R0: Host Longevity Matters

The basic reproduction ratio, R0, is a fundamental concept in epidemiology. It is defined as the total number of secondary infections brought on by a single primary infection, in a totally susceptible population. The value of R0 indicates whether a starting epidemic reaches a considerable part of the population and causes a lot of damage, or whether it remains restricted to a relatively small number of individuals. To calculate R0 one has to evaluate an integral that ranges over the duration of the infection of the host. This duration is, of course, limited by remaining host longevity. So, R0 depends on remaining host longevity and in this paper we show that for long-lived hosts this aspect may not be ignored for long-lasting infections. We investigate in particular how this epidemiological measure of pathogen fitness depends on host longevity. For our analyses we adopt and combine a generic within- and between-host model from the literature. To find the optimal strategy for a pathogen from an evolutionary point of view, we focus on the indicator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0^{{opt}}$$\end{document}R0opt, i.e., the optimum of R0 as a function of its replication and mutation rates. These are the within-host parameters that the pathogen has at its disposal to optimize its strategy. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0^{{opt}}$$\end{document}R0opt is highly influenced by remaining host longevity in combination with the contact rate between hosts in a susceptible population. In addition, these two parameters determine whether a killer-like or a milker-like strategy is optimal for a given pathogen. In the killer-like strategy the pathogen has a high rate of reproduction within the host in a short time span causing a relatively short disease, whereas in the milker-like strategy the pathogen multiplies relatively slowly, producing a continuous small amount of offspring over time with a small effect on host health. The present research allows for the determination of a bifurcation line in the plane of host longevity versus contact rate that forms the boundary between the milker-like and killer-like regions. This plot shows that for short remaining host longevities the killer-like strategy is optimal, whereas for very long remaining host longevities the milker-like strategy is advantageous. For in-between values of host longevity, the contact rate determines which of both strategies is optimal. Electronic supplementary material The online version of this article (10.1007/s10441-018-9315-1) contains supplementary material, which is available to authorized users.

: Within-host dynamics for variations in pathogen replication rate ρ and diversity parameter δ. Three types of within-host behaviour is observed.
[A] One realization for high ρ = 8 and intermediate δ = 10 −6 (type I). Durations of infection vary greatly for type I, reaching from a few days, up to Dmax. Due to the stochasticity observed in type I dynamics, average dynamics is considered.
[B] Average type I behaviour over 100 realizations.
[C] Average type II behaviour is characterized by long-lived infections (almost always) produced by low ρ = 3 and high δ = 10 −3 .
[D] For pathogens that replicate slowly and that are not very diverse (ρ = 3 and δ = 10 −9 ) infections end after only a few days, with the initial strain being dominant and a small probability of mutations occurring.
(2) When within-host pathogen replication is low and diversity is high, the infectious period lasts very long and we notice very little stochastic behaviour, typical of infections such as HIV. We notice low average pathogen loads and low strain numbers at large times, with the initial part of the infectious period dominating overall behaviour for this type II infections.
(3) Finally, when replication is slow and there is little diversity of the infectious agent, we notice short lived infections and low average strain numbers (type III). This behaviour is comparable to childhood like infections, such as measles.

Between-host dynamics
Here, we consider what will happen if we combine the different within-host dynamics, identified in the previous section, with between-host dynamics, given in equations (8-12) in the main text. This is done via fitness landscapes that reflect how duration of infection, cumulative pathogen load, and R 0 behave as functions of the within-host parameters ρ and δ.  Fig. A2 we see that the average duration of an infection is characterised by endemic behaviour at low levels of replication and high levels of diversity. Panel C of Fig. A2 shows the two regions in pathogen space where optimum between-host R 0 is possible. When host longevity is more and more increased, the R 0 landscape more ande more resembles the duration landscape in Panel A of Fig. A2 and optimum R 0 is only achieved by pathogens that replicate slowly.

Appendix B
In this appendix we show how the within-and between-host models, given in equations (1 -12) in the main text, may be written in dimensionless form. We also describe how the between-host dimensionless equation is numerically solved by a process of discretization and numerical approximation. Default parameter values are given in Tables 1 and 2

Within-host model
The within-host model equations (1-7) in the main text are dimensionful. The consequence is that this system of ordinary differential equations yields state variables that may vary greatly in magnitude. E.g., total pathogen load may reach levels of ν ≈ 10 11 , whereas adaptive immunity saturates at η = 10 5 . These huge differences may lead to numerical instabilities and inaccurate results of simulations. That is the reason why we have put all models in dimensionless form, since this leads to state variables and parameters having order of magnitude one. With this approach we also overcome possible problems caused by time scale differences when within-and between-host dynamics are coupled. Non-dimensionalization is not a unique procedure (Groesen and Molenaar 2007). In the present case it suffices to apply a straightforward scaling procedure: where is the maximum obtained for the viral strain loads. Below we derive a reasonable value for . The parameter η is chosen to scale the X i 's, as this parameter denotes the critical load above which immunity saturates. C 0 is the maximum number of resource cells and therefore a logical choice for scaling C. Time is scaled using the parameter ρ. This enables us to reduce the number of parameters by 1.

Between-host model
The between-host model is described in equations (4-8) in the main text with as core the integro-differential equation (8). We put (8) in dimensionless form by scaling the susceptible population via Z = S N and the time variable via t * = tρ, which is consistent with the scaling of the time in the within-host model. Substituting these scalings in (8) we find: where p 6 ≡ ω ρ , and p 7 ≡ φ N . The transmission rate q defined in (4) in the main text has dimension T −1 . By setting β * = α * γ/N , with α * = α/ρ, we get the dimensionless form q * = β * (1 − exp(−ν * /ν * T )). Here, ν * is the total pathogen load from the within-host system and ν * T = ν T / represents the (scaled) infectiousness threshold. To numerically solve the dimensionless integro-differential equation (B5), we use a forward-difference technique to approximate the derivatives, and apply the trapezium rule to approximate the integral terms. This leads to: