Effect of the epidemiological heterogeneity on the outbreak outcomes

Alina Macacu; Dominique J. Bicout

doi:10.3934/mbe.2017041

Article Contents

2017, Volume 14, Issue 3: 735-754. Doi: 10.3934/mbe.2017041

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Effect of the epidemiological heterogeneity on the outbreak outcomes

Alina Macacu and
Dominique J. Bicout^,

Biomathematics and Epidemiology, EPSP -TIMC, UMR 5525 CNRS, Grenoble Alpes University, VetAgro Sup Lyon, 1 avenue Bourgelat -69280 Marcy l'Etoile, France

* Corresponding author: Dominique J. Bicout
* Corresponding author: Dominique J. Bicout

Received: May 07, 2016

Accepted: October 14, 2016

Published: September 2017

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Abstract

Multi-host pathogens infect and are transmitted by different kinds of hosts and, therefore, the host heterogeneity may have a great impact on the outbreak outcome of the system. This paper deals with the following problem: consider the system of interacting and mixed populations of hosts epidemiologically different, what would be the outbreak outcome for each host population composing the system as a result of mixing in comparison to the situation with zero mixing? To address this issue we have characterized the epidemic response function for a single-host population and defined a heterogeneity index measuring how host systems are epidemiologically different in terms of generation time, basic reproduction number $R_0$ and, therefore, epidemic response function. Based on the individual epidemiological characteristics of populations, with heterogeneities and mixing affinities, the response of subpopulations in a multi-host system is compared to that of a single-host system. The case of a two-host system, in which the infection transmission depends solely on the infection susceptibility of the receiver, is analyzed in detail. Three types of responses are observed: dilution, amplification or no effect, corresponding to lower, higher or equal attack rates, respectively, for a host population in an interacting multi-host system compared to the zero-mixing situation. We find that no effect is generally observed for zero heterogeneity. A dilution effect is always observed for all the host populations when their individual $R_{0,i} <1$. Whereas, when at least one of the individual $R_{0,i}>1$, then the hosts "$i$" with $R_{0,i}>R_{0,j}$ undergo a dilution effect while the hosts "$j$" undergo an amplification effect.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25, 37N25.

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Figure 1. Cumulative distribution function (cdf) for the attack rate (left panel) and the reduced extinction time (right panel), for x = 0 and R₀ = 0.5 (dashed line), 1 (dotted line), and 2 (solid line).

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Figure 2. Distributions of attack rate a (left) and of reduced extinction time τ (right) for R₀ = 2 and x = 0

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Figure 3. Mean attack rate A (dashed lines) and mean reduced extinction time T (solid lines) as a function of R₀ for x = 0 (left panel) and x = 0.9 (right panel)

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Figure 4. Probability ω(I_th) of an outbreak occurrence as a function of R₀ for x = 0 and different values of the threshold I_th. Star markers represent the mean attack rate A

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Figure 5. Reduced one-dimensional heterogeneity, H_h/y, as a function of z, for different values of y. The values y = 0.25, 1 and 2.3 correspond to f₁ = 1 − f₂ = 0.8, 0.5 and 0.3, respectively, filled circles to z = h₂/h₁ = 0.53 [with (h₁, h₂) = (1.5, 0.8)], and filled diamonds to z = 0.1, 0.53, 0.625, and 1.

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Figure 6. Sensitivity analysis on $\mathcal{R}_0$ using extended FAST method. For each parameter, the light area represents the main effect and the gray area the interaction effect between parameters

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Figure 7. Contour diagrams in the space {R_{0, 1}, R_{0, 2}} showing level curves of ${\mathcal R}_0$ = 0.5, 1, …, 3.5 (quoted numbers) for different Φ and f₁ and for a total population size N = 5000

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Figure 8. Cumulative distribution function (cdf) for the attack rate for host 1 (dashed line), host 2 (dotted line) and the total population (solid line) in a two-host system. The initial conditions are I₁(0) = 1 and I₂(0) = 0, with parameters x₁ = 0, x₂ = 0.9, f₁ = 1 − f₂ = 0.8, R_{0, 1} = 1.5 and R_{0, 2} = 0.8, corresponding to H_R = 0.043 [filled circle (y, z) = (0.25, 0.53) in Fig. 5], and Φ = 0.5 for a global ${\mathcal R}_0$ = 1.4

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Figure 9. Effects of the heterogeneity and mixing on the outbreak outcome

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Figure 10. Impact of the initial conditions on effects of the heterogeneity and mixing on the outbreak outcome

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Figure 11. Left panel: Probability of minor epidemics as a function of R₀. Triangle markers represent data from stochastic simulations and solid line through the data Eq.(18) for x = 0. Right panel: Mean attack rate as a function of R₀ for x = 0, comparison of simulations (solid line) and the formula in Eq.(16) (dashed line)

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Table 1. Synthetic summary of stochastic simulations for constructing the phase diagram of the outbreak response at individual host level as a function of the combined effects of mixing (Φ ≠ 0) and heterogeneity. Dilution, no effect and amplification responses correspond to η_i < 1, = 1 and > 1, respectively, where η_i in Eq. (12) is the ratio of the equivalent to the bare basic reproduction number. These observations are symmetric with respect to inversion of host 1 and 2, and for each host i the effect on the outbreak response increases when f_i (f_j) decreases (increases), and conversely

heterogeneity	outbreak response
heterogeneity	host 1	host 2
$\begin{array}{l} \bullet \;{H_R} > 0\;\;\;{R_{0,1}}＆ {R_{0,2}} < 1\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{R_{0,1}} < {R_{0,2}}\;{\rm{with}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{at}}\;{\rm{least}}\;{\rm{one}}\;{R_{0,i}} > 1 \end{array} $	dilution	dilution
	amplification	dilution
$\begin{array}{l} \bullet \;{H_R} = 0\;\;\;{x_1} = {x_2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{x_1} < {x_2}\;{\rm{and}}\;\left\{ \begin{array}{l} {R_{0,i}} < 1\\ {R_{0,i}} > 1 \end{array} \right. \end{array}$	no effect	no effect
	dilution	dilution
	no effect	amplification

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