Despite the equivalence of the SI and SIS model fits the data over this limited time range of 1 year, the ramifications of longer time periods for either fit may be appreciated by Figure 4(c) in which the SI model leads to the entire population being infected while the SIS model parameters lead to significantly fewer numbers of infected people at long times. Of course, neither of these models incorporates a recovered population so the “bell” curves occurring in local populations and reported in the media will not be generated from such models and so we now turn to the three-compartment models to generate such curves.

Thus, we examine the original KM equations, Equations (14a)-(14c), albeit with the additional γ₁ term, and seek their solutions, as others have [10] - [17], to no avail. There is simply no obvious analytic solution and those who have offered “algebraic” solutions offer up approximations, as KM did, and/or infinite series of elementary functions. It is, however, quite feasible to perform numerical evaluations of the KM equations and we do so with the aid of a mathematical trick originally provided by Kermack and Mckendrick who wrestled the three differential equations into one. Namely, one divides Equation (14a) by Equation (14c) to obtain:

d S d R = γ 1 − β S γ 2 (24a)

or

d S γ 1 − β S = d R γ 2 (24b)

which may be directly integrated to obtain:

S = γ 1 β − ( γ 2 β − S 0 ) e − β R γ 2 . (25)

Applying the normalization condition and substituting Equation (25) into Equation (14c) one obtains:

d R d t = γ 2 I = γ 2 ( 1 − R − S ) = γ 2 ( 1 − R − γ 1 β + ( γ 1 β − S 0 ) e − β R γ 2 ) (26)

The only difference between this equation and that provided by Kermack and McKendrick nearly 100 years ago is the γ₁ term which, again, simply allows for sliding between SIS and SIR conditions depending on the value of γ₁ which, when non-zero, provides a means for people in the infected pool to go back to the susceptible pool in addition to, perhaps, proceeding to the recovered/dead pool via the rate γ₂. It is now straightforward to utilize Equation (26) to numerically propagate a solution for R iteratively as in:

R i + 1 = R i + ( d R i / d t ) Δ t (27)

where d R i / d t is given by the right side of Equation (26) at each integer time step i and ∆t is the length of each time step. Since numerical evaluation was a luxury Kermack and McKendrick could not really afford, at least not easily in their time, they may be excused for expanding the exponential in Equation (26) to 2n’d order and providing an approximate solution in terms of standard functions. The advent of modern computing machines allows us to directly implement Equation (27) into Matlab (The Mathworks, Needham, MA) as a recursive relation, allowing us to generate the time courses of R, and subsequently I, as exemplified by the curves in Figure 5 which are intended to show how the standard SIR type curves from the KM model, with vanishing γ₁, morph into the analytically tractable SIS like curves as one introduces stronger γ₁ values in relation to γ₂. The curves shown utilized a β value 0.1 days⁻¹, an I₀ of 0.01, a time sampling ∆t of 0.25 days and a total of 800 time samples from 0 to 200 days. The red curve is a pure SIR curve with γ 1 = 0 and γ 2 = 0.04 days⁻¹, the magenta curve is a pure SIS curve with γ 2 = 0 and γ 1 = 0.04 days⁻¹. The dashed black line through the magenta curve was generated with the analytic solution to the SIS model embodied in Equation (23) and confirms the accuracy of the numerical solution (magenta curve) in this regime. The numerically generated black and blue curves are SIS/SIR mixtures with γ₁ value of 0.04 days⁻¹ and γ₂ values of 0.005 and 0.01 days⁻¹, respectively. The introduction of positive γ₂ values into pure SIS models is seen to result in a “flattening” of the pure SIS model curves whose steady state values after many days are not representative of the curves typically reported in the media for COVID-19 epidemic outbreaks. The thick red curve in Figure 5, the pure SIR model with vanishing γ₁, is more representative of the curves reported in the media for COVID-19 outbreaks.

An application of the linear model and the KM model with SIR conditions to some real data is provided in Figure 6(a) where the number of cases reported in Massachusetts for the first 100 days following the initial lockdown in March of 2020 are plotted along with fits to both the linear model (red curve) and the KM model (blue curve). The parameters for the linear and KM model fits are provided in Table 2, where SF is a Scaling Factor found from a three-parameter fit

(β, γ₂ and SF) to the linear model and was used for the KM fit as well. Note that only for the linear model with its analytic solution could the standard non-linear regression analyses of Matlab be used for the three-parameter fit while for the numerical KM curve shown, manual adjustment of the free parameters β and γ₂ was performed to minimize the r² value with f₀ and SF fixed at 0.95 and 4798, respectively. The r² values of 0.83 and 0.87 for the linear and KM models, respectively, are quite similar, albeit slightly better for the KM model (Table 2) with the residuals for both fits shown in Figure 6(b).