Systems Biology of Epidemiology: From Genes to Environment

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1 Systems Biology of Epidemiology: From Genes to Environment MBI CTW: From Within-Host Dynamics to the Epidemiology of Infectious Disease Juan B. Gutierrez UGA s Department of Mathematics & Institute of Bioinformatics April 10, / 40

2 Collaborators MaHPIC Project. PI: Mary Galinski (Emory), Jessica Kissinger (UGA), Alberto Moreno (Emory). NIAID Grant HHSN C ( ). ICEMR Project. PI: Socrates Herrera (Centro de Investigación Caucaseco, Colombia), Myriam Arevalo (Centro de Investigación Caucaseco, Colombia), Martha Quinones (Universidad Nacional de Colombia). NIAID Grant U19AI ( ). 2 / 40

3 The Pioneers Left to right: Ronald Ross, Robert Koch, Giovanni Battista Grassi. With the use of a microscope, the pathogens that inspired them look identical. 3 / 40

4 Epidemics are Driven by Cellular and Molecular Interactions The circulation of pathogens is the result of hosts shedding infectious agents in the environment. The subsequent pathogen uptake and colonization of suitable hosts is the fundamental cause of epidemics. Due to a host s immune response, exposure to low levels of a pathogen may not result in infection; furthermore, pathogens may be unable to find a within-host substrate before degradation. Therefore, the onset of infection requires the direct or indirect (vector-driven) uptake of a minimum dose of infectious agents (virus, bacteria, fungi, protozoa, or helminths). 4 / 40

5 Epidemics are Driven by Cellular and Molecular Interactions The minimum infectious dose that colonizes a hosts with 50% probability of success is known as ID50. It has a very large range of variation, from a minimum of one infectious agent for e.g. Coxiella burnetii to > for Gardnerella vaginalis (Gama JA, Abby SS, Vieira-Silva S, Dionisio F, Rocha EPC (2012) Immune Subversion and Quorum-Sensing Shape the Variation in Infectious Dose among Bacterial Pathogens. PLoS Pathog 8(2): e doi: /journal.ppat ). Hence, ID50 can be interpreted as an Allee effect for the onset of infection. 5 / 40

6 Epidemics are Driven by Cellular and Molecular Interactions Microdiversity among strains of the vast majority of pathogens is extensive; each genotype infecting a host can present significant differences in virulence, immunogenicity, and antigenic variation. Pathogens in circulation are not uniform; instead, they form a distribution defined by the expression of different genetic, pathogenic and population dynamic traits. The circulation of pathogens with different genotypes is multifactorial and can depends heavily on human movement dynamics, and, in some situations, vector availability and competence. 6 / 40

7 Immunoepidemiology Desowitz RS, Saave JJ. The Application of the Haemagglutination Test to a Study of the Immunity to Malaria in Protected and Unprotected Population Groups in Australian New Guinea. Bull. Wld Hlth Org. 32: (1965). The first explicit reference to immuno-epidemiology seems to be: Desowitz RS, Saave JJ, Stein B. The application of the indirect haemagglutination test in recent studies on the immuno-epidemiology of human malaria and the immune response in experimental malaria. Mil Med.,131: (1966). Since then, there have been multiple attempts to link within-host dynamics and between-host transmission. 7 / 40

8 Challenges of omic data Thousands of variables, but significantly less samples. Traditionally, a t-test produces a p-value. The null hypothesis (there is no relationship) is rejected (there is relationship) when p < Multiple tests have to use a smaller p-value. Naive example: 15,000 genes per sample. Then, the null hypothesis would be rejected when p < 0.05/15, We could use only a dozen variable out of 15,000 with this approach. False Discovery Rate (FDR) is used instead. FDR procedures are designed to control the expected proportion of incorrectly rejected null hypotheses. In the previous example, with FDR = 5%, we could use 150 variables. Benjamini, Yoav, and Yosef Hochberg. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B (Methodological) (1995): / 40

9 Molecular Changes vs. Physiological State Image source: MaHPIC Project, Gutierrez Lab 9 / 40

10 Molecular Changes vs. Physiological State Image source: MaHPIC Project, Gutierrez Lab 10 / 40

11 Molecular Changes vs. Physiological State Image source: 11 / 40

12 Molecular Changes vs. Physiological State Image source: 12 / 40

13 Molecular Changes vs. Physiological State Image source: MaHPIC Project, Gutierrez Lab 13 / 40

14 Molecular Changes vs. Physiological State Image source: MaHPIC Project, Gutierrez Lab 14 / 40

15 Introduction Omic Data P0 Between-Host Implementation Within-Host Dynamics Image source: Yan, Gutierrez, in preparation (2013) 15 / 40

16 Within-Host Dynamics Image source: Yan, Gutierrez, in preparation (2013) 16 / 40

A Change in Perspective Image source: http://www.biology.

17 A Change in Perspective Image source: 17 / 40

18 A Shift in Paradigm A simple model that takes into account the immune system. Let p be pathogen population density, and m represent immune cell population. ṗ = νp(1 p)(p σ) µmp, ṁ = mp ηm, (1) ξ + p ν is the rate of growth of the pathogen population. σ is the Median Infective Dose (MID50). µ is the rate of removal of pathogens by the immune response. ξ is the pathogenic load that elicits half the immune response. η is the rate of removal of immune cells. 18 / 40

19 A Shift in Paradigm A simple model that takes into account the immune system. Let p be pathogen population density, and m represent immune cell population. Image source: Gutierrez (2013, in preparation) 19 / 40

20 Basic Propagation Number Now consider the system p = D p + νp(1 p)(p σ) µmp, t (2) m t = mp ηm, ξ + p (3) Multiply Equation 2 through by p and integrate both sides over Ω to yield p p t dx = D p p dx+ν (1 p)(p σ)p 2 dx µ mp 2 dx. (4) Ω Green s first identity, p p dx = p p n ds Ω Ω Ω Ω Ω p 2 dx = Ω Ω p p n ds p 2 dx, Ω is applied, where n is the outward pointing unit vector normal to the surface element ds, and p/ n is the directional derivative. Note that under Dirichlet and Neumann boundary conditions, the boundary integral becomes zero. 20 / 40

21 Basic Propagation Number Therefore, Also note that Ω Ω p pdx = p 2 dx = p 2 2. (5) Ω p p t dx = 1 p 2 dx = 1 2 t Ω 2 p 2 2. (6) t Substitute Equations 5 and 6 into Equation 4 to obtain 1 p 2 2 = D p 2 2 +ν (1 p)(p σ)p 2 dx µ mp 2 dx, (7) 2 t from where 1 p t Ω D p ν Ω (1 p)(p σ) dx. (8) Ω 21 / 40

22 Basic Propagation Number Use Poincare s inequality p 2 2 C p 2 2 to yield 1 p 2 2 D 2 t C p ν (1 p)(p σ) dx Ω D C p ν (p σ p 2 + σp) dx Ω ( p 2 2 ν + D ) + ν(1 + σ)p νσω C where P is the total amount of pathogen in the environment, and Ω is the size of the domain. This equation can be rearranged as an ODE on p 2 2 : ( p ν + D ) p ν[σ(ω P) P] 0 C 22 / 40

23 Basic Propagation Number The solution to the differential inequality ṗ + ap + b 0 subject to p(0) = p 0 is p b ( a + e at p 0 + b ). a Then, there is a time given by t 0 = ln ( p 0 + b a a such that the following uniform estimate holds p a b a ) 23 / 40

24 Basic Propagation Number In terms of the original differential inequality, p 2 σ(ω P) P 2 1 ( ) 1 + D νc The interpretation of this result is that to minimize the density of pathogen, and hence stop transmission: Increase MID50. Decrease pathogen in the environment. Decrease size of the affected domain. Increase diffusivity. Decrease rate of infection. 24 / 40

25 Between-Host Dynamics Carey, H. C. (1867). Principles of social science (Vol. 3). JB Lippincott & Company. Ch 2 pg 41: As planets gravitate towards each other, man tends towards his fellow man ; Man, the molecule of society Reilly, W. J. (1931). The law of retail gravitation. Any city draws trade from its neighboring land. The trade between cities is inversely proportional to the square of the distance between the two cities and proportional to the population of each city. Elder, RF (Ed.) The American economic review. Volume 21 (1931), pp Zipf, G. K. (1946). The P 1 P 2 /D hypothesis: on the intercity movement of persons. American sociological review, 11(6), Wilson, A. G. (1967). A statistical theory of spatial distribution models. Transportation research, 1(3), / 40

26 Diffusion on a Manifold Implicit movement characterized by a gravity model can be made explicit through the use diffusion on a low-dimensional manifold reconstructed from effort of transportation between individuals. Given a planar graph representing the distribution of human population in a geographic area, where the vertices represent the population centers, and the edges represent the transportation network, then it is possible to assign weights to the edges so that each weight represents the time required to travel between two vertices. 26 / 40

27 Diffusion on a Manifold Figure: Manifold that results from considering the quality of roads between towns (left), and the spline smoothing (right) based on bivariate splines. 27 / 40

28 Diffusion on a Manifold Consider T u M, the tangent space of M for each point u in the following way p(x, t) t = D (C( T u M ) p(x, t)) + F (p(x, t), m(x, t)), x = (x, y) Ω R 2, t 0, where T u M represents a metric of the tangent space of M, such that 1 C(S) =. 1 + S 2 28 / 40

29 Key Aspects of Infectious Disease Can Hardly Be Studied with Existing Paradigms Microdiversity among strains of the vast majority of pathogens is extensive; each genotype infecting a host can present significant differences in virulence and immunogenicity. Simultaneous infection: Simultaneous presence of several distinct pathogen genomes, from the same or multiple species. Antigenic diversity: Antigenic differences between pathogens in a population. Antigenic variation: Ability of a pathogen to change antigens presented to the immune system during an infection. 29 / 40

30 Asymptomaticity is Very Important Image source: / 40

31 Vector Behavior is Very Important Image source: CLAIM Project, / 40

32 Vector Behavior is Very Important Image source: CLAIM Project, / 40

33 The Malaria Case 33 / 40

34 The Malaria Case Consider the following populations: Human: A set H = {h i }. Mosquito: A set M = {m j }. Plasmodium: A set of classes P = {p kl } representing population density of different genotypes k and different life stages l. 34 / 40

35 J p ik1 e i ṗ ik1 = λ ji h(d ij )p nk1 τ 2 p ik1 τ 3 p ik1 ω 1 (9) n=1 1 + γ 1 p ik1 ( ṗ ik2 =τ 2 p ik1 + φ 2 p ik2 1 p ) ik2 + p ik3 p ik2 e i τ 4 p ik2 ω 2 (10) C L 1 + γ 2 p ik2 ( ṗ ik3 =τ 3 p ik1 + φ 3 p ik3 1 p ) ik2 + p ik3 p ik3 e i ζ 4 p ik3 ω 3 C L 1 + γ 3 p ik3 (11) ( ṗ ik4 =τ 4 p ik2 + ζ 4 p ik3 + φ 5 p ik5 1 p ) ik4 p ik4 e i τ 5 p ik4 τ 6 p ik4 ω 4 C M 1 + γ 4 p ik4 (12) ( ṗ ik5 =τ 5 p ik4 + φ 5 p ik5 1 p ) ik5 K p + ρ p is5 p ik5 ik5 e i ω 5 C RBC 1 + γ s=1,s j 5 p ik5 (13) ḟ ik = 1 ( 2 τ 6p ik4 1 f ) ik + m ik J p ikf e i λ ij h(d nj )f nk ω f C G n=1 1 + γ f p ikf (14) ṁ ik = 1 ( 2 τ 6p ik4 1 f ) ik + m ik J p ikm e i λ ij h(d nj )m nk ω m C G n=1 1 + γ mp ikm (15) 1..5,f,m ė i = κ l l=1 p ikl e i ɛe i (16) 1 + γ l p ikl I ṗ jk1 = λ ij h(d ij ) ( ) ( ) f qk + m qk τ1 p jk1 δ 6 fqk + m qk q=1 (17) 35 / 40

36 The Malaria Case Human: A set H = {h i }, with I = total number of humans, Mosquito: A set M = {m j }, with J = total number of mosquitoes. Plasmodium: A set of classes P = {p kl } representing population density of different genotypes k and different life stages l. ṗ ik1 =... ṗ jk1 = J p ik1 e i λ ji h(d ij )p nk1 τ 2 p ik1 τ 3 p ik1 ω γ 1 p ik1 n=1 I λ ij h(d ij ) (f qk + m qk ) τ 1 p jk1 δ 6 (f qk + m qk ) q=1 New data to be collected: Human movement and robust detection of anopheline and Plasmodium species is required, e.g. using environmental mitochondrial DNA. 36 / 40

37 From Cells to Continent Image source: Gutierrez, CLAIM Project (2013) 37 / 40

38 Summary Within-host and between-models can be linked through: A variable in the between-host models that represents time since infection. Within-host dynamics determining pathogen shedding, and network dynamics to determine contact between hosts. Within-host dynamics takes int account immune response to generate host classes with different degrees of immunogenicity. etc. 38 / 40

39 The Bottom Line Traditional methods to study epidemiology can t deal with emerging information about large families of pathogens. This research tries to condensate in a unified framework within-host dynamics and epidemiological processes. Within-host dynamics has to take into account the immune system and the cascade of molecular signaling, hence the systems biology in this approach to epidemiology. 39 / 40

40 THANKS FOR YOUR ATTENTION! Juan B. Gutierrez 40 / 40

MULTI-SCALE MODELING OF MALARIA: FROM ENDEMICITY TO ELIMINATION

MULTI-SCALE MODELING OF MALARIA: FROM ENDEMICITY TO ELIMINATION or the DEATH of SEIR models Juan B. Gutierrez UGA s Department of Mathematics & Institute of Bioinformatics December 14, 2012 1 / 19 Collaborators