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A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.

Epidemic models have been widely studied in the last decades involving several inter-actuating subpopulations with mutual coupled dynamics. Important properties which have to be required to the epidemic models for their well-posedness are their solution positivity under any given nonnegative initial conditions as well as their stability conditions around one of the equilibrium points. Thus, relevant background literature has dealt with the study of these issues for different types of epidemic models, in both continuous and discrete-time [

The remaining main body is organized in two more sections referred to the model analysis in the delay-free case and under the presence of delays. The positivity and stability of the solutions in both cases are formally discussed in a general context, rather than for specific models, based on the linearization analysis on the infective compartment for the disease-free and endemic equilibrium points. Illustrative examples referred to the formal links of the presented mathematical framework to particular epidemic models are discussed. Two given appendices are given which contain auxiliary technical results on stability of Metzler and

If

(2)

(3)

It turns out that

The logic conjunction (“And”) and logic disjunction (“Or”) of propositions are defined by the symbols “

For vectors

A general compartmental disease model of dimension

The conditions of Assumptions 1 are invoked for the nonnegativity of the state trajectory solution for initial conditions in the first orthant and for the local stability of the disease-free equilibrium point in the absence of infection, i.e., under a zero reproduction number. See [

(b) The minus transition matrix

The above assumptions are very relevant from a physical point of view having in mind that

Note that Assumption 3(b) is of a clear interpretation for the linearization around the disease-free equilibrium point while its equivalent form of Assumption 2 is easy to test since it is not needed to calculate the allocation of the eigenvalues of the matrix

The following properties hold.

(i) The linearization of the epidemic model (

(ii) A necessary condition for an epidemic model (

(iii) The linearized infective substate around the disease-free equilibrium point has a nonnegative solution which is uniformly bounded for all time with

(1)

(2)

A joint sufficiency-type condition for any of the above Conditions (1)-(2) to hold is that

(iv) The linearized infective substate around the disease-free equilibrium point is nonnegative for any given initial conditions

(1)

(2)

(3)

The solution of the linearized infective substate around the disease-free equilibrium point is

Note that if

The following properties hold.

(i) The state trajectory solution of (

Property (i) follows directly from (

Generally speaking, Theorem

On the other hand, it turns out that the nonnegativity condition

The following properties hold.

(i) The state trajectory solution of (

It is obvious that, for any given

Let

(i) There exists an open set

(ii) Assume, furthermore, that the transition and transmission matrices of the disease-free equilibrium point satisfy that

(b) the endemic equilibrium point exists (while it is unique under the sufficiency-type conditions of Theorem

Assume that

(1) If

(2) If

It has not been proved that the endemic equilibrium point is necessarily unique, independently of the concrete epidemic model, in the whole state space. A detailed proof of the global uniqueness of the disease-free equilibrium point is given in Theorem

It is well-known from the background literature on epidemic models, subject to a unique disease-free equilibrium point and a unique endemic equilibrium one, that, typically, if the reproduction number

A particular case of the SEIADR (susceptible-exposed-symptomatic infectious-asymptomatic infectious-infectious corpses-recovered) proposed in [

It has been proved in Theorem

Assume the following.

(1) The endemic and the disease-free transition matrices

(2) The disease-free and endemic equilibrium points exist and they are unique for any

(3) The disease-free and the endemic transmission matrices are nonnegative, i.e.,

(4)

(5)

Then, the endemic equilibrium point is locally asymptotically stable if

Note that

Note that for Example

Consider the SEIADR epidemic model of Example

Figure

Maximum Eigenvalues of the Jacobians of the model of Example

Consider a true mass action (i.e., the nonlinear infective terms of susceptible-infected products are normalized with the total population) variant of the SEIADR epidemic model of Examples

A global stability theorem now follows by combining the preceding results and the analysis of Poincaré indices and the alternate stability characteristics of limit cycles surrounding singular points in any hyperplanes of the state space. It is proved, in particular, that the local asymptotic stability of the disease-free equilibrium point for a reproduction number less than one (implying also the unattainability of the endemic equilibrium point) leads to the global asymptotic stability of the whole state towards the disease-free equilibrium point. Note that Proposition

Maximum real part of the eigenvalues versus the reproduction number.

Zoom of the maximum real part of the eigenvalues versus the reproduction number close to the stability region boundary.

Under Assumptions 1

(2)

(3)

(i) Then, the total population, i.e., the sum of all the subpopulations, is uniformly bounded for all time. Furthermore, any trajectory solution is nonnegative and uniformly bounded for all time for any given finite initial condition

(ii) Assume that for

The nonnegativity of the solutions and the local asymptotic stability of the disease-free equilibrium point, which is unique, follow directly from Theorem

A very important remark is that the mathematical hypothesis of boundedness of the total population for the cases of reproduction numbers exceeding unity of Theorem

A compartmental disease model of dimension

(b) The disease-free equilibrium point

A simple motivation of the model (

Sufficient conditions for the stability of the linearized system independent of the delay sizes around the disease-free equilibrium point can be obtained for the cases when either

The following properties hold.

(i) The infective linearized subsystem obtained from (

(ii) If

Eq. (

Some basic results, given in Theorem

The following properties hold under Assumptions 4

(i) The linearized epidemic model (

(ii) The linearized infective substate around the disease-free equilibrium point has a nonnegative solution trajectory and it is globally exponentially stable independent of the delays for any admissible function of initial conditions

(iv) The whole linearized model around the unique disease-free equilibrium point has a nonnegative state trajectory solution for any given admissible nonnegative function of initial conditions which converges asymptotically to the unique disease-free equilibrium point.

Note that the solution of the linearized infective substate around the disease-free equilibrium point is

Taking into account (

The linearized infective substate around the disease-free equilibrium point is positive and exponentially stable independent of the set of

Taking into account the expression (

The linearized infective substate around the disease-free equilibrium point is exponentially stable independent of the set of

One gets the characteristic equation from (

It follows from Corollary

The infective linearized system around the disease-free equilibrium point has two important particular cases which define auxiliary linearized systems which are useful for the local stability analysis independent of the delays, namely,

(a)

(b)

Note that the disease reproduction number

Thus, the asymptotic stability independent of the delays of the linearized system around the disease-free equilibrium point holds if

(1) For a set of delays

(3) In the inequalities of the above cases (a) and (b), if

The following three results discuss the stability and positivity independent of the delays of the linearized system around the disease-free equilibrium point if either

If Assumption 5 holds then the following properties hold.

(i) If constraint (

(ii) The

The positivity and stability are guaranteed for the linearized infective substate from Theorem

Under Assumption 5, if

(i)

(ii) the

It follows under a similar reasoning as that in the proof of Corollary

If Assumption 5 holds,

It follows under a similar reasoning as that in the proof of Corollary

If Assumption 5 holds, and

Note that, since Theorem

A case of interest might be when

Consider the subsequent linearized model around the disease-free equilibrium point:

(1)

(2)

(3)

(4)

(5) If

The following result relies on sufficiency-type conditions of global asymptotic stability independent of the delays of the linearized infective substate around the disease-free equilibrium point including the case when the transmission matrices for the various delays are all identical.

The following properties hold:

(i) Assume that

(ii) If

If

Condition 2 of Theorem

A related result to Theorem

Let

(i) There exists an open set

(ii) Assume, furthermore, that

(b) the endemic equilibrium point exists being defined by

Since

To prove Property (ii), note that

(a) If

(b) If

The above result can be reformulated directly under the replacements

Consider the epidemic model (

(1) any of the sets of conditions of either Theorem

(2) there are a unique disease-free equilibrium point and a unique endemic equilibrium point,

(3)

(4)

(5) the epidemic model is subject to any bounded absolutely continuous function of initial conditions

(i) Then, any trajectory solution is nonnegative and uniformly bounded independent of the delays while it is globally convergent to the disease-free equilibrium point if

If furthermore, the following condition holds:

(6) if

(ii) Then, provided that

If

Note that the equilibrium points do not depend on the delays but only on the delay-free and delayed matrices. However, the reproduction numbers are dependent, in general, on each particular set of delays although the threshold

(2) Note also that Theorem

(3) From Remark

Some technical results on positivity and stability of matrices and their perturbed counterparts are given. Those results are then used concerning the stability of the linearized system around the disease-free equilibrium point.

Assume that

(i)

(ii)

(iii)

(iv) There exists

Assume that

(i) Assume that

(ii) Assume that

Since

To prove Property (ii), note that

On the other hand, since

The following properties hold.

(i) Let

(ii) Let

Let

Assume that all the entries of the Jacobian matrix around any equilibrium point are additive functions and that all the equilibrium points have the same associate transition matrix

(i) Let

Assume that there exist two distinct equilibrium points with infective and noninfective variables

Previously reported data used to support this study are cited at relevant places within the text referred to the references in the manuscript.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the Spanish Government for Grants DPI2015-64766-R, RTI2018-094336-B-I00, and DPI2016-77271-R (MINECO/FEDER, UE).