A Mathematical Framework for Predicting Lifestyles of Viral Pathogens

Abstract

Despite being similar in structure, functioning, and size, viral pathogens enjoy very different, usually well-defined ways of life. They occupy their hosts for a few days (influenza), for a few weeks (measles), or even lifelong (HCV), which manifests in acute or chronic infections. The various transmission routes (airborne, via direct physical contact, etc.), degrees of infectiousness (referring to the viral load required for transmission), antigenic variation/immune escape and virulence define further aspects of pathogenic lifestyles. To survive, pathogens must infect new hosts; the success determines their fitness. Infection happens with a certain likelihood during contact of hosts, where contact can also be mediated by vectors. Besides structural aspects of the host-contact network, three parameters appear to be key: the contact rate and the infectiousness during contact, which encode the mode of transmission, and third the immunity of susceptible hosts. On these grounds, what can be said about the reproductive success of viral pathogens? This is the biological question addressed in this paper. The answer extends earlier results of the author and makes explicit connection to another basic work on the evolution of pathogens. A mathematical framework is presented that models intra- and inter-host dynamics in a minimalistic but unified fashion covering a broad spectrum of viral pathogens, including those that cause flu-like infections, childhood diseases, and sexually transmitted infections. These pathogens turn out as local maxima of numerically simulated fitness landscapes. The models involve differential and integral equations, agent-based simulation, networks, and probability.

Keywords: Contact networks; Evolution of viral infections; Infection types; Infectious disease modeling; Phylodynamic patterns; Viral fitness.

Fig. 1

Static patterns. The figure is a ${90}^{\circ }$-rotated sketch of Figure 2A in Grenfell et al. (2004). It indicates the locations of the five static patterns (lying on a parabola) in the pathogen parameter space of Grenfell et al. (2004), which is formed by the immune pressure and the net viral adaptation rate. Furthermore, the figure indicates the monotonic behavior of the strength of selection (blue) and the viral abundance (red) with respect to the immune pressure (y-axis) (Color Figure Online)

Fig. 2

Infection types. This figure is adopted from Figure 3 in Lange and Ferguson (2009). The top row shows the fitness landscapes (due to between-host replication, $R_0$) over pathogen space ($=$antigenic variation$\delta$$\times$intra-host replication$\rho$) for flu-like infections (FLI), sexually transmitted infections (STI), and childhood diseases (ChD). The bottom row shows the corresponding between-host characteristics: total virus count ($\mathrm{\Sigma }v$), duration of infection (D), and the initial peak load ($\mathrm{\Sigma }v\times D$  for the 1st peak), respectively. The maxima of these surfaces define three evolutionary strategies (or lifestyles, as we also refer to them). While having the maxima at the same location in pathogen space, the surfaces of the top and bottom rows are similar too (Color Figure Online)

Fig. 3

Modeling framework. Systematically, for all viruses represented by our pathogen parameter space, we simulate the within-host evolution and calculate the average load over time v(t). The load curve is used to define a time-dependent transmission rate, $\beta (v(t))$. Based on this rate, the between-host dynamics is simulated for a totally susceptible host-contact network. The total number of infected individuals then determines the basic reproduction number $R_0$, our model for viral fitness (Color Figure Online)

Fig. 4

Within-host replication. a The replication of viral strain i into multiple identical copies (about ${\nu }_1$) and the mutant strain $i+1$. Before specific immunity develops, there is a cross-reactive immune response from the earlier strain i. Cross-reactive immunity is exerted based on the loci-allele structure indicated in (b). Its strength depends on the antigenic distance between the involved strains; cf. Eq. (3). The distance is associated with the number of mutations required to transform one strain into the other (Color Figure Online)

Fig. 5

Screening effect. The Sketch illustrates the reduced number ($4<5$) of susceptibles (red) in the contact neighborhood of one infected individual (red dots), caused by one secondary infection in the host network ($\phi \ne 0$). The screened  individual (blue) cannot be infected by the initially infected individual anymore. This reduces the basic reproductive number in comparison with an idealistic network-free scenario ($\phi =0$) (Color Figure Online)

Fig. 6

Fitness and antigenic variation. The figure illustrates the definition of the three infection types (A, B, C) based on antigenic variation (medium, high, low) and maximal fitness. The lower left-hand side panel shows the fitness landscape on transmission space, ${max}_{\delta ,\rho }{R}_0(\widehat{\beta },\alpha ;\delta ,\rho )$. The corresponding top panel indicates the fitness maxima ${\widehat{R}}_0(\widehat{\beta })$ for the simulated transmissibilities $\widehat{\beta }$ (black dots). The lower right-hand side panel shows the antigenic variation ($lg\delta$) over transmission space $(\widehat{\beta },\alpha )$. Here, the gray curve $\widehat{\delta }(\widehat{\beta })$ selects the $\delta$-values that correspond to fitness maxima. These $\delta$-values are shown in the corresponding top panel; they suggest a three-type classification (Color Figure Online)

Fig. 7

Reconstructed static patterns. The top panels show the mean viral load [called abundance in Grenfell et al. (2004)] over the pathogen space (left) and the fitness maximum over transmissibility (right). The bottom panels show the ratio of effective to total strain numbers [representing the strength of selection in Grenfell et al. (2004)] over the pathogen space (denoted as in Grenfell et al. (2004); left) and the fitness maximum over transmissibility (right). In all four diagrams, the black data points (produced by numerical simulation) coincide. In comparison with Grenfell et al. (2004), five static patterns are identified with particular (ranges of) transmissibility (top and button, right) (Color Figure Online)

Fig. 8

Static pattern versus infection types. For five transmissibilities $\widehat{\beta }$, fitness landscapes $R_0(\delta ,\rho )$ are plotted over pathogen space (3D and via contour) and associated with static patterns and infection types. For two of the transmissibilities (bold), the infection type is not clearly differentiated (between A–B and B–C, marked by “?”). They likely correspond to the static patterns (2) and (4) (Color Figure Online)

Fig. 9

Parameter space. a Hue values ($\text{red}\sim \widehat{\alpha }$, $\text{green}\sim 1/\widehat{\alpha }$, $\text{blue}\sim \chi$) that uniquely depend on cross-immunity $\chi$ and the infectiousness bound $\widehat{\alpha }$. b The mapping (18) from parameter- to pathogen space; points of the same color—representing the same parameter values $(\chi ,\widehat{\alpha })$—are connected by a thin line (Color Figure Online)

Fig. 10

Extrapolation and extremal parameter pairs. a The eight panels show the fitness maxima in pathogen space for seven transmissibilities $\widehat{\beta }$ and a cumulative combination of them; the colors uniquely represent parameter pairs $(\chi ,\widehat{\alpha })$ as defined in Fig. 9a. The average $(\widehat{\delta },\widehat{\rho })$-values—taken over the $6\times 4$ parameter pairs $(\chi ,\widehat{\alpha })$—are indicated by black dots; in the cumulative panel, they are connected by black lines. b Intensity-weighted average locations of the four extreme parameter pairs $(\chi ,\widehat{\alpha })$ are shown in pathogen space (red, green, violet, cyan in Fig. 9a). Each of the seven quadrilaterals corresponds to one transmissibility ($lg\widehat{\beta }=-3.5,\cdots ,-0.5$). The quadrilaterals change their orientation about halfway, when the fitness maxima over $\widehat{\beta }$ show a discontinuity (cf. Fig. 7) (Color Figure Online)

Fig. 11

Infection type reconstruction. a The eight panels show the fitness maxima in pathogen space for seven transmissibilities $\widehat{\beta }$ and a cumulative combination of them; the colors uniquely represent parameter pairs $(\chi ,\widehat{\alpha })$ as defined by the correspondences (20). The average $(\widehat{\delta },\widehat{\rho })$-values (taken over all colors) are indicated by black dots, which in the cumulative panel are connected by lines. b The infection types A, B, C (colored blue, red, green, resp.) are located in pathogen space, as well as the static patterns (1,...,5) and the transmissibility $\widehat{\beta }$; the resulting color distribution is approximated well by the cumulative diagram “$lg\widehat{\beta }=-3.5,\dots ,-0.5$” in (a) (Color Figure Online)