1. DISEASE EFFECTS ON REPRODUCTION CAN CAUSE POPULATION CYCLES IN SEASONAL ENVIRONMENTS <br />By: Matthew J. Smith, Andrew White, Jonathan A. Sherratt, Sandra Telfer, Michael Begon and Xavier Lambin<br />Presented by: Ravi Shanker Pandey, Hector Cuesta-Arvizu, Oleg Kolgushev and Jorge Reyes-Silveyra<br />October 2011<br />
2. Introduction<br />What are the effects of disease on host population dynamics?<br /><ul><li>Fecundity Rates
3. Survival Rates
4. Combination of both</li></li></ul><li>Introduction<br />Even more complex:<br /><ul><li>Disease effects do not operate independently
5. Environmental variation
6. Coexistence of parasites in the same host
7. Seasonal variation </li></li></ul><li>Introduction<br /> Understanding host-parasite seasonality variations are crucial for prediction of emergence of zoonoses or species survival<br />
8. Disease dynamics Model<br />Mathematic modeling of disease dynamics of seasonally reproducing host populations<br />Effects of disease that induce<br /><ul><li>A period of no reproduction
9. Reduce fecundity after recover</li></ul>Other models do not consider no reproduction period or seasonality<br />
10. Disease Dynamics Model<br />Data available for multiple species and populations:<br /><ul><li>Population growth rates
11. Survivorship rates and seasonality in reproduction
12. Long-term host dynamics
13. Impact of disease on host fecundity is known in some cases</li></li></ul><li>Populations<br />Wide variety of pathogens have been detected in microtine rodent populations:<br /><ul><li>Most effects are unknown
14. Increase mortality rates
15. Rodent population crashes</li></li></ul><li>Cowpox Effects<br /><ul><li>Delays reproduction until next breeding season for juvenile females
16. Reproductive maturity is key in survival
17. Increase or decrease survival depending on the population
18. Significance for effects in long term has not been studied before</li></li></ul><li>Multi year cycle<br /><ul><li>Do disease regulate or induce multiyear cycles in seasonally reproducing populations by influencing reproductive timing and reproductive output?
19. Mathematical model with representative parameters
20. Studied in a unique population of rodents
21. Delayed density-dependent season length and sero prevalence are predicted as epiphenomena of disease-induced multiyear cycles?</li></li></ul><li>Methods <br />Parameter Definition and Model Structure:<br />Host- population density is divided into four classes:-<br /><ul><li>S = Susceptible to microparasitic infection</li></li></ul><li>Methods <br />Parameter Definition and Model Structure:<br />Host- population density is divided into four classes:-<br /><ul><li>S = Susceptible to microparasitic infection
22. I = Infected individual that cannot reproduce</li></li></ul><li>Methods <br />Parameter Definition and Model Structure:<br />Host- population density is divided into four classes:-<br /><ul><li>S = Susceptible to microparasitic infection
23. I = Infected individual that cannot reproduce
24. Y = Recovered and immune individual that cannot reproduce</li></li></ul><li>Methods <br />Parameter Definition and Model Structure:<br />Host- population density is divided into four classes:-<br /><ul><li>S = Susceptible to microparasitic infection
25. I = Infected individual that cannot reproduce
26. Y = Recovered and immune individual that cannot reproduce
27. Z = Recovered individual that can reproduce,but at lower rate</li></li></ul><li>Parameters<br /><ul><li>A(t) = a T < t < T+L; => reproductive season</li></ul> = 0 T+L < t < T+1 => non-reproductive season<br /><ul><li>L : reproductive season length,
28. N : S + I + Y + Z; Total population density,
29. b : disease-free per capita death rate, b < a;
34. τ : rate of leaving immune but non-reproductive class.
35. f : Reduced birth rate following recovery; 0 < f < 1.</li></li></ul><li>Parameters<br />& total average reproductive delay = (1/γ) + (1/τ);<br />Average recovery time (y or t =0.5 - 52 per year) : 2 years - 1 week;<br />Five combinations of the rodent specific parameters from publish data sets were choosen to represent estimates from variety of rodent population:<br /><ul><li>UK field voles in grassland habitat (Kielder Forest),
36. UK bank voles in mixed wodland (Manor Wood),
37. Field voles in Fennoscandian grassland (Northan Finland),
38. Japanese grey-sided voles in natural woodland (Hokkaido),
39. French common voles in agricultural habitat (South-western France).</li></li></ul><li> Methods<br />The change in densities of the host classes over continuous time t, is modelled with the four ordinary differential equations:<br />
40. Methods<br />The change in densities of the host classes over continuous time t, is modelled with the four ordinary differential equations:<br /><ul><li>dS/dt = A(t)(S + fZ)(1 - qN) - βSI – bs, (1)</li></li></ul><li> Methods<br />The change in densities of the host classes over continuous time t, is modelled with the four ordinary differential equations:<br /><ul><li>dS/dt = A(t)(S + fZ)(1 - qN) - βSI – bs, (1)
41. dI/dt = βSI – (b+α+γ)I, (2)</li></li></ul><li> Methods<br />The change in densities of the host classes over continuous time t, is modelled with the four ordinary differential equations:<br /><ul><li>dS/dt = A(t)(S + fZ)(1 - qN) - βSI – bs, (1)
42. dI/dt = βSI – (b+α+γ)I, (2)
43. dY/dt = γI – (b+τ)Y, (3)</li></li></ul><li> Methods<br />The change in densities of the host classes over continuous time t, is modelled with the four ordinary differential equations:<br /><ul><li>dS/dt = A(t)(S + fZ)(1 - qN) - βSI – bs, (1)
44. dI/dt = βSI – (b+α+γ)I, (2)
45. dY/dt = γI – (b+τ)Y, (3)
46. dZ/dt = τy – bZ (4)</li></li></ul><li>Infection Threshold:<br />For the infected population to increase:<br />dI/dt > 0 => βSI > (b+α+γ)I <br /> => S > (b+α+γ)/β = SC<br />dI/dt = 0 => βSI = (b+α+γ)I <br /> If we say; R = βS/ (b+α+γ);<br />then I can only increase if R > 1.<br />
50. dZ/dt = τY – bZ, (8)</li></li></ul><li>Non-reproductive season dynamics :<br />case 1: When disease is absent (I=Y=Z=0);<br />monotonically S -> 0,<br />case 2: When disease is present; <br /> All population component densities exponential decay to zero: <br /> S = I= Y = Z= 0.<br />
54. dZ/dt = τY – bZ. (12)</li></li></ul><li>Reproductive season dynamics :<br />case 1: S = I= Y = Z= 0,<br />
55. Reproductive season dynamics :<br />case 1: S = I= Y = Z= 0,<br />case 2: When disease is absent (I=Y=Z=0);<br /> monotonically S -> K, <br />
56. Reproductive season dynamics :<br />case 1: S = I= Y = Z= 0,<br />case 2: When disease is absent (I=Y=Z=0);<br /> monotonically S -> K,<br />case 3: When disease is endemic in population ; <br /> S' = (b+α+γ)/β,<br /> Y' = Iγ/(b+τ) , & Z' = Iγτ/b(b+τ)<br />
57. If the reproductive season is less than a critical length (L < bla) then the host population will decay to cero over time.<br />If L > bla then the long-term dynamics will be annual cycles that are repeated exactly each year.<br />The Effects of Seasonal Birth Rates on Long-term Host – Parasite Dynamics<br />
58. Types of long-term multiyear dynamics.<br />1.- disease die out over time.<br /> S-host < Sc<br /> S-host >Sc<br />2.- disease remain endemic <br />en in the susceptible population <br />The Effects of Seasonal Birth Rates on Long-term Host – Parasite Dynamics<br />
59. The Effects of Seasonal Birth Rates on Long-term Host – Parasite Dynamics<br />3.- disease again remain endemics, but the population density oscillations are not repeated exactly every year.<br />Regularity repeated multiyear cycles or quasi-periodic multiyear circles (m-r) <br />
60. Why does the model predict a variety of multiyear dynamics?<br />(a-x) transition from regular annual cycles to multiyear cycles.<br />(a-f) with the population densities of all four population components peaking once each year.<br />(g-l) low-amplitude 2-year multiyear cycles <br />
61. Why does the model predict a variety of multiyear dynamics?<br />(a-f) to (g-l) reveals that the critical reproductive lag at which this transition occurs 𝜏𝑐 is dependent upon the initial population densities used in simulations. <br /> <br />
62. Why does the model predict a variety of multiyear dynamics?<br />(s-x) the population dynamics to settle on regular 2-year cycles, In this case the infected population density peaks every 2 years exactly (v). <br />
63. THE IMPORTANCE OF SEASONAL FORCING INDETERMINING THE MULTIYEAR DYNAMICS<br /><ul><li>Critical reproductive season length necessary for the voles to exist L = b/a = 0·35
64. 0·35 < L < 0·39 the hosts exist </li></ul> at low densities, no endemic<br /><ul><li>The disease persists, but low</li></ul> impact on the host population <br /> 0·39 < L < 0·41.<br /><ul><li>The disease induces multiyear </li></ul>cycles for all τ, 0·41 < L < 0·57.<br /><ul><li>L = 0·78 transition between </li></ul> outbreaks 1/year (lower L) <br /> and 2/year (higher L).<br /><ul><li>L > 0·94 multiyear cycles do</li></ul> not occur for any values of τ.<br />
65. SYSTEMATIC SAMPLING OF PARAMETER SPACE: HOWCOMMON COULD DISEASE-INDUCED CYCLES BE?<br />What sorts of values of τ and f necessary to induce multiyear cycles? Figure shows how the dominant period or amplitude of the multiyear cycles predicted by the model are affected by variation in four of the disease parameters for four sets of rodent parameter values. <br />Varying β has less effect on the model predictions than varying the other parameter values for endemic in the population. <br />Regular annual cycles are predicted when the rate of recovery of reproductive function is fast and f is large (1/τ = 7 days, f = 1) and multiyear cycles are predicted when the rate of recovery of reproductive function is slow and f is small (1/τ ≥ 1 year, f = 0). <br />Between these two corners of 1/τ – f space there is a bifurcation structure with a region of low amplitude. <br />Increasing the time taken to recover reproductive function results in multiyear population cycles being predicted at higher values of f. <br />Increasing α increases the size of the region predicting multiyear cycles in the individual 1/τ against f plots. <br />The size of parameter space predicting multiyear cycles sometimes initially contracting, and then expanding, as 1/γ is increased from 7 days to 2 years. In rodent populations, long reproductive delays following infection (1/τ and 1/γ ≥ 1 year) would mean that most rodents would die before recovering their reproductive ability. A small f would mean that even if individuals did recover they would not make a large per capita contribution to the susceptible population. Low values of τ, γ and f therefore cause the time taken for the susceptible population to recover from a disease to be relatively long and this can induce multiyear cycles. However, a rapid recovery rate from infection (1/γ ≤ 1 month) can also increase the time it takes the susceptible population to exceed SC. This is because increasing γ also increases SC, and if the recovery rate is sufficiently rapid (1/γ ≤ 1 month) SC can get close to, but still be less than, K. In these scenarios, density dependence slows down the growth of the susceptible population as it approaches SC. This increases the date at which the infection threshold is exceeded, and this extra time delay is sufficient to cause multiyear cycles in these cases. <br />
66. Kielder forest field voles<br />A relatively high proportion of simulations with the Kielder Forest and Manor Wood parameters (when disease is endemic) predict multiyear cycles. This is associated with their lower (estimated) maximum per capita growth rates (r = 2·5 and 1·8, respectively) which increase the time taken for the susceptible population to recover following a disease outbreak. <br />
67. French common voles<br />The high maximum population density of the French common vole populations (K = 2000 voles ha–1) allows them to support endemic infections with lower infection rates. In this scenario, multiyear cycles are associated with slow rates of recovery of reproductive function and small f, for all values of β analyzed. <br />
68. Northern Fennoscandian field voles<br />Lowest estimated maximum population densities and highest estimated maximum growth rates - Endemic infections are predicted only at high infection rates for these parameter combinations. Multi-year cycles are also less commonly predicted for these populations and are usually seen only when the rate of recovery of reproductive function is very slow or f is extremely low (1/τ = 2 years, f = 0). <br />
69. DETAILED ANALYSIS INTO THE POSSIBLE EFFECTS OFCOWPOX IN THE KIELDER FOREST SYSTEM<br />The model predicts that if cowpox virus infection has a sufficiently large impact on rodent fecundity following recovery from infection ( f = 0·2) then the host population will exhibit multiyear cycles for all reproductive lags (Fig). <br />However, as f is increased from 0·2, the delay in reproduction required to produce multiyear cycles increases until, beyond f = 0·7, the model does not predict multiyear cycles for any values of τ analyzed. <br />Therefore, for the range of τ and f used in Fig., the model predicts that in order to induce multiyear cycles in the Kielder Forest field voles, a disease such as cowpox would have to induce significant chronic reductions in fecundity following recovery of reproductive function ( f < 0·4).<br />Plots of 1/τ against f for different values of the other disease parameters did not differ qualitatively from those in Fig.<br />
70. DOES THE MODEL PREDICT DELAYEDDENSITY-DEPENDENT REPRODUCTIVE TIMINGAND SEROPREVALENCE IN CASES OFDISEASE-INDUCED MULTIYEAR CYCLES?<br />The effective delay in the onset of the reproductive season is the difference between ‘effective onset’ and the actual onset of the reproductive season. <br />The strength of linear correlations between the effective delay in onset of the reproductive season and population densities and at varying times in the past was measured. <br />Investigated, whether seroprevalence at the start of the reproductive season, measured as the proportion of the total population in the I, Y and Z classes, was correlated significantly with the population density at various times in the past. Figure illustrates, (Kielder Forest ) that both delayed density-dependent reproductive timing and seroprevalence can be predicted by the model when it predicts quasi- periodic multiyear cycles. <br />Figure (a) shows that the effective date at which the reproductive season starts is correlated most highly with the population density around 6 months previously. <br />Figure (b) plots the data for the time at which this most significant correlation occurs. <br />Seroprevalence is also correlated most significantly with the population density around 6 months previously. This means that the model predicts a high seroprevalence (over 90% of individuals with cowpox antibodies in (d) and a long delay in the onset of the reproductive season (around 4 months in b) at the actual onset of the reproductive season, if total rodent density 6 months previously was high, and vice versa.<br />
71. Discussion<br /><ul><li>A period of no reproduction following recovery from infection has been incorporated in this studies.
72. This period and reduced fecundity can destabilize the multiyear dynamics of host populations.
73. Seasonality is key determinant of multiyear dynamics of rodent populations and incorportaion of such seasonality in other parameters would probably alter the predicted dynamics and could also lead to multiyear cycle.
74. Inclusion of demographic and environemntal stochasticity could result in local extinction of disease or of disease and the host population together.</li></li></ul><li>Discussion<br /><ul><li>Two previously unexplained phenomenaare predicted by model and it support the hypothesis that disease may play a role in multiyear population dynamics in Kielder Forest voles.
75. Empirical studies should also investigate for other diseases and other rodent population to generalize these findings.
76. The incorporation of different life stages could change model predictions and stabilize multiyear host dynamics, as only a fraction of total population would be affected significantly.
77. Could repeated multiyear cycles in certain microtine rodents be caused by diseases acting singly or in combination? </li></li></ul><li>
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