Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites

Dhirasakdanon, Thanate; Thieme, Horst R.

doi:10.1051/mmnp/20105606

Cited by 7 publications

(5 citation statements)

References 44 publications

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“…Increased levels of virulence have been shown, both theoretically and experimentally, to favor horizontal transmission but to reduce the probability of vertical transmission, as virulence is generally negatively correlated to host fitness [1,2,4]. Thieme and collaborators [40,41,42] studied a model for a directly-transmitted pathogen in which two competing strains were able to coexist in a single host population, but in addition to the direct transmission, the coexistence required that one strain be transmitted purely vertically and that both strains induce either a reduced fertility or additional mortality in hosts. In our model, we did not consider infected hosts to be submitted to an additive mortality rate due to infection compared to susceptible hosts, as is usually done in models of pathogen evolution.…”

Section: Discussionmentioning

confidence: 99%

The role of the ratio of vector and host densities in the evolution of transmission modes in vector-borne diseases. The example of sylvatic Trypanosoma cruzi

Pelosse

Kribs-Zaleta

2012

Journal of Theoretical Biology

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Pathogens may use different routes of transmission to maximize their spread among host populations. Theoretical and empirical work conducted on directly-transmitted diseases suggest that horizontal (i.e., through host contacts) and vertical (i.e., from mother to offspring) transmission modes trade off, on the ground that highly virulent pathogens, which produce larger parasite loads, are more efficiently transmitted horizontally, and that less virulent pathogens, which impair host fitness less significantly, are better transmitted vertically. Other factors than virulence such as host density could also select for different transmission modes, but they have barely been studied. In vector-borne diseases, pathogen transmission rate is strongly affected by host-vector relative densities and by processes of saturation in contacts between hosts and vectors. The parasite Trypanosoma cruzi which is transmitted by triatomine bugs to several vertebrate hosts is responsible for Chagas' disease in Latin America. It is also widespread in sylvatic cycles in the southeastern U.S. in which it typically induces no mortality costs to its customary hosts. Besides classical transmission via vector bites, alternative ways to generate infections in hosts such as vertical and oral transmission (via the consumption of vectors by hosts) have been reported in these cycles. The two major T. cruzi strains occurring in the U.S. seem to exhibit differential efficiencies at vertical and classical horizontal transmissions. We investigated whether the vector-host ratio affects the outcome of the competition between the two parasite strains using an epidemiological two-strain model considering all possible transmission routes for sylvatic T. cruzi. We were able to show that the vector-host ratio influences the evolution of transmission modes providing that oral transmission is included in the model as a possible transmission mode, that oral and classical transmissions saturate at different vector-host ratios and that the vector-host ratio is between the two saturation thresholds. Even if data on parasite strategies and demography of hosts and vectors in the field are crucially lacking to test to what extent the conditions needed for the vector-host ratio to influence evolution of transmission modes are plausible, our results open new perspectives for understanding the specialization of the two major T. cruzi strains occurring in the U.S. Our work also provides an original theoretical framework to investigate the evolution of alternative transmission modes in vector-borne diseases.

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Section: Discussionmentioning

confidence: 99%

The role of the ratio of vector and host densities in the evolution of transmission modes in vector-borne diseases. The example of sylvatic Trypanosoma cruzi

Pelosse

Kribs-Zaleta

2012

Journal of Theoretical Biology

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“…Recently attention has focused on understanding the factors that lead to coexistence or to competitive exclusion [10,11,34]. In the absence of multiple infections and the presence of complete cross-immunity, only the parasite strain persists that has the maximal basic reproduction ratio.…”

Section: Introductionmentioning

confidence: 99%

Global analysis of multi-strains SIS, SIR and MSIR epidemic models

Bichara

Iggidr

Sallet

2013

J. Appl. Math. Comput.

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International audienceWe consider SIS, SIR and MSIR models with standard mass action and varying population, with $n$ different pathogen strains of an infectious disease. We also consider the same models with vertical transmission. We prove that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of each individual basic reproduction ratio. Actually we also define an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle.On considère les modèles SIS, SIR et MSIR avec la loi de l'action de masse standard et une population non constante, avec n différentes souches de pathogènes. Nous considérons aussi les même modèles avec transmission verticale. On prouve que sous une condition générique, le principe de compétition exclusive est vérifié. Pour chaque souche, un nombre de reproduction de base est associé. Il correspond au cas où seule cette souche existe. Le nombre de reproduction de base du système complet est le maximum de tous les nombres de reproduction de base pris individuellement. Nous définissons aussi un seuil équivalent pour chaque souche. La souche qui gagne la compétition est celle qui maximise le nombre de reproduction de base. C'est aussi la souche la plus virulente, i.e., c'est la souche pour laquelle l'équilibre endémique donne le minimum des individus susceptibles dans la population hôte. C'est le principe de pessimisation

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“…where J ii is the determinant of the 2 × 2 matrix formed from J by deleting the i th row and column; see, for example, [8,Thm.12]. By (4.13),…”

Section: Type Reproduction Numbersmentioning

confidence: 99%

Stability and Persistence in a Model for Bluetongue Dynamics

Gourley¹,

Thieme²,

Driessche³

2011

SIAM J. Appl. Math.

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Abstract.A model for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors, is formulated as a delay differential equation system of six equations. Midges are assumed to have a pre-adult stage of constant duration, and a general incubation period for bluetongue. A linear stability analysis leads to identification of a basic reproduction number that determines if the disease introduced at a low level dies out, or is uniformly weakly persistent in the midges. Stronger conditions sufficient for global stability of the disease free equilibrium are derived. The control reproduction numbers, which guide control strategies for midges, cattle or sheep, are determined in the special case in which the incubation period for midges is exponentially distributed. The possibility of backward bifurcation is briefly discussed as is an equilibrium situation in which the disease wipes out sheep populations that are introduced in small numbers.

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Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites

Cited by 7 publications

References 44 publications

The role of the ratio of vector and host densities in the evolution of transmission modes in vector-borne diseases. The example of sylvatic Trypanosoma cruzi

The role of the ratio of vector and host densities in the evolution of transmission modes in vector-borne diseases. The example of sylvatic Trypanosoma cruzi

Global analysis of multi-strains SIS, SIR and MSIR epidemic models

Stability and Persistence in a Model for Bluetongue Dynamics

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