Threshold conditions for infection persistence in complex host-vectors interactions

Luiz Fernandes Lopez; Francisco Antonio Bezerra Coutinho; Marcelo Nascimento Burattini; Eduardo Massad

doi:10.1016/S1631-0691(02)01534-2

Threshold conditions for infection persistence in complex host-vectors interactions
[Conditions seuil de persistence d’infection dans un système complexe d’interactions hôte–vecteur.]

Présenté par : Pierre Auger

Luiz Fernandes Lopez ¹ ; Francisco Antonio Bezerra Coutinho ¹ ; Marcelo Nascimento Burattini ¹ ; Eduardo Massad ¹

¹ School of Medicine, The University of São Paulo and LIM01/HCFMUSP, Av. Dr. Arnaldo 455, São Paulo 01246-903, SP, Brazil

Comptes Rendus. Biologies, Volume 325 (2002) no. 11, pp. 1073-1084.

Résumés

Anglais
Français

As classically defined by Macdonald in the early 1950s, for the case of diseases with one vector and one host, the Basic Reproduction Number, R₀, is defined as the number of secondary infections caused by a single infective of the same type (vector or host) during its infectiousness period in an entirely susceptible population. In the case of a disease which has one vector and one host, it is easy to show that R₀ coincides with the threshold for the establishment of an endemic state: if $R_{0} > 1 (< 1)$ , the disease can invade (cannot invade) the host population. In this paper we examine various epidemic situations in which there are more than one vector and/or host. We show that in those more complex systems it is not possible to deduce a single R₀ but rather a threshold for infection persistence which is a composite of several quantities closely related to the classical expression of R₀. Another definition of R₀ given by Diekmann, Heesterbeek and Metz, and denoted in this paper R^NGO₀ is discussed and applied as an alternative to calculate the thresholds for infection establishment.

Dans le cas d’une infection à un seul vecteur et à un seul hôte, le taux de reproduction de base R₀ a été défini par Macdonald dans les années 1950 par le nombre d’infections secondaires causées par un seul individu infecté (hôte ou vecteur) pendant toute la durée de sa période infectueuse, au contact d’une population entièrement saine. Dans le cas d’une infection à un seul hôte et un seul vecteur, il est facile de montrer que R₀ correspond à un seuil d’installation d’un état endémique: Si $R_{0} > 1 (< 1)$ , l’épidémie envahit (disparaît de) la population hôte. Dans cet article, nous étudions diverses situations d’épidémies à plusieurs hôtes et/ou vecteurs. Nous montrons que, dans ces systèmes plus complexes, il n’est plus possible de définir un seul R₀, mais plutôt un seuil de persistence de l’infection, qui est un agrégat de plusieurs quantités fortement connectées à l’expression classique de R₀. Nous discutons une autre définition du R₀, initialement proposée par Diekmann, Heesterbeck et Metz, notée R^NGO₀, qui est proposée comme une alternative pour calculer le seuil d’installation d’une infection.

Métadonnées

Reçu le : 2002-08-26
Accepté le : 2002-09-24
Publié le : 2002-11-01

PMID

DOI : 10.1016/S1631-0691(02)01534-2

Keywords: basic reproductive number, vector-borne infections, yellow fever
Mot clés : taux de reproduction de base, infection transmise par un vecteur, fièvre jaune

Affiliations des auteurs :

Luiz Fernandes Lopez ¹ ; Francisco Antonio Bezerra Coutinho ¹ ; Marcelo Nascimento Burattini ¹ ; Eduardo Massad ¹

¹ School of Medicine, The University of São Paulo and LIM01/HCFMUSP, Av. Dr. Arnaldo 455, São Paulo 01246-903, SP, Brazil

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     pages = {1073--1084},
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Luiz Fernandes Lopez; Francisco Antonio Bezerra Coutinho; Marcelo Nascimento Burattini; Eduardo Massad. Threshold conditions for infection persistence in complex host-vectors interactions. Comptes Rendus. Biologies, Volume 325 (2002) no. 11, pp. 1073-1084. doi : 10.1016/S1631-0691(02)01534-2. https://comptes-rendus.academie-sciences.fr/biologies/articles/10.1016/S1631-0691(02)01534-2/

Version originale du texte intégral

1 Introduction

Often unnoticed by practicing physicians in the temperate zone, arthropod-borne diseases account for a huge proportion of the spectrum of human maladies worldwide, and the problem appears to be growing 〚1〛. Despite the enormous effort of the medical and scientific community, controlling disease agents transmitted by arthropod vectors has proven to be difficult. The list of emerging and re-emerging infections is enormous but it is worth citing just a few: dengue, malaria, yellow fever, various mosquito-borne encephalitis, leishmaniasis, and Lyme disease.

Most of the techniques used for the control and eradication of vector-borne diseases were developed in the early 20th century. Rules for source reduction, insecticides, biological control, vaccination, chemotherapy and personal protection were all laid down nearly a century ago 〚2〛. Many of these techniques are still effective, others succeeded initially but failed later for a variety of reasons. Investigators must now incorporate new approaches that will allow them to move to the next level of control to alleviate the effects of vector-borne diseases on human and animal health 〚2〛.

The central parameter related to the intensity of transmission of infection is the so called basic reproduction number (R₀), defined by Macdonald 〚3〛 as the number of secondary infections produced by a single infective in an entirely susceptible population (see next section). Originally applied in the context of malaria, R₀ is a function of the vector population density as related to the host population, m, the average daily biting rate of the vector, a, the host susceptibility, b, the vector mortality rate, μ, the parasite extrinsic incubation period in days, n, and the parasitemia recovery rate, r, according to the (now) historical equation:

R_{0} = \frac{m a^{2} b \exp [- μ n]}{r μ}

(actually, Macdonald denoted R₀ as z₀ in his original paper). From the definition of the basic reproduction number it can be demonstrated that if R₀ is not greater than one, that is, when an index case (the first infective individual) is not able to generate at least one new infection, the disease dies out. Hence, in the original Macdonald analysis, R₀ coincides with the threshold for the infection persistence. For an interesting historical account of R₀ see 〚4〛.

This paper is organized as follows. In section 2 we revisit the Macdonald analysis of malaria transmission and deduce the general expression of R₀ for any one vector – one host epidemic system. We show, using a dynamical system approach, that R₀ coincides with the threshold for the infection persistence.

In section 3 we examine the definition of the basic reproduction number given by Dieckmann, Heesterbeek and Metz 〚5〛 as compared with the classical Mcdonald analysis.

In section 4 we examine more complex vector-hosts systems, exemplifying with a 2 vectors–2 hosts system, and show that there are four basic reproduction numbers of the Macdonald type, one for each kind of index case. Obviously, those R₀'s do not coincide with the threshold for epidemic persistence. However, we show that the threshold can be expressed in terms of those basic reproduction numbers. There is, however, only one basic reproduction number of the type defined by Dieckmann, Heesterbeek and Metz 〚5〛. As we shall see the same threshold can be deduced from it.

In section 5 we discuss an even more complex epidemic system, namely the yellow fever case, which comprises three different kinds of hosts and two vectors. In this case there are five R₀'s, one for each kind of index case. Also in this particular case, those R₀'s do not coincide with the threshold for epidemic persistence. Again, the threshold is given by a combination of them.

Finally, in the discussion section we summarize our findings.

2 The classical Macdonald analysis

In his 1952 seminal paper, Macdonald 〚3〛 addressed the problem of a system involving one vector (Anopheles mosquitoes) and one host (man). As mentioned above, his definition of R₀ is the number of secondary infections in the first generation, that is, produced by a single infectee along his entire infectiousness period. We shall deduce an explicit expression for R₀ from an intuitive perspective to show that it coincides with the threshold for the establishment of the disease. We do this because, as shown in the next section for more complex systems, this approach does not work in such a simple way.

Let us begin by assuming that the index case is a human host. The question to be answered is how many human secondary infections this index case produces in his/her entire infectiousness period.

Let N_m be the number of female mosquitoes. Let a be the average daily biting rate female anophelines inflict in the human population. The number of bites in the human population per unit of time is, therefore, N_ma. Let N_h be the number of humans and r be the rate of recovery from parasitemia in the human cases. Therefore, the index case produces $\frac{N_{m} a}{N_{h} R} c_{h \to m}$ infected mosquitoes, where c_h→m is the probability that a mosquito gets the infection after biting an infective human. These $\frac{N_{m} a}{N_{h} r} c_{h \to m}$ infected mosquitoes, in turn, produce $a \frac{N_{m} a}{N_{h} r} c_{h \to m} \frac{1}{μ} b_{m \to h} e^{- μ δ}$ new human cases in the first generation, where $\frac{1}{μ}$ is the average life expectancy of mosquitoes, b_m→h is the probability that a human gets the infection after being bitten by an infective mosquito and $e^{- μ δ}$ is the fraction of the infected mosquito population that survives through the extrinsic incubation period δ of the parasite. Note that, once infective a mosquito is assumed to remain so for life. Therefore, the expression for R₀ is 〚3〛:

R_{0} = a \frac{N_{m} a}{N_{h} r} c_{h \to m} \frac{1}{μ} b_{m \to h} e^{- μ δ}

Similarly, if we begin with an infective mosquito as an index case, and compute the number of infected mosquitoes this index case produces in the first generation we get the same expression.

Let us now see how this deduction can be performed by a dynamical system approach.

Let Y_h be the number of infected humans, and Y_v the number of infected vectors. We can write

\frac{d Y_{h}}{d t} = \frac{Y_{v} a}{N_{h}} b_{v \to h} S_{h} - r Y_{h} \frac{d Y_{v}}{d t} = \frac{S_{v} (t - δ) a}{N_{h}} c_{h \to m} e^{- μ δ} Y_{h} (t - δ) - μ Y_{v}

where S_h and S_v are the number of susceptible humans and vectors, respectively.

To deduce the threshold for the disease to establish in the human population we analyse the stability of the trivial solution S_h=N_h, S_v=N_v, Y_v=Y_h=0, that is, the solution representing the absence of the infection. Linearising the system (3) around the trivial solution, we get

\frac{d y_{h}}{d t} = y_{v} a b_{v \to h} - r y_{h} \frac{d y_{v}}{d t} = \frac{N_{v} a}{N_{h}} c_{h \to m} e^{- μ δ} y_{h} (t - δ) - μ y_{v}

where y_v and y_h are small deviations from zero. From the system (4), assuming solutions of the type

y_{h} = A e^{λ 𝑡}

and

y_{v} = B e^{λ 𝑡}

, we get the following characteristic equation for λ:

|\begin{matrix} - (λ + r) & a b_{v \to h} \\ \frac{N_{v} a}{N_{h}} c_{h \to m} e^{- μ δ} e^{- λ δ} & - (λ + μ) \end{matrix}|

λ^{2} + (μ + r) λ + μ r - \frac{N_{v} a}{N_{h}} c_{h \to m} e^{- μ δ} e^{- λ δ} a b_{v \to h} = 0

It follows (see Appendix) that the roots of equation (5) or (6) have negative real parts if

μ r - \frac{N_{v} a}{N_{h}} a b_{v \to h} c_{h \to m} e^{- μ δ} > 0

The above result is the same as that obtained by the intuitive McDonald’s approach.

This still holds true for slightly more complex systems, such as those with one vector and two hosts populations or two vectors with one host populations. In these cases, the expression for R₀ is partitioned in a sum with the individual terms of each component of the transmission chain 〚6〛.

3 The next generation operator

In a classical paper, Dieckman et al. 〚5〛 propose a new definition of the basic reproduction number for infections and we now study how it compares with the classical Macdonald definition described above.

Those authors define R₀ as being the greatest eigenvalue of an operator which they call ‘the next generation operator’ (NGO). The case of vector-transmitted infections was analysed in a recent book by Dieckman and Heeterbeek 〚7〛.

In this section we give the next generation operator for the case of one-vector/one-host, exemplified by malaria. In this case, the next generation operator reduces to a two-by-two matrix

NGO = (\begin{matrix} A_{v \to v} & A_{v \to h} \\ A_{h \to v} & A_{h \to h} \end{matrix})

The elements have the following interpretation. The element A_v→h, for instance, means the number of infected humans generated by a single infected vector during its infectious period. Therefore, we have:

A_{v \to v} = 0 A_{v \to h} = a \frac{1}{μ} b_{m \to h} e^{- μ δ} A_{h \to h} = 0 A_{h \to v} = \frac{N_{m} a}{N_{h} r} c_{h \to m}

Essentially, the NGO is the mean infectious output over all possible progressions of the infection within the host individual and therefore contain any process that influences output of infectious material to others. In particular, the NGO was derived without appealing to the dynamical system (3).

In this case the greatest eigenvalue of the NGO matrix, that is, R^NGO₀, is

R_{0}^{NGO} = \sqrt{a \frac{N_{m} a}{N_{h} r} c_{h \to m} \frac{1}{μ} b_{m \to h} e^{- μ δ}}

which is the square root of the Macdonald R₀. It follows from the general theory of the Next Generation Operator 〚5〛 that if

R_{0}^{NGO} < 1 (R_{0}^{NGO} > 1)

the disease cannot (can) invade the host population.

4 Difficulties with more complex systems

The simple one vector/one host system characteristic of a great number of vector-borne diseases such as malaria, however, is not unique in the biology of the diseases transmitted by arthropods 〚2〛.

In some cases, more than one vector, or more than one host, and in some cases, more than one parasite may be involved. Examples of the complexity of the vector-borne infections abound (see, for instance any zoonoses textbooks, for instance Palmer, Soulsby and Simpson 〚8〛).

In some situations, several hosts may act as reservoirs of infection, being, therefore, intermediate steps in the chain of transmission. In the dengue/yellow fever complex, we have two (Haemagogus and Sabethes mosquitoes) or three (Haemagogus, Sabethes and Aedes mosquitoes) vectors for yellow fever, one common vector for dengue and yellow fever (Aedes mosquitoes), and at least two hosts (one human and one primate reservoir). As we demonstrate in the following sections, in such complex situations, a single threshold in the Macdonald’s R₀ style simply does not exist.

4.1 The intuitive approach

Let us consider now a more complex system, namely one with two vectors and two host populations. This hypothetical system comprises an interaction between a human (N_h) and an animal (N_m) population, mediated by two species of vectors, referred to as (N_A) and (N_H). We shall try to deduce an explicit expression for R₀ from the intuitive perspective used by Macdonald 〚3〛.

Let us begin by assuming that there is an index case in the human population. The question to be answered is how many human and animal secondary infections this index case produces in his/her entire infectiousness period.

Let a_A be the average daily biting rate female mosquitoes of type A inflict in both host populations. The number of bites in the human population per unit of time is, therefore, $N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}}$ . Let r_h be the rate of recovery from parasitemia in the human cases. Therefore, the index case produces $N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A}$ infected mosquitoes, where $\frac{1}{N_{h}}$ is the fraction of the bites inflicted in the index case, and c_h→A is the probability that a mosquito gets the infection after biting an infective human. Those $N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A}$ infected mosquitoes, in turn, produce

N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A} a_{A} \frac{1}{μ_{A}} b_{A \to h} e^{- μ_{A} δ_{A}}

new human cases in the first generation, where

\frac{1}{μ_{A}}

is the average life expectancy of mosquitoes, b_A→h is the probability that a human gets the infection after being bitten by an infective mosquito and e^-μ_Aδ_A is the fraction of the infected mosquito population that survives through the extrinsic incubation period δ_A of the parasite. Note that, once infective, a mosquito is assumed to remain so for life.

By the same token, the same index case also produces

N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to H} a_{A} \frac{1}{μ_{H}} b_{H \to h} e^{- μ_{H} δ_{H}}

new human cases in the first generation, through the vector population H.

On the other hand, the same human index case also produces

N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A} a_{A} \frac{1}{μ_{A}} b_{A \to m} e^{- μ_{A} δ_{A}}

secondary animal cases due to vector population of type A, and

N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to H} a_{A} \frac{1}{μ_{H}} b_{H \to m} e^{- μ_{H} δ_{H}}

secondary animal cases due to vector population of type H.

Moreover, an animal index case also produces a similar amount of secondary animal and human infections due to each vector population.

Therefore, the human index case generates:

R_{hAh} + R_{hHh}

where

R_{hAh} = N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A} a_{A} \frac{1}{μ_{A}} b_{A \to h} e^{- μ_{A} δ_{A}} R_{hHh} = N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to H} a_{A} \frac{1}{μ_{H}} b_{H \to h} e^{- μ_{H} δ_{H}}

human cases through the mosquitoes vectors A and H, respectively.

Moreover, the human index case also generates:

R_{hAm} + R_{hHm}

where

R_{hAm} = N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to A} a_{A} \frac{1}{μ_{A}} b_{A \to m} e^{- μ_{A} δ_{A}} R_{hHm} = N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{h}} \frac{1}{r_{h}} c_{h \to H} a_{A} \frac{1}{μ_{H}} b_{H \to m} e^{- μ_{H} δ_{H}}

animal cases through the mosquitoes vectors A and H, respectively.

In the case when the index case is an animal, we find that it generates:

R_{mAh} + R_{mHh}

where

R_{mAh} = N_{A} a_{A} \frac{N_{m}}{N_{h} + N_{m}} \frac{1}{N_{m}} \frac{1}{r_{m}} c_{m \to A} a_{A} \frac{1}{μ_{A}} b_{A \to h} e^{- μ_{A} δ_{A}} R_{mHh} = N_{H} a_{H} \frac{N_{m}}{N_{h} + N_{m}} \frac{1}{N_{m}} \frac{1}{r_{m}} c_{m \to H} a_{A} \frac{1}{μ_{H}} b_{H \to h} e^{- μ_{H} δ_{H}}

human cases through the mosquitoes vectors A and H, respectively.

It also generates:

R_{mAm} + R_{mHm}

where

R_{mAm} = N_{A} a_{A} \frac{N_{m}}{N_{h} + N_{m}} \frac{1}{N_{m}} \frac{1}{r_{m}} c_{m \to A} a_{A} \frac{1}{μ_{A}} b_{A \to m} e^{- μ_{A} δ_{A}} R_{mHm} = N_{H} a_{H} \frac{N_{m}}{N_{h} + N_{m}} \frac{1}{N_{m}} \frac{1}{r_{m}} c_{m \to H} a_{A} \frac{1}{μ_{H}} b_{H \to m} e^{- μ_{H} δ_{H}}

animal cases through the mosquitoes vectors A and H, respectively.

Since, strictly speaking, the expression for the basic reproduction number should be calculated starting from an index case and counting the first generation number of cases of the same kind of the index case we have two basic reproduction numbers, depending on the kind of index case.

If the index case is a human being we tentatively set

R_{0_{h}} = R_{hAh} + R_{hHh}

On the other hand, if the index case is an animal, we tentatively set

R_{0_{m}} = R_{mAm} + R_{mHm}

However, we still have to investigate the case where the index case is a mosquito.

Assume that the index case is a mosquito of type A. It produces

a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{μ_{A}} b_{A \to h}

infected humans and

a_{A} \frac{N_{m}}{N_{h} + N_{m}} \frac{1}{μ_{A}} b_{A \to m}

animal cases. Those, in turn, produce

R_{AhA} + R_{AmA}

where

R_{AhA} = N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{m} + N_{h}} \frac{1}{r_{h}} c_{h \to A} a_{A} \frac{1}{μ_{A}} b_{A \to h} e^{- μ_{A} δ_{A}} R_{AmA} = N_{A} a_{A} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{m} + N_{h}} \frac{1}{r_{h}} c_{m \to A} a_{A} \frac{1}{μ_{A}} b_{A \to m} e^{- μ_{A} δ_{A}}

infected mosquitoes of type A.

They also produce

R_{AhH} + R_{AmH}

where

R_{AhH} = N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{m} + N_{h}} \frac{1}{r_{h}} c_{h \to H} a_{A} \frac{1}{μ_{A}} b_{A \to h} e^{- μ_{𝐴} δ_{𝐴}} R_{AmH} = N_{H} a_{H} \frac{N_{h}}{N_{h} + N_{m}} \frac{1}{N_{m} + N_{h}} \frac{1}{r_{h}} c_{m \to H} a_{A} \frac{1}{μ_{A}} b_{A \to m} e^{- μ_{𝐴} δ_{𝐴}}

infected mosquitoes of type H.

So, if we stick to the convention that R₀ is the number of secondary cases of the same type of the index case, we have two more expressions:

R_{0_{A}} = R_{AhA} + R_{AmA}

and

R_{0_{H}} = R_{HhH} + R_{HmH}

We, therefore, have a problem. We have four tentative R₀ and we do not know if they are somehow linked with the threshold condition. Due to this fact we turn now to the dynamical system approach for the problem. First, however, we summarize and simplify our notation by defining the following transmission coefficients

ϐ_{Ah} = a_{A} \frac{N_{h}}{N_{h} + N_{m}} b_{A \to h} ϐ_{Hh} = a_{H} \frac{N_{h}}{N_{h} + N_{m}} b_{H \to h} ϐ_{Am} = a_{A} \frac{N_{m}}{N_{h} + N_{m}} b_{A \to m} ϐ_{Hm} = a_{H} \frac{N_{m}}{N_{h} + N_{m}} b_{H \to m} ϐ_{hA} = a_{A} \frac{N_{A}}{N_{h} + N_{m}} c_{h \to A} e^{- μ_{A} δ_{A}} ϐ_{mA} = a_{A} \frac{N_{A}}{N_{h} + N_{m}} c_{m \to A} e^{- μ_{A} δ_{A}} ϐ_{hH} = a_{H} \frac{N_{A}}{N_{h} + N_{m}} c_{h \to H} e^{- μ_{H} δ_{H}} ϐ_{mH} = a_{H} \frac{N_{A}}{N_{h} + N_{m}} c_{m \to H} e^{- μ_{H} δ_{H}}

Also, for future use, let us summarize the following parameters as:

\begin{matrix} R_{hAh} & = R_{AhA} & = \frac{ϐ_{hA} ϐ_{Ah}}{r_{h} μ_{A}} \\ R_{hHh} & = R_{HhH} & = \frac{ϐ_{hH} ϐ_{Hh}}{r_{h} μ_{H}} \\ R_{mAm} & = R_{AmA} & = \frac{ϐ_{mA} ϐ_{Am}}{r_{m} μ_{A}} \\ R_{mHm} & = R_{HmH} & = \frac{ϐ_{mH} ϐ_{Hm}}{r_{m} μ_{H}} \\ R_{hHm} & = \frac{ϐ_{hH} ϐ_{Hm}}{r_{h} μ_{H}} \\ R_{mAh} & = \frac{ϐ_{mA} ϐ_{Ah}}{r_{m} μ_{A}} \\ R_{mHh} & = \frac{ϐ_{mH} ϐ_{Hh}}{r_{m} μ_{H}} \\ R_{hAm} & = \frac{ϐ_{hA} ϐ_{Am}}{r_{h} μ_{A}} \\ R_{AhH} & = \frac{ϐ_{Ah} ϐ_{hH}}{r_{m} μ_{H}} \\ R_{HmA} & = \frac{ϐ_{Hm} ϐ_{mA}}{r_{m} μ_{H}} \\ R_{AmH} & = \frac{ϐ_{Am} ϐ_{mH}}{r_{m} μ_{A}} \\ R_{HhA} & = \frac{ϐ_{Hh} ϐ_{hA}}{r_{h} μ_{H}} \end{matrix}

Note also that we have

R_{hHm} R_{mAh} = R_{AhH} R_{HmA} R_{mHh} R_{hAm} = R_{AmH} R_{HhA}

Those parameters have a clear biological significance. For instance, R_hAh is the basic reproduction number of a human index case who generates further human infections through the infective mosquito A. Note that in R_hAh the mosquito population H and the population of animals m exist but are supposed not to contribute to R_hAh. However, the animal population appear in the parameter R_mHm through the factors $\frac{N_{H}}{N_{H} + N_{m}}$ and $\frac{N_{A}}{N_{h} + N_{m}}$ .

4.2 The dynamical system approach

The dynamical system associated with such a situation is:

{\begin{matrix} \frac{d Y_{h}}{d t} = Y_{A} a_{A} \frac{S_{h}}{N_{h} + N_{m}} b_{A \to h} \\ + Y_{H} a_{H} \frac{S_{h}}{N_{h} N_{m}} b_{H \to h} - r_{h} Y_{h} \\ \frac{d Y_{m}}{d t} = Y_{A} a_{A} \frac{S_{m}}{N_{h} + N_{m}} b_{A \to m} \\ + Y_{H} a_{H} \frac{S_{m}}{N_{h} N_{m}} b_{H \to m} - r_{m} Y_{m} \\ \frac{d Y_{A}}{d t} = N_{A} a_{A} \frac{Y_{h} (t - δ_{A})}{N_{h} + N_{m}} c_{h \to A} e^{- μ_{A} δ_{A}} \\ + N_{A} a_{A} \frac{Y_{m} (t - δ_{A})}{N_{h} + N_{m}} c_{m \to A} e^{- μ_{A} δ_{A}} - μ_{A} Y_{A} \\ \frac{{dY}_{H}}{d t} = N_{H} a_{H} \frac{Y_{h} (t - δ_{H})}{N_{h} + N_{m}} c_{h \to H} e^{- μ_{H} δ_{H}} \\ + N_{H} a_{H} \frac{Y_{m} (t - δ_{H})}{N_{h} + N_{m}} c_{m \to H} e^{- μ_{H} δ_{H}} - μ_{H} Y_{H} \end{matrix}

Linearising the system (13) around the solution without disease, namely S_h=N_h, S_m=N_m, S_A=N_A, S_H=N_H, Y_h=Y_m=Y_A=Y_H=0, we get the following system:

{\begin{matrix} \frac{d y_{h}}{d t} = y_{A} ϐ_{Ah} + y_{H} ϐ_{Hh} - r_{h} y_{h} \\ \frac{d y_{m}}{d t} = y_{A} ϐ_{Am} + y_{H} ϐ_{Hm} - r_{m} y_{m} \\ \frac{d y_{A}}{d t} = y_{h} (t - δ_{A}) ϐ_{hA} + y_{m} (t - δ_{A}) ϐ_{mA} - μ_{A} y_{A} \\ \frac{d y_{H}}{d t} = y_{h} (t - δ_{H}) ϐ_{hH} + y_{m} (t - δ_{H}) ϐ_{mH} - μ_{H} y_{H} \end{matrix}

where

y_{i} (i = h, m, A, H)

are small deviations from zero. From the system (14) we get the following characteristic equation:

|\begin{matrix} - (λ + r_{h}) & 0 & ϐ_{Ah} & ϐ_{Hh} \\ 0 & - (λ + r_{m}) & ϐ_{Am} & ϐ_{Hm} \\ ϐ_{hA} e^{- λ δ_{A}} & ϐ_{mA} e^{- λ δ_{A}} & - (λ + μ_{A}) & 0 \\ ϐ_{hH} e^{- λ δ_{H}} & ϐ_{mH} e^{- λ δ_{H}} & 0 & - (λ + μ_{H}) \end{matrix}| = 0

The trivial solution will then be stable if all the roots of the characteristic equation (15) have negative real parts. However, it is almost hopeless to study such a complex characteristic equation as (15), and a sensible procedure should be to neglect all the terms $e^{- λ δ_{i}} (i = A, H)$ , in which case equation (15) would became a fourth order algebraic equation which can be analysed using the Routh–Hurwitz criteria 〚9〛. This can be justified very laboriously using perturbation theory as is done for the simple case of one-host/one-parasite in the appendix. However, it is easier to apply the Next Generation Operator approach to calculate the threshold conditions for the establishment of the epidemic. This is the subject of the next section.

4.3 The next generation operator approach

For the reasons explained in the Appendix, we can neglect the time delays from the very beginning. In this case, the linearised system (14) becomes:

{\begin{matrix} \frac{d y_{h}}{d t} = y_{A} ϐ_{Ah} + y_{H} ϐ_{Hh} - r_{h} y_{h} \\ \frac{d y_{m}}{d t} = y_{A} ϐ_{Am} + y_{H} ϐ_{Hm} - r_{m} y_{m} \\ \frac{d y_{A}}{d t} = y_{h} ϐ_{hA} + y_{m} ϐ_{mA} - μ_{A} y_{A} \\ \frac{d y_{H}}{d t} = y_{h} ϐ_{hH} + y_{m} ϐ_{mH} - μ_{H} y_{H} \end{matrix}

which can be written in the form

\frac{d \bar{𝐲}}{d t} = (𝐓 - 𝐃) \bar{𝐲}

where T is a positive matrix and D is a diagonal positive matrix (see Dieckman and Heesterbeck 〚7〛, p. 105). The next generation operator in this case is given by a four by four matrix:

NGO = {TD}^{- 1} = (\begin{matrix} 0 & 0 & \frac{ϐ_{hA}}{μ_{A}} & \frac{ϐ_{hH}}{μ_{H}} \\ 0 & 0 & \frac{ϐ_{mA}}{μ_{A}} & \frac{ϐ_{mH}}{μ_{H}} \\ \frac{ϐ_{Ah}}{r_{h}} & \frac{ϐ_{Am}}{r_{m}} & 0 & 0 \\ \frac{ϐ_{Hh}}{r_{h}} & \frac{ϐ_{Hh}}{r_{m}} & 0 & 0 \end{matrix})

Although in this case we derived the NGO from the linearised form of the system (14), given by equation (16), this is not necessary. The NGO can be derived independently and, in particular, it does not depend on the assumption made in going from (14) to (16) of neglecting the time delays.

According to the Dieckmann et al. 〚5〛 (see also 〚10〛) theorem, the R₀ given by the Next Generation Operator, R^NGO₀, is the largest eigenvalue of matrix (18). We then have

R_{0}^{NGO} = \sqrt{\frac{1}{2} (B + \sqrt{B^{2} + 4 A})}

where

B = R_{AhA} + R_{HhH} + R_{AmA} + R_{HmH} = R_{hAh} + R_{hHh} + R_{mAm} + R_{mHm}

and

A = R_{hHm} R_{mAh} + R_{mHh} R_{hAm} - R_{mHm} R_{hAh} - R_{mAm} R_{hHh}

If R^NGO₀<1, then the solution without disease of system (13) is stable. Therefore, the threshold condition is given, then, by

R_{0}^{NGO} = \sqrt{\frac{1}{2} (B + \sqrt{B^{2} + 4 A})} \geq 1

which implies

A + B \geq 1

T = (1 - R_{hAh} - R_{hHh} - R_{mAm} - R_{mHm} - R_{hHm} R_{mAh} - R_{mHh} R_{hAm} + R_{mHm} R_{hAh} + R_{mAm} R_{hHh}) \leq 0

The same conclusion can be obtained, more laboriously, by applying the Routh–Hurwitz criteria to equation (15), and taking the exponential terms e^-λδ_i (i = A, H) as 1.

Some particular cases can be analysed. Suppose that all R_iJi are zero except one. Then, T becomes negative if this non-zero R_iJi is greater than 1, and the disease can establish itself in the host population. Another more interesting case occurs when, for instance, R_hAh=R_hHh=0. One might think, at first sight, that in this case a human index case cannot trigger a human epidemic. However, it can be seen that the threshold, which determines whether T is positive or negative is given by

T = (1 - R_{mAm} - R_{mHm} - R_{hHm} R_{mAh} - R_{mHh} R_{hAm})

which has a simple interpretation. In this case the fourth term in the brackets, R_hHmR_mAh, describes the basic reproduction number for a human index who transmit the infection to the animal population through the mosquito type H. The infected animal then passes the disease, through mosquito type A back to the human hosts. Therefore, even when R_hAh=R_hHh=0, a human index case can trigger a human epidemic. For this, it suffices that R_hHmR_mAh>1. The other terms in the expression (20) refer to transmission among the animal population by all the possible ways and all can be less than 1 without preventing the disease establishing itself.

By the same token, if we make R_mAmR_mHm=0 (which apparently implies that an animal index case cannot trigger an animal epidemic), then

T = 1 - R_{hAh} - R_{hHh} - R_{mAh} R_{hHm} - R_{hAm} R_{mHh}

which can be negative when R_mAhR_hHm>1, even when the remaining components R_iJi are all less than 1.

Another attempt to interpret our condition is as follows. Suppose that when the index case is h, and R_mAm=R_mHm=0. Then, we have that T reduces to (21). Suppose now that when the index case is m, and R_hAh=R_hHh=0. In this case T reduces to

T = (1 - R_{mAm} - R_{mHm} - R_{mAh} R_{hHm} - R_{mHh} R_{hAm})

To investigate the cases where the index cases are mosquitoes, we rewrite T using the relations given by equations (11) and (12). We get:

T = (1 - R_{AhA} - R_{HhH} - R_{AmA} - R_{HmH} - R_{AhH} R_{HmA} - R_{AmH} R_{HhA} + R_{HmH} R_{AhA} + R_{AmA} R_{HhH})

If we now assume that the index cases is a mosquito of the type A, we put R_HhH=R_HmH=0 to obtain

T = (1 - R_{AhA} - R_{AmA} - R_{AhH} R_{HmA} - R_{AmH} R_{HhA})

Similarly, when the index case is a mosquito of type H we put R_AhA=R_AmA=0 to obtain

T = (1 - R_{HhH} - R_{HmH} - R_{HmA} R_{AhH} - R_{HhA} R_{AmH})

5 The yellow fever model

Urban yellow fever is transmitted from person to person by peridomestic Aedes aegypti mosquitoes 〚11〛. By contrast, jungle yellow fever is a zoonosis, transmitted from monkeys to humans by mosquitoes that breed in tree-holes of the genuses Haemagogus and Sabethes in the rain forest ecosystem of South America. The jungle form is only partly controlled by vaccination of rural residents and provides a source of infection to population centres infested with Ae aegypti. In the early 20th century, when it was discovered that the yellow fever virus was transmitted in its urban cycle by Aedes aegypti, measures of control were introduced leading to its disappearance. Progressive neglect of the disease, however, led to a new outbreak in Africa in 1927 〚12〛 during which the etiological agent was isolated; some years later a vaccine was discovered and yellow fever disappeared again. Unfortunately, reinfestation with the Aedes vector, which began in the 1970s, is now virtually complete, and vector control is substantially more difficult than before. The threat of urban yellow fever is greatest in towns near forests, but improved transport links increase the likelihood of spread by viremic people to non-endemic areas 〚13〛.

Classified as one of the viral hemorrhagic fevers, yellow fever is unique in its severity, in particular because of its hepatic impairment. Yellow fever is currently endemic and epidemic in tropical areas of the Americas and Africa 〚14–16〛.

From the point of view of its dynamics, yellow fever differs from other vector borne infection by involving three host populations and two vector populations and, therefore, as we shall see, the threshold condition for non-existence of the infection is quite complicated. We recently calculated the risk of urban yellow fever in a dengue infested area 〚17〛.

5.1 The dynamical system approach

Let $Y_{c} (t)$ be the number of infected human individuals living in cities, $Y_{f} (t)$ the number of infected ‘fishermen’ (those individuals who either live part of time in forest areas or who eventually frequent those areas for leisure or other purposes), $Y_{m} (t)$ the number of infected non-human primates, $Y_{A} (t)$ the number of infected mosquitoes of the Aedes genus, and $Y_{H} (t)$ the number of infected mosquitoes of the Haemagogus or Sabetes genuses. Let also N_i $(i = c, f, m, A, H)$ be the total number of individuals in each population considered.

The linearised equations for Y_i around the solution without the disease are:

{\begin{matrix} \frac{d Y_{c}}{d t} = Y_{A} a_{A} \frac{N_{c}}{N_{c} + N_{f}} b_{A \to c} - r_{c} Y_{c} \\ \frac{d Y_{f}}{d t} = Y_{A} a_{A} \frac{N_{f}}{N_{c} + N_{f}} b_{A \to f} \\ + Y_{H} a_{H} \frac{N_{f}}{N_{c} + N_{f}} b_{H \to f} - r_{f} Y_{f} \\ \frac{d Y_{m}}{d t} = Y_{H} a_{H} \frac{N_{m}}{N_{m} + N_{f}} b_{H \to m} - r_{m} Y_{m} \\ \frac{d Y_{A}}{d t} = N_{A} a_{A} \frac{Y_{c} (t - δ_{A})}{N_{c} + N_{f}} g_{c \to A} e^{- μ_{A} δ_{A}} \\ + N_{A} a_{A} \frac{Y_{f} (t - δ_{A})}{N_{m} + N_{f}} g_{f \to A} e^{- μ_{A} δ_{A}} - μ_{A} Y_{A} \\ \frac{d Y_{H}}{d t} = N_{H} a_{H} \frac{Y_{f} (t - δ_{H})}{N_{m} + N_{f}} g_{f \to H} e^{- μ_{H} δ_{H}} \\ + N_{H} a_{H} \frac{Y_{m} (t - δ_{H})}{N_{m} + N_{f}} g_{m \to H} e^{- μ_{H} δ_{H}} - μ_{H} Y_{H} \end{matrix}

where

a_{i} (i = A, H)

is the average number of bites per unit time that the vectors inflict on the hosts; b_i→j (i = A, H and j = c, f) is the fraction of the potentially infective bites

(a_{i} Y_{i})

per unit time, which are actually infective; g_j→i (i = A, H and j = c, f) is the fraction of the bites inflicted on infective hosts that are really infective to the vectors. Finally, r_j (j = c, f, m) is the rate with which the hosts leave the infective compartment (either by death or by recovery from the infection), δ_i (i = A, H) is the incubation period of the virus in the mosquito population (that is, the extrinsic incubation period, the time elapsed between infection and infectiousness in the vectors) and e^-μ_iδ_i is the natural mortality rate of the vectors (therefore μ_i (i = A, H) is the fraction of mosquitoes which survives throughout the extrinsic incubation period).

A few words about the system (26). Take for instance the second equation: in it, the term Y_Aa_A represents the total number of potentially infective bites Aedes mosquitoes inflict in the fraction $\frac{N_{f}}{N_{c} + N_{f}}$ of the ‘fishermen’ hosts, and b_A→f is the fraction of those bites which are actually infective for the host. Therefore, the complete term $Y_{A} a_{A} \frac{N_{f}}{N_{c} + N_{f}} b_{A \to f}$ is the number of ‘fishermen’ who get the infection per unit of time from the Aedes population. Note that we have linearised the original equations. In its complete form, the fraction $\frac{N_{f}}{N_{c} + N_{f}}$ would be $\frac{S_{f}}{N_{c} + N_{f}}$ , where S_f is the number of susceptible ‘fishermen’. By the same token, the second term $Y_{H} a_{H} \frac{N_{f}}{N_{c} + N_{f}} b_{H \to f}$ is the number of ‘fishermen’ who get the infection per unit of time from the Haemagogus population. And finally, the last term r_fY_f represents the number of ‘fishermen’ removed from the infectious state by death or recovery.

Similarly, the equations for the mosquitoes, for instance, the fourth equation could be explained in words as follows. The first term N_Aa_A represents the total number of bites Aedes mosquitoes inflict in the fraction $\frac{Y_{c} (t - δ_{A})}{N_{c} + N_{f}}$ of infected citizens $(t - δ_{A})$ days ago, g_c→A is the fraction of those bites that are actually infective for the mosquitoes, and e^-μ_Aδ_A is the fraction of mosquitoes that survives throughout the extrinsic incubation period, therefore becoming infective to the hosts. Therefore, the complete term $N_{A} a_{A} \frac{Y_{c} (t - δ_{A})}{N_{c} + N_{f}} g_{c \to A} e^{- μ_{A} δ_{A}}$ represents the number of Aedes per unit time which gets the infection from the ‘citizens’ hosts. Note that the original non-linearised system should be $S_{A} (t - δ_{A}) a_{A} \frac{Y_{c} (t - δ_{A})}{N_{c} + N_{f}} g_{c \to A} e^{- μ_{A} δ_{A}}$ , where $S_{A} (t - δ_{A})$ is the number of susceptible Aedes mosquitoes (t-δ_A) days ago. The second term of the fourth equation $N_{A} a_{A} \frac{Y_{f} (t - δ_{A})}{N_{m} + N_{f}} g_{f \to A} e^{- μ_{A} δ_{A}}$ , by the same reasoning, represents the number of Aedes per unit time which gets the infection from the ‘fishermen’ hosts. And finally, the last term μ_AY_A represent Aedes removed from the infective condition by mortality.

Calculating the threshold condition for yellow fever

In order to simplify the notation let us define the following transmission coefficients

ϐ_{Ac} = a_{A} \frac{N_{c}}{N_{c} + N_{f}} b_{A \to c} ϐ_{Af} = a_{A} \frac{N_{f}}{N_{c} + N_{f}} b_{A \to f} ϐ_{Hf} = a_{H} \frac{N_{f}}{N_{c} + N_{f}} b_{H \to f} ϐ_{Hm} = a_{H} \frac{N_{m}}{N_{m} + N_{f}} b_{H \to m} ϐ_{cA} = a_{A} \frac{N_{A}}{N_{c} + N_{f}} g_{c \to A} e^{- μ_{A} δ_{A}} ϐ_{fA} = a_{A} \frac{N_{A}}{N_{m} + N_{f}} g_{f \to A} e^{- μ_{A} δ_{A}} ϐ_{fH} = a_{H} \frac{N_{H}}{N_{m} + N_{f}} g_{f \to H} e^{- μ_{H} δ_{H}} ϐ_{mH} = a_{H} \frac{N_{H}}{N_{m} + N_{f}} g_{m \to H} e^{- μ_{H} δ_{H}}

To calculate the threshold condition we examine the stability of the trivial solution of system (26). We get the following characteristic equation, written as a determinant:

|\begin{matrix} - (r_{c} + λ) & 0 & 0 & ϐ_{Ac} & 0 \\ 0 & - (r_{f} + λ) & 0 & ϐ_{Af} & ϐ_{Hf} \\ 0 & 0 & - (r_{m} + λ) & 0 & ϐ_{Hm} \\ ϐ_{cA} e^{- λ δ_{A}} & ϐ_{fA} e^{- λ δ_{A}} & 0 & - (μ_{A} + λ) & 0 \\ 0 & ϐ_{fH} e^{- λ δ_{H}} & ϐ_{mH} e^{- λ δ_{H}} & 0 & - (μ_{H} + λ) \end{matrix}| = 0

Again, the trivial solution will then be stable if all the roots of the characteristic equation (27) have negative real parts. As mentioned above, it is almost hopeless to study such a complex characteristic equation as (27), and a sensible procedure should be to neglect all the terms e^-λδ_i (i = A, H), in which case equation (27) would became a fifth order algebraic equation which can be analysed using the Routh–Hurwitz criteria 〚9〛. This can be justified by using perturbation theory as done in the appendix. However, once again, it is easier to apply the Next Generation Operator approach to calculate the threshold conditions for the establishment of the epidemic.

5.2 The next generation operator approach

The next generation operator approach for the above model of yellow fever is given by

NGO = (\begin{matrix} 0 & 0 & 0 & \frac{ϐ_{Ac}}{μ_{A}} & 0 \\ 0 & 0 & 0 & \frac{ϐ_{Af}}{μ_{A}} & \frac{ϐ_{Hf}}{μ_{H}} \\ 0 & 0 & 0 & 0 & \frac{ϐ_{Hm}}{μ_{H}} \\ \frac{ϐ_{cA} e^{- μ_{A} δ_{A}}}{r_{c}} & \frac{ϐ_{fA} e^{- μ_{A} δ_{A}}}{r_{f}} & 0 & 0 & 0 \\ 0 & \frac{ϐ_{fH} e^{- μ_{H} δ_{H}}}{r_{f}} & \frac{ϐ_{m} e^{- μ_{H} δ_{H}}}{r_{m}} & 0 & 0 \end{matrix})

According to the Dieckmann et al. theorem, the R₀ given by the Next Generation Operator, R^NGO₀, is the largest eigenvalue of matrix (28). We then have

R_{0}^{NGO} = \sqrt{\frac{1}{2} (B + \sqrt{B^{2} + 4 A})}

where

B = R_{fHf} + R_{fAf} + R_{mHm} + R_{cAc}

and

A = - R_{fAf} R_{mHm} - R_{cAc} R_{mHm} - R_{cAc} R_{fHf}

The threshold condition is given, then, by

R_{0}^{NGO} = \sqrt{\frac{1}{2} (B + \sqrt{B^{2} + 4 A})} \geq 1

which implies

A + B \geq 1

or,

T = 1 - (R_{cAc} + R_{fAf} + R_{fHf} + R_{mHm} - R_{cAc} R_{fHf} - R_{cAc} R_{mHm} - R_{mHm} R_{fAf}) \leq 0

Some particular cases can be of interest. Suppose that transmission to fishermen is zero, that is, R_fAf and R_fHf are equal to zero. Then:

T = 1 - (R_{cAc} + R_{mHm} - R_{cAc} R_{mHm}) \leq 0

Therefore, the threshold occurs when either R_mHm or R_cAc is greater than one. However, if both R_mHm and R_cAc are greater than one, then T may become negative. Therefore we have to apply the Routh–Hurwitz 〚9〛 condition to each particular case. However, there will obviously be an epidemic because the population of humans living in cities becomes decoupled from the population of non-human primates.

Suppose now that we have transmission only to fishermen. In this case, the threshold becomes

T = [1 - (R_{fAf} + R_{fHf})] \leq 0

It is obvious that the threshold occurs when T becomes negative and is given by

R_{fAf} < 1 R_{fHf} < 1

but

R_{fAf} + R_{fHf} > 1

which is intuitively very reasonable.

6 Summary and discussion

The basic reproduction number, as defined by Macdonald 〚3〛 is the average number of secondary cases produced by an index case during its infectiousness period. Note that the Macdonald definition implies secondary cases of the same kind as the index case. Furthermore, the basic reproduction number in a disease which involves only one host and one vector coincides with the threshold that breaks the stability of the trivial solution, as shown in this paper. Another definition of R₀ is the largest eigenvalue of the Next Generation Operator. The dominant eigenvalue of the NGO gives the average multiplication factor with which successive generations grow. It is therefore precisely the average number of cases in the next generation of infecteds produced by a typical case in the present generation. Thus, a threshold is immediately apparent: the disease cannot invade the population if the dominant eigenvalue of the NGO is less than 1.

Suppose now an indirectly transmitted disease which involves more than one type of host and/or vector. Then, as we have shown in this paper, several basic reproduction numbers can be defined. Notwithstanding, there is only one threshold for the epidemic persistence that can be written as a function of the many R₀'s and some other quantities, which are not R₀, but are similar. In fact, they are the number of secondary cases of host(s)/vector(s) of a different kind of the index case.

In section 2, we revisited the classical Macdonald model, calculated the basic reproduction number and showed, by using a dynamical system approach, that the threshold and the R₀ are the same quantity.

In section 3, we discussed in detail the case of an hypothetical infection system with two hosts (denoted h and m) and two vectors (denoted A and H). We showed that this system has four R₀, namely

R_{hAh} = R_{AhA} R_{hHh} = R_{HhH} R_{mAm} = R_{AmA} R_{mHm} = R_{HmH}

We also calculated the threshold for the epidemic persistence, which is given by a quantity T (see equation (19)), namely:

T = (1 - R_{hAh} - R_{hHh} - R_{mAm} - R_{mHm} - R_{hHm} R_{mAh} - R_{mHh} R_{hAm} + R_{mHm} R_{hAh} + R_{mAm} R_{hHh})

If T is greater than zero there is no epidemic persistence. Note that T involves terms that are R₀ as defined above (like R_hAh or R_hHh), but it involves as well terms that are not R₀ but refers to the number of human or mosquitoes secondary infections produced by a single infective mosquitoe or human, respectively (like R_hHm or R_mAh).

Acknowledgements

The authors would like to thank Professor Hans Heesterbeek for his helpful comments on the manuscript. Eventual remaining errors are our own responsibility.

Appendix

We write equation (6) as:

λ^{2} + a_{1} λ + a_{21} - a_{22} e^{- λ δ} = 0

where

a_{1} = μ + {ra}_{21} = μ {ra}_{22} = \frac{N_{v} a}{N_{h}} c_{h \to m} e^{- μ δ} a b_{v \to h}

Assume a root

z = x + i y

We can separate equation (34) in its real and imaginary parts. We get:

x^{2} - y^{2} + a_{1} x + a_{21} - a_{22} e^{𝑥 δ} \cos (y δ) = 0

and

2 x y + a_{1} y - a_{22} e^{𝑥 δ} \sin (y δ) = 0

Let us solve (35) and (36) perturbatively around the solution obtained with δ=0. We write

x = x_{0} + x_{1} δ + x_{2} δ^{2} + . . .

and

y = y_{0} + y_{1} δ + y_{2} δ^{2} + . . .

Replacing (37) and (38) in (35) and (36), we get, keeping only terms of zero order in δ:

x_{0}^{2} + a_{1} x_{0} + a_{21} - a_{22} = 02 x_{0} y_{0} + a_{1} y_{0} = 0

whose solutions are

y_{0} = 0 x_{0} = \frac{- a_{1} \pm \sqrt{a_{1}^{2} - 4 (a_{21} - a_{22})}}{2}

and

x_{0} = - \frac{a_{1}}{2} y_{0} = \pm \frac{\sqrt{4 (a_{21} - a_{22}) - a_{1}^{2}}}{2}

However, the second solution (41) represents a root with negative real part, and therefore is uninteresting. We can see that in the zero order in δ a root crosses the imaginary axis through the real axis when a₂₁-a₂₂→0.

The terms in first order in δ give

2 x_{0} x_{1} - 2 y_{0} y_{1} + a_{1} x_{1} - a_{22} x_{0} = 0

and

2 x_{0} y_{1} + 2 x_{1} y_{0} + a_{1} y_{1} - a_{22} y_{0} = 0

Replacing y₀=0 in equations (42) and (43), we get:

2 x_{0} x_{1} + a_{1} x_{1} - a_{22} x_{0} = 02 x_{0} y_{1} + a_{1} y_{1} = 0

whose solution is

y_{1} = 0 x_{1} = \frac{a_{22} x_{0}}{2 x_{0} + a_{1}}

Then we see that when $(a_{21} - a_{22}) \to 0$ , $x_{1} \to 0$ , together with x₀.

The terms in second order in δ, after substituting y₀=0 and y₁=0, give

x_{1}^{2} + 2 x_{0} x_{2} + a_{1} x_{2} - a_{22} x_{1} - \frac{1}{2} a_{22} x_{0}^{2} = 02 x_{0} y_{2} + a_{1} y_{2} = 0

whose solution in

y_{2} = 0 x_{2} = \frac{a_{22} x_{1} - x_{1}^{2} + \frac{1}{2} a_{22} x_{0}^{2}}{2 x_{0} + a_{1}}

Again we see that when $(a_{21} - a_{22}) \to 0$ , $x_{2} \to 0$ together with x₀ and x₁.

So, up to second order in δ, a root crosses the imaginary axis through the real axis when $(a_{21} - a_{22}) \to 0$ . It is not difficult to see that all terms in the expansion vanish when $(a_{21} - a_{22}) \to 0$ , showing that a₂₁-a₂₂=0 is the threshold for the infection to persist.

Bibliographie

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