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Traditional biomedical approaches treat diseases in isolation, but the importance of synergistic disease interactions is now recognized. As a first step we present and analyze a simple coinfection model for two diseases simultaneously affecting a population. The host population is affected by the

Coinfection is the simultaneous infection of a host by multiple pathogen species. The global incidence of coinfection among humans is huge [

The case of positive parasite interactions falls into the concept of

In this work we deal with a particular, but very common, type of coinfection. We consider the interactions of two diseases, the first one of the type called

The importance of opportunistic diseases for public health [

Specifically, we want to know whether or not the coinfection by a secondary disease produces epidemiological scenarios not allowed by the primary disease submodel. In the latter case, it is of interest to assess if coinfection has any influence on the actual outcome of the model, even if it is only among those allowed by the primary disease submodel. On the other hand, and in any case, we look for identifying mechanisms to modulate the epidemiological outcome.

A primary disease enabling secondary infections has typically a long illness period. It must produce a persistent alteration of the immune response which weakens the body’s ability to clear secondary diseases. On the other hand, a compromised immune system presents an opportunity that a secondary pathogen must rapidly take advantage of. As a simplified approximation of the general case we suppose that the primary disease is a long-term infection that evolves slowly compared to the opportunistic disease which has a rapid evolution and, thus, can be considered a short-term infection. This difference in the acting speed of both infections is reflected in our model in two different issues. Firstly, we assume that demography has an impact in the primary disease, due to its slow evolution, whereas it is negligible for the opportunistic disease which evolves in a short period of time. Secondly, the system of differential equations, in terms of which we express our model, possesses two time scales: the slow one encompassing the demography and the primary disease evolution and the fast one associated with the opportunistic disease evolution.

The inclusion of two time scales in the system has the advantage of allowing its reduction. The asymptotic behavior of the solutions of the initial three-dimensional system can be studied through a planar system. The reduction of the system is undertaken with the help of aggregation methods [

The model is presented in Section

We build up in this section a model of coinfection that describes the interaction between two diseases, one of primary type whereas the second one is of opportunistic type. Only the individuals infected by the primary disease are susceptible of being infected by the opportunistic disease. Moreover, the interaction of both diseases occurs at different time scales, the evolution of the opportunistic disease being much faster than that of the primary one. The model is written in terms of a slow-fast ordinary differential equations model. After building the slow-fast model, the separation of time scales allows us to apply approximate aggregation techniques [

The primary disease dynamics is described by a

We denote by

We consider demographic effects with only horizontal transmission of the disease. In many mathematical models, from a demographic point of view, the differences between susceptible and infected individuals are reduced to an additional disease-related death rate or disease-induced reduction in fecundity [

The primary disease submodel is given by the equations

The opportunistic disease spreads only through the individuals infected by the primary disease. We consider that the opportunistic disease dynamics is also described by a

The fast evolution of the opportunistic disease, compared to primary disease and demography, suggests not including demographic effects and choosing the frequency-dependent transmission form. Let

The opportunistic disease submodel is represented by the equations

Finally, we construct the model encompassing both diseases. It has the form of a system with three state variables: susceptible

In the slow part of system (

Concerning the part of demography, we keep the same intrinsic per capita fertility rate of uninfected individuals

The complete two-time-scale system reads as follows:

In this section we take advantage of the two time scales to reduce the dimension of the complete system (

The fast equilibria

Note that the reduced system (

We proceed in this section to analyze the reduced system (

We first note that

In the next result we prove that, as expected, if the susceptible fertility rate

If

Let us call

Henceforth, we assume that

Assuming

Let

Calling

In addition to the trivial equilibrium

Let

if

if

if

To prove the two first items it suffices to calculate the matrix of the linearization of system (

To prove the last assertion we first note that there exist no interior equilibria because the right-hand side of the

Up to now we have obtained the condition of nonextinction of the population,

From now on we are also assuming that

In the next results we search for conditions ensuring the endemicity of the infection. To express them in a simpler form we define another parameter

Using parameters

The equation of the

Possible profiles of the

The

Let

Condition

There are two situations for the infection to become endemic. The first one is allowing invasion, that is,

Let

The assumptions on parameters yield the existence of unique interior equilibrium (see Figure

Note that the condition

Condition

The invariant region

In any case, what condition

Related to Proposition

Indeed, we introduce

Let

if

if

if

Note that the asymptotic stability of

Next, we focus on showing the relation between

Note that when

Concerning statement

We now assume

We have set up a model aimed at ascertaining the impact of an opportunistic disease outbreak in a population already affected by a primary disease by assuming that both diseases evolve within different time scales. For the discussion of results, let us remember the two main aims stated in the Introduction. On the one hand, we wanted to know whether the coinfection by a secondary disease produces epidemiological scenarios not allowed by the primary disease submodel or not. In the latter case, it is of interest to determine if coinfection has any influence on the actual outcome of the model, even if just among those allowed by the primary disease submodel.

The answer to the first question is negative, as we have pointed out at the end of Section

Nevertheless, the effect of the opportunistic disease must be taken into account. In Section

Possible epidemic outcomes. Yellow: disease-free. Orange: endemic primary infection. Gray: disease-free or endemic coinfection depending on initial values. Red: endemic coinfection. (a) Parameter values:

In Figure

Summing up, both the irruption of an opportunistic disease and the competitive pressure of individuals being in different epidemiological state may affect the evolution of the primary disease outbreak. The effect can be determined by means of the parameters

Related to our second objective, our results point out two different kinds of mechanisms to modulate the outcome of the model, each of them feasible within certain ranges of the parameter values.

On the one hand, having control on parameters

On the other hand, the results in Propositions

A final comment has to do with the selection of the transmission form of the opportunistic disease. Preliminary calculations show that considering DDT instead of FDT leads to equivalent results. This means that even if the nullclines are different, the possible outcomes (say the dynamical scenarios) of the corresponding aggregated model are the same.

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

M. Marvá and R. Bravo de la Parra are partially supported by Ministerio de Ciencia e Innovación (Spain), Projects MTM2011-24321 and MTM2011-25238. E. Venturino is partially supported by the Project “Metodi numerici in teoria delle popolazioni” of the Dipartimento di Matematica “Giuseppe Peano.”