American Institute of Mathematical Sciences

June  2017, 14(3): 735-754. doi: 10.3934/mbe.2017041

Effect of the epidemiological heterogeneity on the outbreak outcomes

 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

Received  May 08, 2016 Accepted  October 15, 2016 Published  December 2016

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.

Citation: Alina Macacu, Dominique J. Bicout. Effect of the epidemiological heterogeneity on the outbreak outcomes. Mathematical Biosciences & Engineering, 2017, 14 (3) : 735-754. doi: 10.3934/mbe.2017041
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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).
Distributions of attack rate a (left) and of reduced extinction time τ (right) for R0 = 2 and x = 0
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)
Probability ω(Ith) of an outbreak occurrence as a function of R0 for x = 0 and different values of the threshold Ith. Star markers represent the mean attack rate A
Reduced one-dimensional heterogeneity, Hh/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.
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
Contour diagrams in the space {R0, 1, R0, 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
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, R0, 1 = 1.5 and R0, 2 = 0.8, corresponding to HR = 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
Effects of the heterogeneity and mixing on the outbreak outcome

The reduced equivalent reproduction number, η1, for host 1, and global reproductive number ${\mathcal R}_0$ (from Eq.(10)) as a function of assortative mixing Φ for various values of heterogeneity, HR, and R0, 2. The initial conditions are I1(0) = 1 and I2(0) = 0, with parameters x1 = x2 = 0, and f1 = 1−f2 = 0.5. Values of HR correspond to filled diamonds along the line y = 1 in Fig. 5 with HR = 0 (R0, 1 = R0, 2 = 0.8 for z = 1), 0.05 (R0, 1 = 0.8; R0, 2 = 0.5 for z = 0.625), 0.092 (R0, 1 = 0.8; R0, 2 = 1.5 for z = 1.88) and 0.67 (R0, 1 = 2; R0, 2 = 0.2 for z = 0.1).

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

Reduced equivalent reproduction numbers ηi (i = 1, 2) and global reproductive number ${\mathcal R}_0$ (from Eq.(10)) as a function of the assortative mixing Φ for the two hosts for different heterogeneity. The initial conditions are I1(0) = 1 and I2(0) = 0 (filed symbols for η1 and open symbols for η2), and I1(0) = 0 and I2(0) = 1 (open symbols for η1 and filled symbols for η2) with the parameters x1 = 0, x2 = 0.9, R0, 1 = 1.5 and R0, 2 = 0.8. Values of HR correspond to filled circles at z = R0, 2/R0, 1 = 0.53 in Fig. 5 with HR = 0.043 (f1 = 1− f2 = 0.8 for y = 0.25), 0.093 (f1 = 1− f2 = 0.5 for y = 1) and 0.101 (f1 = 1 − f2 = 0.3 for y = 2.3)

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)
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 ηi < 1, = 1 and > 1, respectively, where ηi 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 fi (fj) decreases (increases), and conversely
 heterogeneity outbreak response 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
 heterogeneity outbreak response 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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