**Date Published:** February 19, 2018

**Publisher:** Springer Netherlands

**Author(s):** L. M. Viljoen, L. Hemerik, J. Molenaar.

http://doi.org/10.1007/s10441-018-9315-1

**Abstract**

**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}
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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}
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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.**

**Partial Text**

In general, successful parasites engage in a dynamic co-evolutionary interaction with their host population. As such, they do not eradicate their hosts. Instead, they live in some kind of stable hostility, resulting in an arms race between parasite and host (Haraguchi and Sasaki 1996). For vertebrate hosts and their obligatory directly transmitted pathogenic microbes this arms race is taking place between the hosts’ immune system on the one hand, and the complex genetic systems that the pathogens develop on the other hand. The main tools pathogens have at their disposal are antigenic diversification and variation in replication rates. These two mechanisms are essential to the pathogen’s continued struggle to evade the host’s immune system that is adapting to control the proliferation of infectious organisms (Deitsch et al. 2009).

We first discuss the modelling concepts underlying the present study. The model used is taken from Lange and Ferguson (2009). The numerical implementation is optimized to allow for many and long computer simulations. We started with replicating their results (see Fig. A1 in Appendix A) to check for computational correctness. Although the modelling principles used in our analysis are thus not new, we prefer to present them in this paper in a self-contained way to avoid unnecessary reference to the literature.

We combined and implemented the within- and between-host models in Eqs. (1–8) and, as a check, first recalculated the results as reported in Lange and Ferguson (2009). We found the same results; they are given in Appendix A and B. Next, we focussed at finding how the evolutionary optimal documentclass[12pt]{minimal}

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begin{document}$$R_0^{{opt}}$$end{document}R0opt depends on contact rate documentclass[12pt]{minimal}

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begin{document}$$alpha$$end{document}α between hosts and host longevity documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax.

A lot of studies relate pathogen success or failure to R0 by investigating how changes in the parameters governing model dynamics lead to changes in R0. The commonly considered factors, like force of infection, transmission probabilities and contact neighbourhoods are important. This certainly leads to more insight pertaining to the success or failure of infectious disease progression. Here, we have shown that the factor “remaining longevity of the host”, also influences the basic reproduction number of an epidemic and thus can essentially determine whether an outbreak is to be expected or not. We used a specific model from the literature to investigate the relation between R0 and remaining host longevity documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax. The within-host model used allows for variation in the diversity of the infectious agent, as well as for different rates of replication. Coupling it with the population-level model, we are able to show how the optimal strategy for the pathogen depends on the parameters. It should be noted that these optimal strategies will develop in the end and that during the transient phase suboptimal strategies could be observed. We found that optimal strategies depend heavily on documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax, as is clearly shown in Fig. 2. We conclude that the value of documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax should be chosen in correspondence with the specific host-pathogen system under consideration. The value of documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax may vary from very short, e.g., in the cases of fly or mosquito, to very long, e.g., for humans. One of the consequences of this insight is that the results given in Lange and Ferguson (2009), from which we adopted the modelling ideas, are valid for systems in which the hosts live at most 2 years after the start of the infection, because they fixed documentclass[12pt]{minimal}

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begin{document}$$D_{{max}}$$end{document}Dmax at that period, assuming that it would not change their conclusions. The present work shows that varying (remaining) host longevity can change conclusions drastically: in our long run simulations to determine the optimal R0 (Eq. 11), one of the three different regimes that Lange and Ferguson (2009) distinguished never pops up as the optimum strategy.

Appendix A This appendix gives some typical results obtained through numerical simulations. We show typical types of within-host behaviour, as well as the fitness landscapes that result from the combination of the two models.

Source:

http://doi.org/10.1007/s10441-018-9315-1