On the redundancy of birth and death rates in homogenous epidemic SIR models

The dynamics of fractional population sizes y_i=Y_i/N in homogeneous compartment models with time dependent total population N is analyzed. Assuming constant per capita birth and death rates the vector field Y_i'=V_i(Y) naturally projects to a vector field F_i(Y) tangent to the leaves of constant population N. A universal formula for the projected field F_i is given. In this way, in many SIR-type models with standard incidence all demographic parameters become redundant for the dynamical system y_i'=F_i(y). They may be put to zero by shifting remaining parameters appropriately. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to be already covered by papers 20 years ago.


Introduction
The classic SIR model had been introduced by Kermack and McKendrick in 1927 [8] as one of the first models in mathematical epidemiology. The model divides a population into three compartments with fractional sizes S (Susceptibles), I (Infectious) and R (Recovered), such that S +I +R = 1. The flow diagram between compartments as given in Fig. 1 leads to the dynamical systemṠ = −βSI,İ = βSI − γI,Ṙ = γI.
(1.1) Here γ denotes the recovery rate and β the effective contact rate (i.e. the number of contacts/time of a susceptible leading to an infection given the contacted was infectious). Members of R are supposed to be immune forever. By (1.1) S decreases monotonically causing eventually βS < γ andİ < 0. At the end the disease dies out, I(∞) = 0, and one stays with a nonzero final size S(∞) > 0, thus providing a model for Herd immunity.
To construct models featuring also endemic scenarios one needs enough supply of susceptibles to keep the incidence βSI ongoing above a positive threshold. The literature discusses three basic methods to achieve this, see Fig. 2.
• Hethcotes classic endemic model adds to the SIR model balanced birth and death rates µ and assumes all newborns susceptible. This leads to a bifurcation from a stable disease-free equilibrium point to a stable endemic scenario when raising the basic reproduction number r 0 = β/γ above one [5,6,7].
• The SIRS model adds to the SIR model an immunity waning flow α R R from R to S, leading to the same result. • The SIS model considers recovery without immunity, i.e. a recovery flow γ S I from I to S while putting R = 0. Again this leads to the same result. In what follows the reader is assumed to be familiar with the basic notions in these models. For a comprehensive and self-contained overview of history, methods and results on mathematical epidemiology see the textbook by M. Martcheva, [11], where also an extensive list of references to original papers is given.
As a starting point for this paper observe from Fig. 3 that Hethcote's model could equivalently be reformulated by disregarding birth and death rates and instead introducing a combined SI(R)S ≡ SIRS/SIS model with flow rates γ S = α R = µ. More generally, adding Hethcote's balanced birth and death rates µ to a SI(R)S model with independent parameters (γ S , α R ) apparently becomes equivalent to considering the SI(R)S model without birth and death rates and with shifted parametersγ S = γ S + µ andα R = α R + µ [12]. The aim of this letter is to generalize this observation to homogeneous 3-compartment models with A) positive susceptibility of the R-compartment describing incomplete immunity (in which case it makes sense to rename S ≡ S 1 and R ≡ S 2 ), B) a non-trivial birth matrix and a time varying population size N due to compartment dependent constant per capita birth and death rates.
As a result we will see that for coinciding birth-minus-death rates in compartments S 1 ≡ S and S 2 ≡ R in the dynamics of fractional variables all demographic parameters become redundant by shifting remaining parameters appropriately. In particular, transmission coefficients β i describing S i -susceptibility are replaced byβ i = β i − ∆µ I , where ∆µ I denotes the excess mortality in compartment I. Hence,β i may possibly become negative.
This result leads to a unifying normalization prescription by always considering these models without vital dynamics and, instead, with two distinguished and possibly also negative incidence ratesβ i ∈ R. When normalized this way, seemingly different models in the literature become isomorphic at coinciding shifted parameters. As an example, recent results of [1] already follow from earlier results of [9] (forβ 2 > 0) and [10] (forβ 2 < 0).

Compartment models
For simplicity, all maps are supposed to be C ∞ . Let V = R n and V : V → V a homogeneous vector field, V (λY ) = λV (Y ) for all λ ∈ R + and Y ∈ V. Denote V * the dual of V and ·|· : V * ⊗ V → R the dual pairing. Let φ t : V → V be the local flow of V . For functions f : V → R we denote their time derivative along φ t byḟ := d/dt| t=0 (f • φ t ) = ∇f |V . Let 0 ∈ P ⊂ V be a cone and N : P → R + be a homogeneous function, N (λY ) = λN (Y ), satisfying ∇N = 0 on P. In this case the local flow φ t naturally projects to a local flow ψ t leaving the The vector field F : P → R n generating ψ t is given by Clearly, F is also homogeneous and putting y := Y /N we haveẏ = F (y).
Now let's specialize to compartment models, where Y i gives the population in compartment i, N (Y ) := i Y i the total population and P := R n ≥0 \{0}. To guarantee P being forward invariant one also needs We call ν i the total birth-minus-death rate in compartment i.
In such models one usually decouples the time development of N and analyzes the dynamics of fractional variables y = Y /N ,ẏ = F (y). The main observation of this paper states, that in many standard models the correction term N −1Ṅ Y in Eq. (2.1) can be absorbed by redefining the parameters determining V .
Let us apply this to vector fields V of the form Here µ j ≥ 0 is the mortality rate in compartment j, B ij Y j ≥ 0 denotes the number of newborns from compartment j landing in compartment i, and the parameters M ij and Λ ijk determine the population flow from compartment j to i, say due to infection transmission, recovery, loss of immunity, vaccination, etc. Thus, δ j := i B ij is the total birth rate in compartment j and ν j = i L ij = δ j − µ j . Also, forward invariance of the nonnegative orthant P for zero birthrates requires a) M to be essentially the new parametersM andΛ have the same properties as M and Λ and we get Hence, in the dynamics for fractional variables y = Y /N all birth and death rates may be absorbed by redefining M and Λ. Note that standard models typically satisfy Λ ijj = 0, which is consistent with Q ijj = 0. On the other hand,Λ iik =Λ iki might change sign as compared to epidemiological requirements.
for some function f and constant excess mortalities ∆µ i , then V is no longer homogeneous but Hence, the function f does not appear in the definition ofM andΛ in (2.4), implying that F in Eq. (2.5) is independent of f and still homogeneous, whenceẏ = F (y).

The 3-compartment master model
As a kind of master example consider an abstract SI(R)S-type model consisting of three compartments, S 1 , S 2 and I, with total population N = S 1 + S 2 + I. Members of I are infectious, members of S 1 are highly susceptible (not immune) and members of S 2 are less susceptible (partly immune). The flow diagram between compartments is completely symmetric with respect to permuting 1 ↔ 2 and depicted in Fig. 4. All parameters are assumed nonnegative. Also p I ≤ 1, q 1 + q 2 = 1, β 1 + β 2 > 0 and γ 1 + γ 2 > 0.
(3.6) to conclude from (2.4) In summary, denoting fractions of the total population by S i = S i /N and I = I/N and assuming the condition ν 1 = ν 2 =: ν the dynamics for fractional variables becomes So, for ν 1 = ν 2 = ν all birth and death rates become redundant and may be absorbed by redefining β i , α i and γ i . The price to pay is thatβ i = β i + ν I − ν might become negative. Hence, the space of admissible parameters for the system (3.8) becomes 1 : Concerning the dynamics of fractional variables, any two models mapping to the same set of shifted parameters a ∈ A become isomorphic. In particular, the case of constant population, ν i = ν I = 0, yieldsβ i = β i . In summary we get

Examples from the literature
For simplicity, from now on let's assume the rate of not infected newborns to be compartment independent, δ 1 = δ 2 = (1 − p I )δ I = δ, implying B = δN . Also, in this case one may without loss assume p I = 0 by redefining µ I . Hence ν 1 = ν 2 ⇔ µ 1 = µ 2 =: µ and in this case ∆ν I = ∆µ I := µ I − µ gives the excess mortality in the infectious compartment.
Below there is a list of prominent examples from the literature. Table 1 maps these examples to the present set of parameters.

SIRS
The 8-parameter constant population SI(R)S model with vaccination and two recovery flows I → S 1 and I → S 1 . Hence δ = µ i = µ I and β 2 = 0. HaCa The 6-parameter core system in [4], with transmission and recovery rates β i , γ i > 0, a vaccination term α 1 > 0 and a constant population with balanced birth and death rates, δ = µ i = µ I > 0 and q 1 = 1. KZVH The 7-parameter vaccination models of [9] adding an immunity waning rate α 2 > 0 to the model of [4]. LiMa The 8-parameter SIS-model with vaccination and varying population size of [10] keeping only γ 2 = β 2 = 0 and assuming µ 1 = µ 2 = µ. 2 AABH The 8-parameter SIRS-type model analyzed recently by [1], keeping only γ 1 = q 2 = 0 and all other parameters positive. The authors allow a varying population size by first discussing the general case of all mortality rates being different and then concentrate on µ 1 = µ 2 = δ and ∆µ I > 0. 1 The caseβ1 =β2 will be ignored, since in this case putting S = S1 + S2 one easily checks that (S, I) obeys the dynamics of a SIS model, which can immediately be solved by separation of variables. Also, due to the permutation symmetry 1 ↔ 2, there is no loss assumingβ1 >β2.
Assuming µ 1 = µ 2 and applying the transformations (3.6) we get a classification in terms of the redundancy-free 6-parameter set A.
The dimensions of these parameter spaces are listed in the last column of Table 1. In summary, we arrive at Corollary 4.1. Consider the dynamics of fractional variables in the models of Table 1, for BuDr and AABH under the restriction µ 1 = µ 2 . Disregarding boundary configurationsγ i = 0 in parameter space A, the following relations hold. i) The model of AABH [1] is isomorphic to the master model (3.8) and covers all other models.

Summary
We have seen in Lemma 2.2 that in a large class of homogeneous compartment models with constant per capita demographic rates and time dependent total population N the dynamics of fractional variables y = Y /N can be rewritten such that all demographic parameters become redundant. This way various prominent SI(R)S-type models with standard incidence, demographic parameters and possibly susceptible R-compartments may be normalized such that the dynamics of fractional variables appears as sub-case of a master model with zero birth and death rates, see Eqs. (4.1)-(4.7). Since apparently none of the original papers has used the identity (2.2) of Lemma 2.2, these relations have not been realized before. The price to pay is that in the normalized master model infection transmission ratesβ i may also be negative. As a particular example, recent results on backward bifurcation in models with time varying total population N (t), coinciding mortality rates µ 1 = µ 2 and an excess mortality 0 < ∆µ I < min{β 1 , β 2 } by AABH [1] are already covered by the isomorphic model with constant population of KZVH [9] published in 2000. The complementary case ∆µ I > min{β 1 , β 2 } turns out to be isomorphic to the variable population SIS model with β 2 = 0 published by LiMa [10] in 2002.
The normalized master model (3.8) will also be the starting point of an ongoing analysis of symmetry operations in these kinds of models giving rise to further parameter reductions, see work in progress in [13,14].