Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality

Figure 1.  Continuously stable strategy of host resistance when the initial trait value $ \beta (0) = 0.0009 $. (a) A trade-off curve $ b(\beta) $ described in (17) with $ b_{0} = 0.01, b_{1} = 2.0 $ and $ b_{2} = 0.1 $ and a tangential critical function curve $ b_{crit}(\beta) $. The filled circle $ \beta^{*} = 0.000646 $ is an evolutionarily singular strategy. The grey area indicates combinations of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium, $ (S^{*}(\beta), I^{*}(\beta)) $, is globally asymptotically stable. (b) A pairwise invasibility plot (PIP) obtained with the trade-off function in (a). The grey areas indicate the mutant host can invade, while in green areas the mutant host becomes extinct. $ \beta^{*} = 0.000646 $ is a continuously stable strategy. (c) An evolutionary time series plot of transmission rate $ \beta $ obtained through simulation of model (11) with initial condition $ \beta (0) = 0.0009 $. (d) The basic reproduction number $ R_{0} $ against evolutionary time. (e) Equilibrium population densities of susceptible hosts $ S^{*}(\beta) $ and infected hosts $ I^{*}(\beta) $ when the transmission rate $ \beta $ evolves. Other parameter values: $ \alpha = 0.01, m_{0} = 0.005, m_{1} = 0.0001, \mu = 0.01, \sigma = 0.00001 $

Figure 2.  Continuously stable strategy of host resistance when the initial trait value $ \beta (0) = 0.0003 $. (a) An evolutionary time series plot of transmission rate $ \beta $ obtained through simulation of model (11) with initial condition $ \beta (0) = 0.0003 $. (b) The basic reproduction number $ R_{0} $ against evolutionary time. (c) Equilibrium population densities of susceptible hosts $ S^{*}(\beta) $ and infected hosts $ I^{*}(\beta) $ when the transmission rate $ \beta $ evolves. (d) The basic reproduction number $ R_{0} $ against transmission rate $ \beta $. Other parameter values are the same as in Fig. 1

Figure 3.  Evolutionary branching strategy of host resistance. (a) A trade-off curve $ b(\beta) $ described in (18) with $ b_0 = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_0 = 0.0005, \sigma_c = 0.0002 $ and a tangential critical function curve $ b_{crit}(\beta) $. The filled circle $ A = 0.000419 $ is an evolutionarily singular strategy. The grey area indicates combinations of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium, $ (S^{*}(\beta), I^{*}(\beta)) $, is globally asymptotically stable. (b) A pairwise invasibility plot (PIP) obtained with the trade-off function in (a). The grey regions indicate the mutant host can invade, while in green regions the mutant host becomes extinct. $ A, B, C $ are three evolutionarily singular strategies. (c) A mutual invasibility plot obtained with the trade-off function in (a). In the grey regions, both $ f(\beta_{m}, \beta)>0 $ and $ f(\beta, \beta_{m})>0 $. Other parameter values: $ \alpha = 0.075, m_{0} = 0.006, m_{1} = 0.00014, \mu = 0.01, \sigma = 0.00001 $

Figure 4.  Bifurcation diagram of evolutionary dynamics when the parameter of tarde-off function is changed. (a) Evolutionarily singular strategy $ \beta^{*} $ versus the maximum birth rate $ b_{1} $ when the trade-off function is the same as in Fig. 1a. (b) Evolutionarily singular strategy $ \beta^{*} $ versus the maximum birth rate $ b_{1} $ when the trade-off function is the same as in Fig. 3a. (c) Evolutionarily singular strategy $ \beta^{*} $ versus the trade-off strength $ b_{3} $ when the trade-off function is the same as in Fig. 3a. (d) Evolutionarily singular strategy $ \beta^{*} $ versus the shape parameter $ \sigma_c $ when the trade-off function is the same as in Fig. 3a. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $ CSS $ denotes the continuously stable strategy, $ EBP $ denotes the evolutionary branching point. The grey region indicates combination of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium $ (S^{*}(\beta), I^{*}(\beta)) $ is globally asymptotically stable. In non-grey region, $ I^{*}(\beta) $ equals to zero, the evolution of host resistance can not occur. In (a), other parameter values are the same as in Fig. 1. In (b), (c) and (d), other parameter values are the same as in Fig. 3.

Figure 5.  Bifurcation diagram of evolutionary dynamics when the demographic parameter is changed. (a) Evolutionarily singular strategy $ \beta^{*} $ versus the pathogen-induced mortality rate $ \alpha $ when the trade-off function is the same as in Fig. 1a. (b) Evolutionarily singular strategy $ \beta^{*} $ versus the natural death rate $ m_{0} $ when the trade-off function is the same as in Fig. 1a. (c) Evolutionarily singular strategy $ \beta^{*} $ versus the strength of density-dependent mortality $ m_{1} $ when the trade-off function is the same as in Fig. 1a. (d) Evolutionarily singular strategy $ \beta^{*} $ versus the pathogen-induced mortality rate $ \alpha $ when the trade-off function is the same as in Fig. 3a. (e) Evolutionarily singular strategy $ \beta^{*} $ versus the natural death rate $ m_{0} $ when the trade-off function is the same as in Fig. 3a. (f) Evolutionarily singular strategy $ \beta^{*} $ versus the strength of density-dependent mortality $ m_{1} $ when the trade-off function is the same as in Fig. 3a. In (a), (b), (c), (d), (e) and (f), the arrows indicate the directions of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $ CSS $ denotes the continuously stable strategy, $ EBP $ denotes the evolutionary branching point. The grey region indicates combination of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium is globally asymptotically stable. In (a), (b) and (c), other parameter values are the same as in Fig. 1. In (d), (e) and (f), other parameter values are the same as in Fig. 3

Figure 6.  Continuously stable coexistence of dimorphic susceptible hosts. The trade-off function is given by (18) with parameters $ b_{0} = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_{0} = 0.0005, \sigma_{c} = 0.0002 $. (a) A trait evolution plot. The singular strategy $ A = 0.000419 $ is an evolutionary branching point. After the branching has occurred, there exist an interior dimorphic singular strategy $ E_{1} = (0.000932, 0.000103) $ and its reflection point $ E_{2} = (0.000103, 0.000932) $. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $ \beta_{1} $ and $ \beta_{2} $. The black curves and red curves indicate the isoclines of traits $ \beta_{1} $ and $ \beta_{2} $, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. (b) A fitness landscape plot when $ (\beta_{1}^{*}, \beta_{2}^{*}) = (0.000932, 0.000103) $. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with the initial condition $ \beta(0) = 0.0005 $. At evolutionary time $ \tau_1 $ evolutionary branching occurs. (d) Equilibrium population densities of susceptible hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. (e) Equilibrium population density of infected hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. Other parameter values: $ \alpha = 0.075, m_{0} = 0.006, m_{1} = 0.00014, \mu = 0.01, \sigma = 0.00001 $

Figure 12.  The trade-off curve $ b(\beta) $. The trade-off function is given by (18). (a) The trade-off curve $ I $ is concave-convex-concave when $ \sigma_c = 0.0002 $. The trade-off curve $ II $ is also concave-convex-concave, but with a wider convex region when $ \sigma_c = 0.0005 $. Other parameter values: $ b_{0} = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_{0} = 0.0005 $. (b) The trade-off curve $ I $ looks like a sigmoidal curve when $ b_{1} = 5.5 $ and $ \sigma_c = 0.008 $. The trade-off curve $ II $ looks also like a sigmoidal curve, but with a larger maximum birth rate when $ b_{1} = 10.0 $ and $ \sigma_c = 0.0085 $. Other parameter values: $ b_{0} = 0.05, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018 $

Figure 7.  Continuously stable coexistence of boundary dimorphism. The trade-off function and parameter values are the same as in Fig. 6 except for $ \sigma_{c} = 0.0005 $. (a) A trait evolution plot. The singular strategy $ A = 0.000492 $ is an evolutionary branching point. After the branching has occurred, there is no interior dimorphic singular strategy. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $ \beta_{1} $ and $ \beta_{2} $. The black curves and red curves indicate the isoclines of traits $ \beta_{1} $ and $ \beta_{2} $, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable. (b) A fitness landscape plot when $ (\beta_{1}^{*}, \beta_{2}^{*}) = (0.001113, 0.0) $. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with the initial condition $ \beta(0) = 0.0006 $. (d) Equilibrium population densities of susceptible hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. (e) Equilibrium population density of infected hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve

Figure 8.  Bifurcation diagram of monomorphic evolutionary dynamics when one of the parameters in the model is changed. The trade-off function is given by (18) with parameters $ b_{0} = 0.05, b_{1} = 5.5, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018 $ and $ \sigma_c = 0.008 $. (a) Evolutionarily singular strategy $ \beta^{*} $ versus the maximum birth rate $ b_{1} $. (b) Evolutionarily singular strategy $ \beta^{*} $ versus the shape parameter $ \sigma_c $. (c) Evolutionarily singular strategy $ \beta^{*} $ versus the natural death rate $ m_{0} $. (d) Evolutionarily singular strategy $ \beta^{*} $ versus the strength of density-dependent mortality $ m_{1} $. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $ CSS $ denotes the continuously stable strategy, $ EBP $ denotes the evolutionary branching point. The grey region indicates combination of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium $ (S^{*}(\beta), I^{*}(\beta)) $ is globally asymptotically stable

Figure 9.  Evolutionary extinction of one emerging branch. The trade-off function is given by (18) with parameters $ b_{0} = 0.05, b_{1} = 5.5, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018 $ and $ \sigma_c = 0.008 $. (a) A pairwise invasibility plot. (b) A trait evolution plot. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $ \beta_{1} $ and $ \beta_{2} $. The black curves and red curves indicate the isoclines of traits $ \beta_{1} $ and $ \beta_{2} $, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with initial condition $ \beta(0) = 0.0015 $. At evolutionary time $ \tau_1 $ evolutionary branching occurs and at evolutionary time $ \tau_2 $ the emerging branch $ \beta_{2} $ become extinct. (d) Equilibrium population densities of susceptible hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. At evolutionary time $ \tau_2 $, the equilibrium population density of susceptible hosts $ S_{2} $ becomes zero. The dashed curve indicates the total equilibrium population density of the two types of susceptible hosts. (e) Equilibrium population density of infected hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. Other parameter values: $ \alpha = 0.075, m_{0} = 0.0055, m_{1} = 0.000098, \mu = 0.01, \sigma = 0.00005 $

Figure 10.  Bifurcation diagram of monomorphic evolutionary dynamics when one of the parameters in the model is changed. The trade-off function and parameter values are the same as in Fig. 9 except for $ b_{1} = 10.0 $ and $ \sigma_c = 0.0085 $. (a) Evolutionarily singular strategy $ \beta^{*} $ versus the maximum birth rate $ b_{1} $. (b) Evolutionarily singular strategy $ \beta^{*} $ versus the shape parameter $ \sigma_c $. (c) Evolutionarily singular strategy $ \beta^{*} $ versus the natural death rate $ m_{0} $. (d) Evolutionarily singular strategy $ \beta^{*} $ versus the strength of density-dependent mortality $ m_{1} $. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $ CSS $ denotes the continuously stable strategy, $ EBP $ denotes the evolutionary branching point. The grey region indicates combination of $ \beta $ and $ b(\beta) $ for which the endemic equilibrium $ (S^{*}(\beta), I^{*}(\beta)) $ is globally asymptotically stable

Figure 11.  Evolutionary extinction of one existing branch. The trade-off function and parameter values are the same as in Fig. 9 except for $ b_{1} = 10.0 $ and $ \sigma_c = 0.0085 $. (a) A pairwise invasibility plot. (b) A trait evolution plot. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $ \beta_{1} $ and $ \beta_{2} $. The black curves and red curves indicate the isoclines of traits $ \beta_{1} $ and $ \beta_{2} $, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curve indicates the evolutionarily singular strategies which are not evolutionarily stable. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with initial condition $ \beta(0) = 0.0006 $. At evolutionary time $ \tau_1 $ evolutionary branching occurs and at evolutionary time $ \tau_2 $ the existing branch $ \beta_{1} $ become extinct. (d) Equilibrium population densities of susceptible hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve. At evolutionary time $ \tau_2 $, the equilibrium population density of susceptible hosts $ S_{1} $ becomes zero. The dashed curve indicates the total equilibrium population density of the two types of susceptible hosts. (e) Equilibrium population density of infected hosts when the traits $ \beta_{1} $ and $ \beta_{2} $ evolve