Philippa Kaur

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Oligomorphic dynamics (OMD) of antigenic escapeWe considered a model of the antigenic escape of a pathogen from host herd immunity on a one-dimensional antigenicity space (x). We tracked the changes in the density (S(t,x)) of hosts that are susceptible to antigenicity variant x of pathogen at time t, and the density (I(t,x)) of hosts that are currently infected and infectious with antigenicity variant x of pathogen at time t:$$frac{{partial Sleft( {t,x} right)}}{{partial t}} = – Sleft( {t,x} right)mathop {smallint }limits_{ – infty }^infty beta sigma left( {x – y} right)Ileft( {t,y} right)dy,$$
(5)
$$frac{{partial Ileft( {t,x} right)}}{{partial t}} = beta Sleft( {t,x} right)Ileft( {t,x} right) – left( {gamma + alpha } right)Ileft( {t,x} right) + Dfrac{{partial ^2Ileft( {t,x} right)}}{{partial x^2}},$$
(6)
where β, α and γ are the transmission rate, virulence (additional mortality due to infection) and recovery rate of pathogens, which are independent of antigenicity. The function σ(x−y) denotes the degree of cross immunity: a host infected by pathogen variant y acquires perfect cross immunity with probability σ(x−y), but fails to acquire any cross immunity with probability 1−σ(x−y) (this is called polarized cross immunity by Gog and Grenfell25). The degree σ(x−y) of cross immunity is assumed to be a decreasing function of the distance |x−y| between variants x and y. When a new variant with antigenicity x = 0 is introduced at time t = 0, the initial host population is assumed to be susceptible to any antigenicity variant of pathogen: S(0,x) = 1. In equation (6), (D = mu sigma _mathrm{m}^2/2) is one half of the mutation variance for the change in antigenicity, representing random mutation in the continuous antigenic space.Susceptibility profile moulded by the primary outbreakWe first analysed the dynamics of the primary outbreak of a pathogen and derived the resulting susceptibility profile, which can be viewed as the fitness landscape subsequently experienced by the pathogen. For simplicity, we assumed that mutation can be ignored during the first epidemic initiated with antigenicity strain x = 0. The density ({{{S}}}_0left( {{{t}}} right) = {{{S}}}({{{t}}},0)) of hosts that are susceptible to the currently prevailing antigenicity variant x = 0, as well as the density ({{{I}}}_0left( {{{t}}} right) = {{{I}}}({{{t}}},0)) of hosts that are currently infected by the focal variant change with time as$$frac{{dS_0}}{{dt}} = – S_0beta I_0,$$
(7)
$$frac{{dI_0}}{{dt}} = S_0beta I_0 – left( {gamma + alpha } right)I_0,$$
(8)
$$frac{{dR_0}}{{dt}} = gamma I_0,$$
(9)
with (S_0left( 0 right) = 1), (I_0left( 0 right) approx 0) and (R_0left( 0 right) = 0). The final size of the primary outbreak,$$psi _0 = R_0left( infty right) = 1 – S_0left( infty right) = exp left[ { – beta mathop {int}limits_0^infty {I_0} left( t right)dt} right],$$is determined as the unique positive root of$$begin{array}{*{20}{c}} {psi _0 = 1 – e^{ – rho _0psi _0},} end{array}$$
(10)
where (rho _0 = beta /left( {gamma + alpha } right) > 1) is the basic reproductive number6. Associated with this epidemiological change, the susceptibility profile (S_xleft( t right) = S(t,x)) against antigenicity x ((x ne 0)) other than the currently circulating variant (x = 0) changes by cross immunity as$$begin{array}{*{20}{c}} {frac{{dS_x}}{{dt}} = – S_xbeta sigma left( x right)I_0,quad left( {x ne 0} right).} end{array}$$
(11)
Integrating both sides of equation (11) from t = 0 to (t = infty), we see that the susceptibility profile (sleft( x right) = S_x(infty )) after the primary outbreak at x = 0 is$$begin{array}{*{20}{c}} {sleft( x right) = exp left[ { – beta sigma left( x right)mathop {int}limits_0^infty {I_0} left( t right)dt} right] = left( {1 – psi _0} right)^{sigma left( x right)} = e^{ – rho _0sigma left( x right)psi _0},} end{array}$$
(12)
where the last equality follows from equation (10). The susceptibility can be effectively reduced by cross immunity when the primary variant has a large impact (that is, when the fraction of hosts remaining uninfected, 1−ψ0, is small) and when the degree of cross immunity is strong (that is, when σ(x) is close to 1). With a variant antigenically very close to the primary variant (x ≈ 0), the cross immunity is very strong ((sigma left( x right) approx 1)) so that the susceptibility against variant x is nearly maximally reduced: (s(x) approx 1 – psi _0). With a variant antigenically distant from the primary variant, σ(x) becomes substantially smaller than 1, making the host more susceptible to the variant. For example, if the cross immunity is halved ((sigma left( x right) = 0.5)) from its maximum value 1, then the susceptibility to that variant is as large as (left( {1 – psi _0} right)^{0.5}). If a variant is antigenically very distant from the primary variant, then (sigma left( x right) approx 0), and the host is nearly fully susceptible to the variant ((sleft( x right) approx 1)).Threshold antigenic distance for escaping immunity raised by primary outbreakOf particular interest is the threshold antigenicity distance xc that allows for antigenic escape, that is, any antigenicity variant more distant than this threshold from the primary variant (x > xc) can increase when introduced after the primary outbreak. Such a threshold is determined from$$frac{{beta sleft( {x_c} right)}}{{gamma + alpha }} = rho _0sleft( {x_c} right) = 1$$or$$begin{array}{*{20}{c}} {sleft( {x_c} right) = left( {1 – psi _0} right)^{sigma left( {x_c} right)} = e^{ – rho _0sigma left( {x_c} right)psi _0} = frac{1}{{rho _0}},} end{array}$$
(13)
where we used equation (12). With a specific choice of cross-immunity profile,$$begin{array}{*{20}{c}} {sigma left( x right) = exp left[ { – frac{{x^2}}{{2omega ^2}}} right],} end{array}$$
(14)
the threshold antigenicity beyond which the virus can increase in the susceptibility profile s(x) after the primary outbreak is obtained, by substituting equation (14) into equation (13)$$exp left[ { – rho _0psi _0exp left[ { – frac{{x_c^2}}{{2omega ^2}}} right]} right] = frac{1}{{rho _0}},$$and taking the logarithm of both sides twice:$$begin{array}{*{20}{c}} {x_c = omega sqrt {2log frac{{rho _0psi _0}}{{log rho _0}}} .} end{array}$$
(15)
OMDIntegrating both sides of equation (6) over the whole space, we obtained the dynamics for the total density of infected hosts, ({{{bar{ I}}}}left( {{{t}}} right) = {int}_{ – infty }^infty {{{{I}}}left( {{{{t}}},{{{x}}}} right){{{dx}}}}):$$begin{array}{*{20}{c}} {frac{{dbar I}}{{dt}} = left[ {beta mathop {smallint }limits_{ – infty }^infty Sleft( {t,x} right)phi left( {t,x} right)dx – left( {gamma + alpha } right)} right]bar Ileft( t right) = left[ {beta bar Sleft( t right) – left( {gamma + alpha } right)} right]bar Ileft( t right)} end{array},$$
(16)
where$$phi left( {t,x} right) = Ileft( {t,x} right)/bar Ileft( t right)$$is the relative frequency of antigenicity variant x in the pathogen population circulating at time t, and$$begin{array}{*{20}{c}} {bar Sleft( t right) = mathop {int}limits_{ – infty }^infty S left( {t,x} right)phi left( {t,x} right)dx} end{array}$$
(17)
is the mean susceptibility experienced by currently circulating pathogens. The dynamics for the relative frequency (phi left( {t,x} right)) of pathogen antigenicity is$$begin{array}{*{20}{c}} {frac{{partial phi }}{{partial t}} = beta left{ {Sleft( {t,x} right) – bar S(t)} right}phi left( {t,x} right) + Dfrac{{partial ^2phi }}{{partial x^2}}.} end{array}$$
(18)
As in Sasaki and Dieckmann27, we decomposed the frequency distribution to the sum of several morph distributions (oligomorphic decomposition) as$$begin{array}{*{20}{c}} {phi left( {t,x} right) = mathop {sum }limits_i p_iphi _ileft( {t,x} right)} end{array},$$
(19)
where pi(t) is the frequency of morph i and (phi _i(t,x)) is the within-morph distribution of antigenicity. By definition, and ({int}_{ – infty }^infty {phi _i} (t,x)dx = 1). Let$$begin{array}{*{20}{c}} {bar x_i = mathop {int}limits_{ – infty }^infty {xphi _ileft( {t,x} right)dx} } end{array}$$
(20)
be the mean antigenicity of a morph and$$begin{array}{*{20}{c}} {V_i = mathop {smallint }limits_{ – infty }^infty left( {x – bar x_i} right)^2phi _ileft( {t,x} right)dx = Oleft( {{it{epsilon }}^2} right)} end{array}$$
(21)
where O is order be the within-morph variance of each morph, which is assumed to be small, of the order of ({it{epsilon }}^2). We denoted the mean susceptibility of host population for viral morph (i) by (bar S_i = {int}_{ – infty }^infty {Sleft( {t,x} right)phi _i(t,x)dx}). As shown in Sasaki and Dieckmann27, the dynamics for viral morph frequency is expressed as$$begin{array}{*{20}{c}} {frac{{dp_i}}{{dt}} = beta left( {bar S_i – bar S} right)p_i + Oleft( {it{epsilon }} right),} end{array}$$
(22)
while the dynamics for the within-morph distribution of antigenicity is$$begin{array}{*{20}{c}} {frac{{partial phi _i}}{{partial t}} = beta left{ {Sleft( {t,x} right) – bar S_i} right}phi _ileft( {t,x} right) + Dfrac{{partial ^2phi _i}}{{partial x^2}}.} end{array}$$
(23)
From this, the dynamics for the mean antigenicity of a morph,$$begin{array}{*{20}{c}} {frac{{dbar x_i}}{{dt}} = V_ibeta left. {frac{{partial S}}{{partial x}}} right|_{x = bar x_i} + Oleft( {{it{epsilon }}^3} right)} end{array}$$
(24)
and the dynamics for the within-morph variance of a morph$$begin{array}{*{20}{c}} {frac{{dV_i}}{{dt}} = frac{1}{2}beta left. {frac{{partial ^2S}}{{partial x^2}}} right|_{x = bar x_i}left{ {Eleft[ {xi _i^4} right] – V_i^2} right} + 2D + Oleft( {{it{epsilon }}^5} right)} end{array}$$
(25)
are derived, where (xi _i = x – bar x_i) and (Eleft[ {xi _i^4} right] = {int}_{ – infty }^infty {left( {x – bar x_i} right)^4phi _ileft( {t,x} right)dx}) are the fourth central moments of antigenicity around the morph mean. Assuming that the within-morph distribution is normal (Gaussian closure), (Eleft[ {xi _i^4} right] = 3V_i^2), and hence equation (25) becomes$$begin{array}{*{20}{c}} {frac{{dV_i}}{{dt}} = beta left. {frac{{partial ^2S}}{{partial x^2}}} right|_{x = bar x_i}V_i^2 + 2D + Oleft( {{it{epsilon }}^5} right).} end{array}$$
(26)
Second outbreak predicted by OMDEquations (22), (24) and (26) are general, but they rely on a full knowledge of the dynamics of the susceptibility profile S(t,x). To make further progress, we used an additional approximation by substituting equation (13), the susceptibility profile, over viral antigenicity space after the primary outbreak at x = 0 and before the onset of the second outbreak at a distant position. We kept track of two morphs at positions x0(t) and x1(t), where the first morph is that caused by the primary outbreak at x = 0, and the second morph is that emerged in the range x > xc beyond the threshold antigenicity xc defined in equation (13) (and equation (15) for a specific form of σ(x)) as the source of the next outbreak.As (sleft( x right) = left( {1 – psi _0} right)^{sigma (x)} = exp [sigma left( x right)log (1 – psi _0)]), we have$$frac{{{{{mathrm{d}}}}s}}{{{{{mathrm{d}}}}x}}left( {bar x_i} right) = left[ {frac{{{{{mathrm{d}}}}sigma }}{{{{{mathrm{d}}}}x}}left( {bar x_i} right)log left( {1 – psi _0} right)} right]sleft( {bar x_i} right),$$and$$frac{{{{{mathrm{d}}}}^2s}}{{{{{mathrm{d}}}}x^2}}left( {bar x_i} right) = left[ {frac{{{{{mathrm{d}}}}^2sigma }}{{{{{mathrm{d}}}}x^2}}left( {bar x_i} right)log left( {1 – psi _0} right) + left{ {frac{{{{{mathrm{d}}}}sigma }}{{{{{mathrm{d}}}}x}}left( {bar x_i} right)log left( {1 – psi _0} right)} right}^2} right]sleft( {bar x_i} right).$$Therefore, the frequency, mean antigenicity and variance of antigenicity of an emerging morph (i = 1) change respectively as$$begin{array}{*{20}{l}} {frac{{dp_1}}{{dt}} = beta left[ {sleft( {bar x_1} right) – sleft( {bar x_0} right)} right]p_1left( {1 – p_1} right),} hfill \ {frac{{dbar x_1}}{{dt}} = V_1beta frac{{{{{mathrm{d}}}}s}}{{{{{mathrm{d}}}}x}}left( {bar x_1} right),} hfill \ {frac{{dV_1}}{{dt}} = beta frac{{{{{mathrm{d}}}}^2s}}{{{{{mathrm{d}}}}x^2}}left( {bar x_1} right)V_1^2 + 2D}. hfill end{array}$$
(27)
The predicted change in the mean antigenicity was plotted by integrating equation (27). As initial condition, we chose the time when a seed of second peak in the range x > xc first appeared, and then computed the mean trait as$$begin{array}{*{20}{c}} {bar xleft( t right) = x_0left( {1 – p_1left( t right)} right) + bar x_1p_1left( t right).} end{array}$$
(28)
In the case of Fig. 2, where β = 2, γ + α = 0.6, D = 0.001 and ω = 2, the final size of epidemic for the primary outbreak, defined as equation (7), was ψ = 0.959, and the critical antigenic distance for the increase of pathogen variant obtained from equation (26) was xc = 2.795. The initial conditions for the oligomorphic dynamics (equation 27) for the second morph were then (p_1left( {t_0} right) = 1.6 times 10^{ – 8}), (bar x_1left( {t_0} right) = 3.239), (V_1left( {t_0} right) = 0.2675) at t0 = 41. In Fig. 2, the predicted trajectory for the mean antigenicity (equation 28) is plotted as a red curve, together with the mean antigenicity change observed in simulation (blue curve).Accuracy of predicting the antigenicity with OMD and the timing of the second outbreakHere we describe how we defined the initial conditions for oligomorphic dynamics, that is, the frequency, the mean antigenicity and the variance in antigenicity of the morph that caused the primary outbreak and the morph that may cause the second outbreak. We then show how the accuracy in prediction of the second outbreak depends on the timing of the prediction.We divided the antigenicity space into two at x = xc, above which the pathogen can increase under the given susceptibility profile after the primary outbreak, but below which the pathogen cannot increase. We then took the relative frequencies of pathogens above xc and below xc, and the conditional mean and variance in these separated regions to set the initial frequencies, means and variances of the morphs at time t0 when we started integrating the oligomorphic dynamics to predict the second outbreak:$$begin{array}{*{20}{c}} {begin{array}{*{20}{l}} {p_0left( {t_0} right) = frac{{mathop {smallint }nolimits_0^{x_c} Ileft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_0^infty Ileft( {t_0,x} right)dx}},} hfill & {p_1left( {t_0} right) = frac{{mathop {smallint }nolimits_{x_c}^infty Ileft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_0^infty Ileft( {t_0,x} right)dx}},} hfill \ {bar x_0left( {t_0} right) = frac{{mathop {smallint }nolimits_0^{x_c} xIleft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_0^{x_c} Ileft( {t_0,x} right)dx}},} hfill & {bar x_1left( {t_0} right) = frac{{mathop {smallint }nolimits_{x_c}^infty xIleft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_{x_c}^infty Ileft( {t_0,x} right)dx}},} hfill \ {V_0left( {t_0} right) = frac{{mathop {smallint }nolimits_0^{x_c} left( {x – bar x_0left( {t_0} right)} right)^2Ileft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_0^{x_c} Ileft( {t_0,x} right)dx}},} hfill & {V_1left( {t_0} right) = frac{{mathop {smallint }nolimits_{x_c}^infty left( {x – bar x_1left( {t_0} right)} right)^2Ileft( {t_0,x} right)dx}}{{mathop {smallint }nolimits_{x_c}^infty Ileft( {t_0,x} right)dx}}.} hfill end{array}} end{array}$$
(29)
We then compared the trajectory for mean antigenicity change observed in simulation (blue curve in Fig. 2) and the predicted trajectory (red curve in Fig. 2) for mean antigenicity (equation 28) by integrating oligomorphic dynamics (equation 27) with the initial condition (equation 29) at time t = t0. Extended Data Fig. 2 shows how the accuracy of prediction, measured by the Kullback–Leibler divergence between these two trajectories, depends on the timing t0 chosen for the prediction. The second outbreak occurs around t = 54.6, where mean antigenicity jumps from around 0 to around 5. The prediction with OMD is accurate if it is made for t0 > 40. Figure 2 is drawn for t0 = 41 where the second peak is about to emerge (see Extended Data Fig. 2). Even for the latest prediction for t0 = 51 in Extended Data Fig. 2, the morph frequency of the emerging second morph was only 0.3% off, so the prediction is still worthwhile to make.Extended Data Fig. 2 shows that the prediction power is roughly constant (albeit with a wiggle) for (5 < t_0 < 30) (the predicted timings are 10–15% longer than actual timing for (5 < t_0 < 30)), and steadily improved for t0 > 30. When the prediction was made very early (t0 < 5), the deviations were larger.OMD for the joint evolution of antigenicity and virulenceLet s(x) be the susceptibility of the host population against antigenicity x. A specific susceptibility profile is given by equation (12), with cross-immunity function σ(x) and the final size ψ0 of epidemic of the primary outbreak. Note that, as above, the susceptibility profile is, in general, a function of time. The density (I(x,alpha )) of hosts infected by a pathogen of antigenicity x and virulence α changes with time, when rare, as$$begin{array}{*{20}{c}} {frac{{partial Ileft( {x,alpha } right)}}{{partial t}} = beta sleft( x right)Ileft( {x,alpha } right) - left( {gamma + alpha } right)Ileft( {x,alpha } right) + D_xfrac{{partial ^2I}}{{partial x^2}} + D_alpha frac{{partial ^2I}}{{partial alpha ^2}}.} end{array}$$ (30) The change in the frequency (phi left( {x,alpha } right) = Ileft( {x,alpha } right)/{int!!!!!int} I left( {x,alpha } right)dxdalpha) of a pathogen with antigenicity x and virulence α follows$$begin{array}{*{20}{c}} {frac{{partial phi }}{{partial t}} = left{ {wleft( {x,alpha } right) - bar w} right}phi + D_xfrac{{partial ^2phi }}{{partial x^2}} + D_alpha frac{{partial ^2phi }}{{partial alpha ^2}},} end{array}$$ (31) where$$begin{array}{*{20}{c}} {wleft( {x,alpha } right) = beta left( alpha right)sleft( x right) - alpha } end{array}$$ (32) is the fitness of a pathogen with antigenicity x and virulence α and (bar w = {int!!!!!int} w left( {x,alpha } right)dxdalpha) is the mean fitness.We decomposed the joint frequency distribution ϕ(x, α) of the viral quasi-species as (oligomorphic decomposition):$$begin{array}{*{20}{c}} {phi left( {x,alpha } right) = mathop {sum }limits_i phi _ileft( {x,alpha } right)p_i,} end{array}$$ (33) where ϕi(x, α) is the joint frequency distribution of antigenicity x and virulence α in morph i (({int!!!!!int} {phi _idxdalpha = 1})) and pi is the relative frequency of morph i ((mathop {sum}nolimits_i {p_i = 1})). The frequency of morph i then changes as$$begin{array}{l}frac{{dp_i}}{{dt}} = left( {bar w_i - mathop {sum }limits_j bar w_jp_j} right)p_i,\ frac{{partial phi _i}}{{partial t}} = left( {wleft( {x,alpha } right) - bar w_i} right)phi _ileft( {x,alpha } right) + D_xfrac{{partial ^2phi _i}}{{partial x^2}} + D_alpha frac{{partial ^2phi _i}}{{partial alpha ^2}},end{array}$$ (34) where (bar w_i = {int!!!!!int} w left( {x,alpha } right)phi _ileft( {x,alpha } right)dxdalpha) is the mean fitness of morph i.Assuming that the traits are distributed narrowly around the morph means (bar x_i = {int!!!!!int} x phi _ileft( {x,alpha } right)dxdalpha) and (bar alpha _i = {int!!!!!int} alpha phi _i(x,alpha )dxdalpha), so that (xi _i = x - bar x_i = O({it{epsilon }})) and (zeta _i = alpha - bar alpha _i = O({it{epsilon }})) where ({it{epsilon }}) is a small constant, we expanded the fitness w(x, α) around the means (bar x_i) and (bar alpha _i) of morph i,$$begin{array}{*{20}{l}} {wleft( {x,alpha } right)} hfill & = hfill & {wleft( {bar x_i,bar alpha _i} right) + left( {frac{{partial w}}{{partial x}}} right)_ixi _i + left( {frac{{partial w}}{{partial alpha }}} right)_izeta _i} hfill \ {} hfill & {} hfill & { + frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_ixi _i^2 + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ixi _izeta _i + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_izeta _i^2 + Oleft( {{it{epsilon }}^3} right).} hfill end{array}$$Substituting this and$$bar w_i = wleft( {bar x_i,bar alpha _i} right) + frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_iV_i^{xx} + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_iV_i^{xalpha } + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_iV_i^{alpha alpha } + Oleft( {{it{epsilon }}^3} right)$$into equation (34), we obtained$$frac{{dp_i}}{{dt}} = left[ {w_i - mathop {sum }limits_j w_jp_j} right]p_i + Oleft( {it{epsilon }} right),$$ (35) $$begin{array}{*{20}{l}} {frac{{partial phi _i}}{{partial t}}} hfill & = hfill & {left[ {left( {frac{{partial w}}{{partial x}}} right)_ixi _i + left( {frac{{partial w}}{{partial alpha }}} right)_izeta _i + frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_ileft( {xi _i^2 - V_i^x} right) + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ileft( {xi _izeta _i - C_i} right)} right.} hfill \ {} hfill & {} hfill & {left. { + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_ileft( {zeta _i^2 - V_i^alpha } right)} right]phi _i + D_xfrac{{partial ^2phi _i}}{{partial x^2}} + D_alpha frac{{partial ^2phi _i}}{{partial alpha ^2}} + Oleft( {{it{epsilon }}^3} right),} hfill end{array}$$ (36) where (w_i = wleft( {bar x_i,bar alpha _i} right)), (left( {frac{{partial w}}{{partial x}}} right)_i = frac{{partial w}}{{partial x}}left( {bar x_i,bar alpha _i} right)), (left( {frac{{partial w}}{{partial alpha }}} right)_i = frac{{partial w}}{{partial alpha }}left( {bar x_i,bar alpha _i} right)), (left( {frac{{partial ^2w}}{{partial x^2}}} right)_i = frac{{partial ^2w}}{{partial x^2}}left( {bar x_i,bar alpha _i} right)), (left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_i = frac{{partial ^2w}}{{partial xpartial alpha }}left( {bar x_i,bar alpha _i} right)) and (left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_i = frac{{partial ^2w}}{{partial alpha ^2}}left( {bar x_i,bar alpha _i} right)) are fitness and its first and second derivatives evaluated at the mean traits of morph i, and$$begin{array}{l}V_i^x = E_ileft[ {left( {x - bar x_i} right)^2} right],\ C_i = E_ileft[ {left( {x - bar x_i} right)left( {alpha - bar alpha _i} right)} right], \ V_i^alpha = E_ileft[ {left( {alpha - bar alpha _i} right)^2} right]end{array}$$ (37) are within-morph variances and covariance of the traits of morph i. Here (E_ileft[ {fleft( {x,alpha } right)} right] = {int!!!!!int} f left( {x,alpha } right)phi _ileft( {x,alpha } right)dxdalpha) denotes taking expectation of a function f with respect to the joint trait distribution (phi _i(x,alpha )) of morph i.Substituting equation (36) into the change in the mean antigenicity of morph i$$frac{{dbar x_i}}{{dt}} = frac{d}{{dt}}{int!!!!!int} x phi _i(x,alpha )dxdalpha = {int!!!!!int} x frac{{partial phi _i}}{{partial t}}dxdalpha = {int!!!!!int} {(bar x_i + xi _i)} frac{{partial phi _i}}{{partial t}}dxi _idzeta _i,$$we obtained$$begin{array}{*{20}{c}} {frac{{dbar x_i}}{{dt}} = left( {frac{{partial w}}{{partial x}}} right)_iV_i^x + left( {frac{{partial w}}{{partial alpha }}} right)_iC_i + Oleft( {{it{epsilon }}^3} right).} end{array}$$ (38) Similarly, the change in the mean virulence of morph i was expressed as$$begin{array}{*{20}{c}} {frac{{dbar alpha _i}}{{dt}} = left( {frac{{partial w}}{{partial x}}} right)_iC_i + left( {frac{{partial w}}{{partial alpha }}} right)_iV_i^alpha + Oleft( {{it{epsilon }}^3} right).} end{array}$$ (39) Equations (38) and (39) from the mean trait change was summarized in a matrix form as$$begin{array}{*{20}{c}} {frac{d}{{dt}}left( {begin{array}{*{20}{c}} {bar x_i} \ {bar alpha _i} end{array}} right) = {{{boldsymbol{G}}}}_{{{boldsymbol{i}}}}left( {begin{array}{*{20}{c}} {left( {frac{{partial w}}{{partial x}}} right)_i} \ {left( {frac{{partial w}}{{partial alpha }}} right)_i} end{array}} right) + O({it{epsilon }}^3),} end{array}$$ (40) where$$begin{array}{*{20}{c}} {{{{boldsymbol{G}}}}_{{{boldsymbol{i}}}} = left( {begin{array}{*{20}{c}} {V_i^x} & {C_i} \ {C_i} & {V_i^alpha } end{array}} right)} end{array}$$ (41) is the variance-covariance matrix of the morph i.Substituting equation (36) into the right-hand side of the change in variance of antigenicity of morph i,$$frac{{dV_i^x}}{{dt}} = frac{d}{{dt}}{int!!!!!int} {xi _i^2phi _idxi _idzeta _i} = {int!!!!!int} {xi _i^2frac{{partial phi _i}}{{partial t}}dxi _idzeta _i}$$and those in the change in the other variance and covariance, we obtained$$begin{array}{*{20}{l}} {frac{{dV_i^x}}{{dt}}} hfill & = hfill & {frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_ileft[ {E_ileft( {xi _i^4} right) - left( {V_i^x} right)^2} right] + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ileft[ {E_ileft( {xi _i^3zeta _i} right) - V_i^xC_i} right]} hfill \ {} hfill & {} hfill & { + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_ileft[ {E_ileft( {xi _i^2zeta _i^2} right) - V_i^xV_i^alpha } right] + 2D_x + O({it{epsilon }}^5),} hfill \ {frac{{dC_i}}{{dt}}} hfill & = hfill & {frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_ileft[ {E_ileft( {xi _i^3zeta _i} right) - V_i^xC_i} right] + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ileft[ {E_ileft( {xi _i^2zeta _i^2} right) - C_i^2} right]} hfill \ {} hfill & {} hfill & { + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_ileft[ {E_ileft( {xi _izeta _i^3} right) - C_iV_i^alpha } right] + O({it{epsilon }}^5),} hfill \ {frac{{dV_i^alpha }}{{dt}}} hfill & = hfill & {frac{1}{2}left( {frac{{partial ^2w}}{{partial x^2}}} right)_ileft[ {E_ileft( {xi _i^2zeta _i^2} right) - V_i^xV_i^alpha } right] + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ileft[ {E_ileft( {xi _izeta _i^3} right) - C_iV_i^alpha } right]} hfill \ {} hfill & {} hfill & { + frac{1}{2}left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_ileft[ {E_ileft( {zeta _i^4} right) - left( {V_i^alpha } right)^2} right] + 2D_alpha + O({it{epsilon }}^5).} hfill end{array}$$ (42) If we assume that antigenicity and virulence within a morph follow a 2D Gaussian distribution for given means, variances and covariance, we should have (E_i(xi _i^4) = 3left( {V_i^x} right)^2,E_i(xi _i^3zeta _i) = 3V_i^xC_i), (E_i(xi _i^2zeta _i^2) = V_i^xV_i^alpha + 2C_i^2), (E_i(xi _izeta _i^3) = 3V_i^alpha C_i) and (E_i(zeta _i^4) = 3left( {V_i^alpha } right)^2), and hence$$frac{{dV_i^x}}{{dt}} = left( {frac{{partial ^2w}}{{partial x^2}}} right)_ileft( {V_i^x} right)^2 + 2left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_iV_i^xC_i + left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_iC_i^2 + 2D_x + O({it{epsilon }}^5),$$ (43) $$frac{{dC_i}}{{dt}} = left( {frac{{partial ^2w}}{{partial x^2}}} right)_iV_i^xC_i + left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_ileft{ {V_i^xV_i^alpha - C_i^2} right} + left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_iC_iV_i^alpha + O({it{epsilon }}^5),$$ (44) $$frac{{dV_i^alpha }}{{dt}} = left( {frac{{partial ^2w}}{{partial x^2}}} right)_iC_i^2 + 2left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_iV_i^alpha C_i + left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_ileft( {V_i^alpha } right)^2 + 2D_alpha + O({it{epsilon }}^5).$$ (45) Equations (43) and (44) were rewritten in a matrix form as$$begin{array}{*{20}{c}} {frac{{dG_i}}{{dt}} = G_iH_iG_i + left( {begin{array}{*{20}{c}} {2D_xV_i^x} & 0 \ 0 & {2D_alpha V_i^alpha } end{array}} right) + O({it{epsilon }}^5),} end{array}$$ (46) where$$begin{array}{*{20}{c}} {H_i = left( {begin{array}{*{20}{c}} {left( {frac{{partial ^2w}}{{partial x^2}}} right)_i} & {left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_i} \ {left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_i} & {left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_i} end{array}} right),} end{array}$$ (47) is the Hessian of the fitness function of morph i.In our equation (30) of the joint evolution of antigenicity and virulence of a pathogen after its primary outbreak, the fitness function is given by (w(x,alpha ) = beta (alpha )s(x) - alpha ,) and hence (w_i = beta left( {bar alpha _i} right)sleft( {bar x_i} right) - bar alpha _i), (left( {frac{{partial w}}{{partial x}}} right)_i = beta left( {bar alpha _i} right)sprime left( {bar x_i} right)), (left( {frac{{partial w}}{{partial alpha }}} right)_i = beta prime left( {bar alpha _i} right)sleft( {bar x_i} right) - 1), (left( {frac{{partial ^2w}}{{partial x^2}}} right)_i = beta left( {bar alpha _i} right)sprimeprime left( {bar x_i} right)), (left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_i = beta prime left( {bar alpha _i} right)sprime left( {bar x_i} right)), (left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_i = beta prime left( {bar alpha _i} right)sprime left( {bar x_i} right)) and (left( {frac{{partial ^2w}}{{partial alpha ^2}}} right)_i = beta primeprime left( {bar alpha _i} right)sleft( {bar x_i} right)), where a prime on β(α) and s(x) denotes differentiation by α and x, respectively. Substituting these into the dynamics for morph frequencies (equation 35), for morph means (equations 38 and 39), and for within-morph variance and covariance (equations 43–45), we obtained$$frac{{dp_i}}{{dt}} = left[ {beta left( {bar alpha _i} right)sleft( {bar x_i} right) - bar alpha _i - mathop {sum }limits_j left( {beta left( {bar alpha _j} right)sleft( {bar x_j} right) - bar alpha _j} right)p_j} right]p_i,$$ (48) $$frac{{dbar x_i}}{{dt}} = beta left( {bar alpha _i} right)sprime left( {bar x_i} right)V_i^x + left{ {beta prime left( {bar alpha _i} right)sleft( {bar x_i} right) - 1} right}C_i,$$ (49) $$frac{{dbar alpha _i}}{{dt}} = beta left( {bar alpha _i} right)sprime left( {bar x_i} right)C_i + left{ {beta prime left( {bar alpha _i} right)sleft( {bar x_i} right) - 1} right}V_i^alpha ,$$ (50) $$frac{{dV_i^x}}{{dt}} = beta left( {bar alpha _i} right)sprimeprime left( {bar x_i} right)left( {V_i^x} right)^2 + 2beta prime left( {bar alpha _i} right)sprime left( {bar x_i} right)V_i^xC_i + beta primeprime left( {bar alpha _i} right)sleft( {bar x_i} right)C_i^2 + 2D_x,$$ (51) $$frac{{dC_i}}{{dt}} = beta left( {bar alpha _i} right)sprimeprime left( {bar x_i} right)V_i^xC_i + beta prime left( {bar alpha _i} right)sprime left( {bar x_i} right)left{ {V_i^xV_i^alpha - C_i^2} right} + beta primeprime left( {bar alpha _i} right)sleft( {bar x_i} right)C_iV_i^alpha ,$$ (52) $$frac{{dV_i^alpha }}{{dt}} = beta left( {bar alpha _i} right)sprimeprime left( {bar x_i} right)C_i^2 + 2beta prime left( {bar alpha _i} right)sprime left( {bar x_i} right)V_i^alpha C_i + beta primeprime left( {bar alpha _i} right)sleft( {bar x_i} right)left( {V_i^alpha } right)^2 + 2D_alpha .$$ (53) Equations (48)–(53) describe the oligomorphic dynamics of the joint evolution of antigenicity and virulence of a pathogen for a given host susceptibility profile s(x) over pathogen antigenicity.Of particular interest is whether antigenicity or virulence evolve faster when they jointly evolve than when they evolve alone. After the primary outbreak at a given antigenicity, for example x = 0, the susceptibility s(x) of the host population increases due to cross immunity as the distance x > 0 from the antigenicity at the primary outbreak increases. Hence, (sprime left( {bar x_i} right) > 0.) Combining this with the positive trade-off between transmission rate and virulence, we see that (left( {partial ^2w/partial xpartial alpha } right)_i = beta prime (bar alpha _i)sprime (bar x_i) > 0), and then from equation (52), we see that the within-morph covariance between antigenicity and virulence becomes positive starting from a zero initial value:$$begin{array}{*{20}{c}} {left. {frac{{dC_i}}{{dt}}} right|_{C_i = 0} = left( {frac{{partial ^2w}}{{partial xpartial alpha }}} right)_iV_i^xV_i^alpha > 0.} end{array}$$
(54)
If all second moments are initially sufficiently small for an emerging morph, a quick look at the linearization of equations (51)–(53) around ((V_i^x,C_i,V_i^alpha ) = (0,0,0)) indicates that both (V_i^x) and (V_i^alpha) become positive due to the random generation of variance by mutation, Dx > 0 and Dα > 0, while the covariance stays close to zero. Then, equation (54) guarantees that the first move of the covariance is from zero to positive, which then guarantees that Ci > 0 for all t. Therefore, the second term in equation (38) is positive until the mean virulence reaches its optimum ((beta prime (alpha )s(x) = 1)). This means that joint evolution with virulence accelerates the evolution of antigenicity. The same is true for virulence evolution: the first term in equation (39) (which denotes the associated change in virulence due to the selection in antigenicity through genetic covariance between them) is positive, indicating that joint evolution with antigenicity accelerates virulence evolution.Numerical exampleFigure 5 shows the oligomorphic dynamics prediction of the emergence of the next variant in antigenicity–virulence coevolution. To make progress numerically, we assumed s(x) to be constant in the following analysis because we are interested in the process between the end of the primary outbreak and the emergence of the next antigenicity–virulence morph. The partial differential equations for the density of host S(t,x) susceptible to the antigenicity variant x at time t, and the density of hosts infected by the pathogen variant with antigenicity x and virulence α are$$begin{array}{l}frac{{partial Sleft( {t,x} right)}}{{partial t}} = – Sleft( {t,x} right)mathop {smallint }limits_{alpha _{{{{mathrm{min}}}}}}^{alpha _{{{{mathrm{max}}}}}} mathop {smallint }limits_0^{x_{{{{mathrm{max}}}}}} beta left( alpha right)sigma left( {x – y} right)Ileft( {t,y,alpha } right)dydalpha ,\ frac{{partial Ileft( {t,x,alpha } right)}}{{partial t}} = left[ {beta left( alpha right)Sleft( {t,x} right) – left( {gamma + alpha } right)} right]Ileft( {t,x,alpha } right) + left( {D_xfrac{{partial ^2}}{{partial x^2}} + D_alpha frac{{partial ^2}}{{partial alpha ^2}}} right)Ileft( {t,x,alpha } right),end{array}$$
(55)
with the boundary conditions (left( {partial S/partial x} right)left( {t,0} right) = left( {partial S/partial x} right)left( {t,x_{{{{mathrm{max}}}}}} right) = 0), (left( {partial I/partial x} right)left( {t,0,alpha } right) = left( {partial I/partial x} right)left( {t,x_{{{{mathrm{max}}}}},0} right) = 0), (left( {partial I/partial x} right)left( {t,x,alpha _{{{{mathrm{min}}}}}} right) = left( {partial I/partial x} right)left( {t,x,alpha _{{{{mathrm{max}}}}}} right) = 0), and the initial conditions (Sleft( {0,x} right) = 1) and (Ileft( {0,x,alpha } right) = {it{epsilon }}delta left( x right)delta left( alpha right)), where (delta ( cdot )) is the delta function and ({it{epsilon }} = 0.01). The trait space is restricted in a rectangular region: (0 < x < x_{{{{mathrm{max}}}}} = 300) and (alpha _{{{{mathrm{min}}}}} = 0.025 < alpha < 10 = alpha _{{{{mathrm{max}}}}}). Oligomorphic dynamics prediction for the joint evolution of antigenicity and virulence was applied for the next outbreak after the outbreak with the mean antigenicity around x = 108 at time t = 102. The susceptibility of the host to antigenicity variant x at t0 = 104.8 after the previous outbreak peaked around time t = 102 came to an end is$$sleft( x right) = Sleft( {t_0,x} right).$$This susceptibility profile remained unchanged until the next outbreak started, and hence the fitness of a pathogen variant with antigenicity x and virulence α is given by$$wleft( {x,alpha } right) = beta left( alpha right)sleft( x right) - (gamma + alpha ).$$We bundled the pathogen variants into two morphs at time t0 at the threshold antigenicity xc, above which the net growth rate of the pathogen variant under the given susceptibility profile s(x) and the mean antigenicity become positive:$$wleft( {x_c,bar alpha left( {t_0} right)} right) = beta left( {bar alpha (t_0)} right)sleft( {x_c} right) - left( {gamma + bar alpha left( {t_0} right)} right) = 0.$$The initial frequency and the moments of the two morphs, the variant 0 with (x < x_c) and the variant 1 with (x > x_c) were then calculated respectively from the joint distribution (I(t_0,x,alpha )) in the restricted region (left{ {left( {x,alpha } right);0 < x < x_c,alpha _{{{{mathrm{min}}}}} < alpha < alpha _{{{{mathrm{max}}}}}} right}) and that in the restricted region (left{ {left( {x,alpha } right);x_c < x < x_{{{{mathrm{max}}}}},alpha _{{{{mathrm{min}}}}} < alpha < alpha _{{{{mathrm{max}}}}}} right}). The frequency p1 of morph 1 (the frequency of morph 0 is given by (p_0 = 1 - p_1)), the mean antigenicity (bar x_i) and mean virulence (bar alpha _i) of morph i, and the variances and covariance, (V_i^x), and (V_i^alpha) Ci of morph i (i = 0,1) follow equations (48)–(53), where the dynamics for the morph frequency (equation 48) is simplified in this two-morph situation as$$frac{{dp_1}}{{dt}} = left[ {beta left( {bar alpha _1} right)sleft( {bar x_1} right) - beta left( {bar alpha _0} right)sleft( {bar x_0} right) - left( {bar alpha _1 - bar alpha _0} right)} right]p_1left( {1 - p_1} right),$$with (p_0left( t right) = 1 - p_1(t)). This is iterated from (t = t_0 = 104.8) to (t_e = 107). The frequency p1 of the new morph, the population mean antigenicity (bar x = p_0bar x_0 + p_1bar x_1), virulence (bar alpha = p_0bar alpha _0 + p_1bar alpha _1), variance in antigenicity (V_x = p_0V_0^x + p_1V_1^x), covariance between antigenicity and virulence (C = p_0C_0 + p_1C_1), and variance in virulence (V_alpha = p_0V_0^alpha + p_1V_1^alpha) are overlayed by red thick curves on the trajectories of moments observed in the full dynamics (equation 55).In Fig. 5a, the dashed vertical line represents the threshold antigenicity xc, above which (R_0 = beta s(x)/(gamma + bar alpha ) > 1) at (t = t_s = 104.8), where oligomorphic dynamics prediction was attempted. Two morphs were then defined according to whether or not the antigenicity exceeded a threshold x = xc: the resident morph (morph 1) is represented as the dense cloud to the left of x = xc and the second morph (morph 2) consisting of all the genotypes to the right of x = xc with their R0 greater than one. The within-morph means and variances were then calculated in each region. The relative total densities of infected hosts in the left and right regions defined the initial frequency of two morphs in OMD. A 2D Gaussian distribution was assumed for within-morph trait distributions to have the closed moment equations as previously explained. Using these as the initial means, variances, covariances of the two morphs at t = ts, the oligomorphic dynamics for 11 variables (relative frequency of morph 1, mean antigenicity, mean virulence, variances in antigenicity and virulence and their covariance in morphs 0 and 1) was integrated up to t = te. The results are shown as red curves in Fig. 5c–h, which are compared with the simulation results (blue curves).Fig. 5c–e respectively show the change in total infected density, mean antigenicity and mean virulence. Red curves show the predictions by oligomorphic dynamics from the initial moments of each morph at t = ts to the susceptibility distribution (s(x) = S(t_s,x)), which are compared with the simulation results (blue curves). The OMD-predicted mean antigenicity, for example, is defined as$$bar xleft( t right) = left( {1 – p_1left( t right)} right)bar x_0left( t right) + p_1left( t right)bar x_1left( t right),$$where p1(t) is the frequency of morph 1, (bar x_0) and (bar x_1) are the mean antigenicities of morphs 0 and 1.The red curves in Fig. 5f–h show the OMD-predicted changes in the variance in antigenicity, variance in virulence and covariance between antigenicity and virulence, which are compared with the simulation results (blue curves). The OMD-predicted covariance, for example, is defined as$$begin{array}{rcl}Cleft( t right) & = & left( {1 – p_1(t)} right)C_0left( t right) + p_1left( t right)C_1left( t right) + p_1left( t right)left( {1 – p_1left( t right)} right)\ && left( {bar x_0left( t right) – bar x_1left( t right)} right)left( {bar alpha _0left( t right) – bar alpha _1left( t right)} right),end{array}$$where (C_0(t)) and (C_1(t)) are the antigenicity–virulence covariances in morphs 0 and 1, and (bar alpha _0(t)) and (bar alpha _1(t)) are the mean virulence of morphs 0 and 1.Selection for maximum growth rateWe next show that a pathogen that has the strategy of maximizing growth rate in a fully susceptible population is evolutionarily stable in the presence of antigenic escape.At stationarity, the travelling wave profiles of (hat I(z)) and (hat S(z)) along the moving coordinate, (z = x – vt), that drifts constantly to the right with the speed v are defined as$$begin{array}{l}0 = Dfrac{{d^2hat Ileft( z right)}}{{dz^2}} + vfrac{{dhat Ileft( z right)}}{{dz}} + beta hat Sleft( z right)hat Ileft( z right) – left( {gamma + alpha } right)hat Ileft( z right),\ 0 = vfrac{{dhat Sleft( z right)}}{{dz}} – beta hat Sleft( z right)mathop {smallint }limits_{ – infty }^infty sigma left( {z – xi } right)hat Ileft( xi right)dxi ,end{array}$$
(56)
with (hat Ileft( { – infty } right) = hat Ileft( infty right) = 0), (hat Sleft( infty right) = 1).Let j(t,x) be the density of a mutant pathogen variant, with virulence α′ and transmission rate β′, that is introduced in the host population where the resident variant is already established (equation 50). For the initial transient phase in which the density of mutants is sufficiently small, we have an equation for the change in (Jleft( {t,z} right) = j(t,x)):$$begin{array}{*{20}{c}} {frac{partial }{{partial t}}Jleft( {t,z} right) = left{ {Dfrac{{partial ^2}}{{partial z^2}} + vfrac{partial }{{partial z}} + beta prime hat Sleft( z right) – left( {gamma + alpha prime } right)} right}Jleft( {t,z} right),} end{array}$$
(57)
with the initial condition (Jleft( {0,z} right) = {it{epsilon }}delta (z)), where ({it{epsilon }}) is a small constant and (delta ( cdot )) is Dirac’s function.Consider a system$$begin{array}{*{20}{c}} {frac{{partial w}}{{partial t}} = left{ {Dfrac{{partial ^2}}{{partial z^2}} + vfrac{partial }{{partial z}} + beta prime – left( {gamma + alpha prime } right)} right}w,} end{array}$$
(58)
with (wleft( {0,z} right) = Jleft( {0,z} right) = {it{epsilon }}delta (z)). Noting that (hat Sleft( z right) < 1), we have (Jleft( {t,z} right) le w(t,z)) for any (t > 0) and (z in {Bbb R}) from the comparison theorem. The solution to equation (52) is$$begin{array}{*{20}{c}} {wleft( {t,z} right) = frac{{it{epsilon }}}{{sqrt {4pi Dt} }}exp left[ {rprime t – frac{{left( {z + vt} right)^2}}{{4Dt}}} right]} end{array},$$
(59)
where (rprime = beta prime – left( {gamma + alpha prime } right)). This follows by noting that (wleft( {t,x} right)e^{ – rprime t}) follows a simple diffusion equation (partial w/partial t = Dpartial ^2w/partial x^2). By rearranging the exponents of equation (53),$$begin{array}{*{20}{l}} {wleft( {t,z} right)} hfill & = hfill & {exp left[ {at – kz} right]frac{{it{epsilon }}}{{sqrt {4pi Dt} }}e^{ – z^2/4Dt}} hfill \ {} hfill & {} hfill & { < frac{{it{epsilon }}}{{sqrt {4pi Dt} }}exp left[ {at - kz} right],} hfill end{array}$$ (60) where$$a = frac{{v^{prime 2} - v^2}}{{4D}},$$ (61) $$k = frac{v}{{2D}}.$$ (62) Here (vprime = 2sqrt {rprime D}) is the asymptotic wave speed if the mutant variant monopolizes the host population. Therefore, if (vprime < v), then (a < 0), and hence (w(t,z)) for a fixed z converges to zero as t goes to infinity; this, in turn, implies that (J(t,z)) converges to zero because (Jleft( {t,z} right) le wleft( {t,z} right)) for all t and z. Therefore, we conclude that any mutant that has a slower wave speed than the resident can never invade the population, implying that a variant that has the maximum wave speed (v = 2sqrt {rD}) is locally evolutionarily stable.Reporting SummaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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Cropland-mapping extent and time intervalsThe global boundaries for the cropland mapping were informed by the US Geological Survey (USGS) Global Food Security-Support Analysis Data at 30 m (GFSAD)11. The cropland mapping extent was defined using the geographic 1° × 1° grid. We included every 1° × 1° grid cell that contains cropland area according to the GFSAD. Small islands were excluded due to the absence of Landsat geometrically corrected data (Supplementary Fig. 1).The cropland mapping was performed at 4-year intervals (2000–2003, 2004–2007, 2008–2011, 2012–2015 and 2016–2019). Use of a long interval (rather than a single year) increased the number of clear-sky satellite observations in the time-series, which improves representation of land-surface phenology and the accuracy of cropland detection. For each 4-year interval, we mapped an area as cropland if a growing crop was detected during any of these years. In this way, we implemented the criterion of the maximum fallow length: if an area was not used as cropland for >4 years, it was not included in the cropland map for the corresponding time interval.Landsat dataWe employed the global 16-day normalized surface reflectance Landsat Analysis Ready Data (Landsat ARD19) as input data for cropland mapping. The Landsat ARD were generated from the entire Landsat archive from 1997 to 2019. The Landsat top-of-atmosphere reflectance was normalized using globally consistent MODIS surface reflectance as a normalization target. Individual Landsat images were aggregated into 16-day composites by prioritizing clear-sky observations.For each 4-year interval, we created a single annualized gap-free 16-day observation time-series. For each 16-day interval, we selected the observation with the highest near-infrared reflectance value (to prioritize observations with the highest vegetation cover) from 4 years of Landsat data. Observations contaminated by haze, clouds and cloud shadows, as indicated by the Landsat ARD quality layer, were removed from the analysis. If no clear-sky data were available for a 16-day interval, we filled the missing reflectance values using linear interpolation.The annualized, 16-day time-series within each 4-year interval were transformed into a set of multitemporal metrics that provide consistent land-surface phenology inputs for global cropland mapping. Metrics include selected ranks, inter-rank averages and amplitudes of surface reflectance and vegetation index values, and surface reflectance averages for selected land-surface phenology stages defined by vegetation indices (that is, surface reflectance for the maximum and minimum greenness periods). The multitemporal metrics methodology is provided in detail19,38. The Landsat metrics were augmented with elevation data39. In this way, we created spatially consistent inputs for each of the 4-year intervals. The complete list of input metrics is presented in Supplementary Table 1.Global cropland mappingGlobal cropland mapping included three stages that enabled extrapolation of visually delineated cropland training data to a temporally consistent, global cropland map time-series using machine learning. At all three stages, we employed bagged decision tree ensembles40 as a supervised classification algorithm that used class presence and absence data as the dependent variables, and a set of multitemporal metrics as independent variables at a Landsat ARD pixel scale. The bagged decision tree results in a per-pixel cropland probability layer, which has a threshold of 0.5 to obtain a cropland map.The first stage consisted of performing individual cropland classifications for a set of 924 Landsat ARD 1° × 1° tiles for the 2016–2019 interval (Supplementary Fig. 1). The tiles were chosen to represent diverse global agriculture landscapes. Classification training data (cropland class presence and absence) were manually selected through visual interpretation of Landsat metric composites and high-resolution data from Google Earth. An individual supervised classification model (bagged decision trees) was calibrated and applied to each tile.At the second stage, we used the 924 tiles that had been classified as cropland/other land and the 2016–2019 metric set to train a series of regional cropland mapping models. The classification was iterated by adding training tiles and assessing the results until the resulting map was satisfactory. We then applied the regional models to each of the preceding 4-year intervals, thus creating a preliminary time-series of global cropland maps.At the third stage, we used the preliminary global cropland maps as training data to generate temporally consistent global cropland data. As the regional models applied at the second stage were calibrated using 2016–2019 data alone, classification errors may arise due to Landsat data inconsistencies before 2016. The goal of this third stage was to create a robust spatiotemporally consistent set of locally calibrated cropland detection models. For each 1° × 1° Landsat ARD tile (13,451 tiles total), we collected training data for each 4-year interval from the preliminary cropland extent maps within a 3° radius of the target tile, with preference to select stable cropland and non-cropland pixels as training. Training data from all intervals were used to calibrate a single decision tree ensemble for each ARD tile. The per-tile models were then applied to each time interval, and the results were post-processed to remove single cropland class detections and omissions within time-series and eliminate cropland patches More

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