DIPARTIMENTO DIELETTRONICA, INFORMAZIONE E BIOINGEGNERIA
DOCTORALPROGRAMME IN INFORMATIONTECHNOLOGY
(SYSTEMS & CONTROLAREA)
ANALYSIS,
CONTROL AND FORECAST OF
SCHISTOSOMIASIS SPATIOTEMPORAL DYNAMICS
VIA NETWORK MODELLING
Doctoral Dissertation of:
Manuela Ciddio
Supervisor:
Prof. Marino Gatto
Tutor:
Prof. Paolo Bolzern
The Chair of the Doctoral Program:
Prof. Andrea Bonarini
S
CHISTOSOMIASIS is a parasitic, water-related disease that is preva-lent in tropical and subtropical areas of the world, causing severe and chronic consequences especially among children. In this dis-sertation, different modelling frameworks are proposed, focusing on the main environmental and socioeconomic aspects considered to be relevant in schistosomiasis spread. These models are used to analyze transmission patterns in different settings, ranging from purely theoretical ones to real case studies. First, the mechanisms that drive the temporal variability of disease severity and prevalence are explored introducing nonlinearities in demographic and epidemiological dynamics. Then, the impacts of differ-ent sources of local and spatial heterogeneity are investigated, together with their implications on effectiveness of possible intervention strategies. Spa-tially explicit network models, properly informed by socioeconomic and environmental data, are thus used to study the spread of schistosomiasis in Senegal, where the urogenital form of the infection is widespread. The analysis is performed by integrating proxies of human mobility (inferred from a very large database of mobile phone traces) with a geospatial anal-ysis which includes georeferenced data on demography, water supply/sani-tation, and schistosomiasis prevalence. Results are presented and discussed in the perspective of using epidemiological models as tools for disease con-trol. In this respect, the effects of intervention strategies based on human development, exposure and contamination prevention, awareness about risk factors, and biological control of snail intermediate hosts are evaluated by means of model simulation.I INTRODUCTION 1
1 Objectives and thesis structure 3
1.1 Research focus . . . 3
1.2 Main results . . . 5
1.3 Thesis outline . . . 6
1.4 Research works . . . 7
2 Schistosomiasis 9 2.1 Epidemiology and pathophysiology . . . 9
2.2 Life cycle and schistosomiasis transmission . . . 12
2.3 Schistosomiasis models: state-of-the-art . . . 13
II THEORETICAL INVESTIGATIONS 23 3 Temporal patterns of schistosomiasis dynamics 25 3.1 Age-of-infection structured model . . . 26
3.1.1 Model formulation . . . 26
3.1.2 Parameter estimation . . . 30
3.1.3 Results . . . 32
3.2 SEI-like compartmental model . . . 34
3.2.1 Model formulation . . . 34
3.2.2 Parameter estimation . . . 35
3.2.3 Equilibria and stability . . . 38
3.2.5 Periodically forced model . . . 45
3.3 Conclusions . . . 49
4 Sources of heterogeneity in schistosomiasis dynamics 53 4.1 Multi-group model formulation . . . 54
4.2 Local heterogeneities in schistosomiasis transmission . . . . 55
4.2.1 Group heterogeneity . . . 56
4.2.2 Source heterogeneity . . . 58
4.3 Intervention strategies for schistosomiasis control . . . 61
4.4 Conclusions . . . 64
III THE CASE STUDY OF SENEGAL 69 5 The study area 71 5.1 Administrative boundaries and population distribution . . . 71
5.2 Water resources, sanitation conditions, poverty . . . 72
5.3 Schistosomiasis in Senegal . . . 75
5.4 Human movement patterns . . . 76
5.4.1 The analysis of Call Detail Records (CDRs) . . . 77
5.4.2 Human mobility fluxes across Senegal . . . 80
6 Country-wide network model 85 6.1 Methods . . . 85
6.1.1 Spatially explicit network model . . . 86
6.1.2 Simplifying hypotheses . . . 88
6.1.3 Model set-ups . . . 90
6.2 Application of the model to Senegal . . . 90
6.2.1 Administrative boundaries and population distribution 90 6.2.2 Human exposure and contamination . . . 90
6.3 Model simulation, calibration and comparison . . . 93
6.3.1 Results . . . 95
6.4 Artificial manipulations of human mobility patterns . . . 95
6.5 The fight against schistosomiasis in Senegal . . . 101
6.5.1 WASH interventions . . . 101
6.5.2 IEC campaigns . . . 103
6.6 Conclusions . . . 103
7 Saint-Louis network model 107 7.1 Multidimensional network model . . . 107
7.2.1 Human population . . . 111
7.2.2 Snail population . . . 111
7.2.3 Adult parasites and larval stages . . . 113
7.2.4 Human mobility and water contact patterns . . . 114
7.2.5 Infection and contamination risk . . . 115
7.2.6 Hydrological connectivity . . . 117
7.3 Model outputs . . . 120
7.4 Results . . . 121
7.4.1 Mean worm burden and prevalence distribution . . . 121
7.4.2 Infection intensity . . . 124
7.4.3 Hydrological regimes . . . 124
7.5 Conclusions . . . 124
IV CONCLUDING REMARKS 131 8 Conclusions and future perspectives 133 V APPENDIX 139 A Bifurcation analysis of the SEI-like compartmental model 141 A.1 Transcritical bifurcation curve . . . 141
A.2 Hopf bifurcation curve . . . 142
B Stability analysis of the DFE of the heterogeneous model 145 B.1 2-group community with access to a single water source . . 145
B.2 One community with access to two water sources . . . 146
C Standard mobility models 149 C.1 Gravity and radiation models . . . 149
C.2 Theoretical models vs. CDR analysis . . . 150
CHAPTER
1
Objectives and thesis structure
1.1
Research focusWaterborne diseases belong to a group of infections caused by pathogenic microorganisms that most commonly are transmitted in contaminated fresh-water. Typically, infection occurs during bathing, washing, drinking, or in-gesting contaminated food. For these reasons, they are especially endemic among low-income populations in developing regions and represent a se-rious public health problem where people lack access to clean water and adequate health care resources.
In this work, novel models are developed for the transmission of schis-tosomiasis, the deadliest among Neglected Tropical Diseases (NTDs). It is caused by a snail-transmitted trematode, which may infect the urinary tract or the intestines. The specific aim is to include in the models some important features of the transmission cycle that are still underestimated, yet crucial, such as the impact of social and environmental conditions. In particular, this is performed by integrating proxies of human mobility with a geospatial analysis which includes georeferenced data on demography, water supply/sanitation, and urogenital schistosomiasis prevalence. These models are used to analyze transmission patterns in different settings,
rang-ing from purely theoretical ones to real case studies. The developed models incorporate both the biological complexity associated with the parasite’s life cycle (including secondary hosts) and the mechanisms that influence the spatiotemporal dynamics of the disease. Specifically, mathematical modelling is used to identify the role of social interactions and physical interconnections between populations in sustaining the transmission of the disease.
The developed modelling frameworks are then applied to Senegal, where the urogenital form of the infection is widespread. In the country, schisto-somiasis represents a major health problem, being the third disease (after malaria and lymphatic filariasis) in terms of years lived with disability. A spatially explicit model is first applied to medium-to-large spatial scales, at which human mobility is retained as the main mechanism for the spa-tial spread of the disease. Over finer spaspa-tial scales, instead, connectivity via hydrological transport and snail dispersal increases the risk of disease propagation. A multidimensional network model accounting for both so-cial and environmental connectivity is thus applied to a set of connected villages in the area of the Lower Basin of the Senegal River.
A further analysis on the effectiveness of some intervention strategies shows the potentialities of this work, and of epidemiological modelling in general. Current measures for schistosomiasis control are, in fact, princi-pally focused on preventive chemotherapy, which however does not confer permanent immunity to humans. In particular, the strategy for the disease control aims at preventing morbidity through regular treatment with praz-iquantel, which is currently the only recommended drug for human infec-tions caused by schistosomes. Experience from China and Egypt shows that preventive chemotherapy (i.e. mass treatment without individual diag-nosis) with high coverage can significantly impact on indices of infection, also reducing future requirements for praziquantel1. However, an efficient
use of resources could become fundamental, especially in the poorest re-gions of the world. In this work, the spatial structure is used to quantita-tively evaluate the impacts of different sanitary and humanitarian efforts, such as preventing exposure and contamination of environmental freshwa-ter, spreading awareness about disease transmission, and biological control of snail hosts. In this respect, a quantitative decision-support tool can help guide resource allocation in the fight against the disease, identifying the focal hotspots of disease transmission to reduce its impact on society.
1.2
Main resultsAs a principal and most significant contribution to the state-of-the-art in the field of schistosomiasis transmission modelling, this study presents a comprehensive approach to the epidemiological problem, underlying the importance of integrating different perspectives, from environmental to so-cial and behavioural. The main results include:
1. the intermediate snail host population dynamics play an important role in generating variations of disease transmission patterns over time, with potential implications for long-term disease dynamics. The de-veloped models show that it is possible to qualitatively reproduce both the intra- and inter-annual variability of prevalence patterns observed in many endemic regions via a classical SEI-like model for snails ecol-ogy;
2. socioeconomic conditions and water availability are fundamental in the definition of the infection risk of different communities, in par-ticular with respect to human habits and quality of snails habitats. Through simple prototypical examples, the impact of different sources of heterogeneities is explored, helping in the identification of high-risk communities and most critical water sources;
3. spatial coupling mechanisms are very important in the spread, per-sistence and infection intensity of schistosomiasis. At larger spatial scales, the case study of Senegal shows that accounting for human mobility is crucial for an accurate reproduction of the observed spatial patterns of schistosomiasis prevalence. In this respect, anonymized Call Detail Records (CDRs) from mobile phone networks represent a useful source of information as a proxy of human movements;
4. hydrologically mediated processes may have relevant impacts on the global prevalence and disease burden. Environmental drivers, prop-erly integrated with CDR-based information, are among the funda-mental components of metapopulation models able to investigate also the role of biophysical mechanisms, such as snail dispersal and larval transport;
5. the implementation of a comprehensive approach is important for fight-ing the disease, since models could contribute to the development of effective intervention strategies. The analysis of different actions sug-gests that it is possible to transform the developed modelling
frame-works into support tools to help decision makers in the design of ef-fective plans for fighting schistosomiasis.
1.3
Thesis outlineThe present dissertation is structured in four main parts. Part I is dedi-cated to introduce schistosomiasis and its worldwide burden. Part II is de-voted to present and discuss some theoretical models useful to investigate sources and impacts of spatial and temporal variability. Part III is used to illustrate application results obtained applying the developed models to the case study of Senegal. Part IV closes the dissertation giving some conclud-ing remarks and future perspectives. As for each Chapter, the path is the following.
Chapter 2 introduces epidemiology and pathophysiology of schistoso-miasis, giving a detailed description of the life cycle of the schistosome responsible for the disease. The simplest model for schistosomiasis, pro-posed by Macdonald in 1965, is described, together with its main results. References to other published work are given in the text.
Chapter 3explores the mechanisms that drive the temporal variability of schistosomiasis severity and prevalence using different local disease mod-els. In particular, the role of the intermediate host population is investigated including different classifications for the snails, considered to be classified according to their infection age or epidemiological status.
Chapter 4 proposes a network model which integrates local epidemio-logical processes and the impacts of different sources of heterogeneity in the human host population (arising from the presence of sub-groups with different infection risk) and in the available water sources (in terms of con-tact preference and suitability as snails habitat). The model is then used to evaluate the effectiveness of possible control strategies.
Chapter 5describes the context for the case studies applications. Demo-graphic, socioeconomic and environmental conditions of Senegal are pre-sented, and human mobility patterns across the country are inferred from mobile phone data analysis.
Chapter 6 illustrates simulation results obtained applying a country-wide spatially explicit network model to Senegal. The model utilizes
geo-referenced demographic and socioeconomic information, and is calibrated against the available prevalence records of urogenital schistosomiasis in the country. The effects of control strategies based on reducing human expo-sure and contamination are also analyzed.
Chapter 7 presents a spatially explicit metapopulation model, in which schistosomiasis spreads within a network of connected villages in the Lower Basin of the Senegal River. Social and environmental interconnections link villages through human mobility and hydrology, underlying the importance of including different pathways for the disease.
Chapter 8presents the concluding remarks of the study and recommen-dations and perspectives for future works.
1.4
Research worksSome of the subjects, the methods and the results presented in this disser-tation can be found in the following published and under review journal papers:
• M. CIDDIO, L. Mari, M. Gatto, A. Rinaldo, R. Casagrandi, “The tem-poral patterns of disease severity and prevalence in schistosomiasis”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015 • M. CIDDIO, L. Mari, S. H. Sokolow, G. A. De Leo, R. Casagrandi,
M. Gatto, “The spatial spread of schistosomiasis: A multidimensional network model applied to Saint-Louis region, Senegal”, Advances in Water Resources, 2016
• L. Mari, R. Casagrandi, M. CIDDIO, E. D. Dia, S. H. Sokolow, G. A. De Leo, M. Gatto. “Big-data-driven modeling unveils country-wide drivers of endemic schistosomiasis”, submitted to Scientific Reports • L. Mari, M. CIDDIO, R. Casagrandi, J. Perez-Saez, E. Bertuzzo, A.
Rinaldo, S. H. Sokolow, G. A. De Leo, M. Gatto, “Heterogeneity in schistosomiasis transmission dynamics”, to be submitted
The main contents have been also presented in some national and inter-national conferences:
• M. CIDDIO, L. Mari, R. Casagrandi, M. Gatto, “A model for schisto-somiasis transmission accounting for infection age in snails: sensitiv-ity and bifurcation analyses”, XXIV SItE Congress, Ferrara, 2014
• M. CIDDIO, L. Mari, M. Gatto, A. Rinaldo, R. Casagrandi, “Impact of environmental conditions on snails dynamics and schistosomiasis transmission”, IECID conference, Sitges (Spain), 2015
• L. Mari, R. Casagrandi, M. CIDDIO, S. H. Sokolow, G. A. De Leo, M. Gatto, “Uncovering the impact of human mobility on schistosomiasis via mobile phone data”, NetMob conference, Cambridge (MA), USA, 2015
• M. CIDDIO, L. Mari, R. Casagrandi, S. H. Sokolow, G. A. De Leo, M. Gatto, “The impact of human mobility on schistosomiasis in Sene-gal: an analysis via mobile phone data”, ECTMIH conference, Basel (Switzerland), 2015
• M. CIDDIO, L. Mari, R. Casagrandi, S. H. Sokolow, G. A. De Leo, M. Gatto, “Human population movement and schistosomiasis trans-mission risk: the case study of Senegal”, EPID conference, Clearwater Beach (FL), USA, 2015
• M. CIDDIO, L. Mari, R. Casagrandi, M. Gatto, “A schistosomiasis transmission model to study the effects of heterogeneity on human and snail prevalence”, 1st SItE-UZI-SIB joint conference, Milano, 2016
CHAPTER
2
Schistosomiasis
2.1
Epidemiology and pathophysiologySchistosomiasis, also known as bilharziasis, is one of the commonest wa-terborne diseases. It is a major parasitic infection in many areas of the developing world, affecting about 250 million individuals in 74 countries and putting at risk about 700 million people in regions where the disease is endemic [1]. In terms of impact, schistosomiasis is second only to malaria as the most devastating parasitic disease [2] and likely the deadliest among Neglected Tropical Diseases (NTDs). Its burden is disproportionately con-centrated in Africa. According to the World Health Organization, many control programs are available which can successfully eliminate the dis-ease. However, schistosomiasis remains a major cause of mortality and morbidity in a number of countries, notably those of sub-Saharan Africa, which accounts for at least 90% of cases worldwide [2], but also in some ar-eas of Asia and Latin America [1]. In particular, schistosomiasis is directly responsible for the death of about 12,000 people yearly [3], being also a co-factor in at least 200,000 deaths annually [4]. In Fig. 2.1, the yearly number of individuals requiring preventing chemotherapy for
schistosomi-asis is shown, according to 2015 estimates available online from WHO1. It is estimated that 120 million of infected individuals worldwide are symptomatic, with 20 million developing severe symptoms from the dis-ease [5] and an estimated disability-adjusted life years (DALYs, i.e. the number of years lost due to ill-health, disability or early death) of 4.5 million [6]. Although the disease has a low mortality rate, the related morbidity caused by schistosomiasis can inflict a heavy health burden on high-prevalence communities. Water has a key role in its transmission and spread. Human-to-environment transmission occurs when infected people contaminate freshwater bodies with their excreta containing parasite eggs. Environment-to-human transmission occurs when people are exposed to in-fested water during routine activities, ranging from agricultural to domestic and from occupational to recreational. Therefore, the disease is especially prevalent in rural communities. Lack of hygiene and certain play habits make school-aged children particularly vulnerable to infection, an aspect which must be regarded with care, because schistosomiasis may induce se-vere health consequences in absence of adequate treatments. Typically, par-asites inside human tissues induce a response that causes local and systemic pathological effects ranging from anaemia, impaired growth and cognitive development, and decreased physical fitness, to organ-specific effects such as fibrosis of the liver, bladder cancer, and urogenital inflammation [7, 8].
There are two major forms of schistosomiasis – intestinal and urogen-ital – caused by six species of blood flukes belonging to the genus Schis-tosoma, of which S. haematobium, S. mansoni and S. japonicum are the three most important ones [7]. These parasites need as obligate interme-diate hosts some species of freshwater snails belonging to the genus Buli-nus(for S. haematobium), Biomphalaria (for S. mansoni) or Oncomelania (for S. japonicum). The geographical distribution of schistosomes is thus defined by the specific range of snail host habitat: S. haematobium and S. mansonioccur in both Africa and the Middle East, whereas only S. man-soniis present in the Americas. S. japonicum is localized in Asia, primarily the Philippines and China [7].
< 0.1 million 0.1 - 0.9 million 1 - 4.9 million 5 - 9.9 million > 10 million No PC required
Non-endemic
Preventive chemotheraphy (PC) required [number of people]
Figur e 2.1: P opulation requiring Prev entiv e Chemotherapy (PC) for sc histosomiasis annually . Estimated number of individuals requiring PC for sc histosomiasis in 2015. The map is elabor ated accor ding to data available online fr om WHO (http:// www.who.int/ neglected_diseases/ preventive_chemotherapy/ sch/ en/ ).
The occurrence of schistosomiasis usually displays both inter- and intra-annual variability, with peaks of infection occurring in different seasons. See e.g. Figs. 2.2-2.3 for some sample patterns from Limpopo Province, South Africa [9], and Wonji, Ethiopia [10].
1998 1999 2000 2001 2002 2003 2004 0 25 75 100 J F M A M J J A S O N D 0 25 50 75 100 Pre va le n ce (% ) 2001 2002 2003 (a) (b) 50
Figure 2.2: Sample patterns of yearly and monthly occurrence of schistosomiasis in South Africa. a) Overall yearly prevalence of urinary schistosomiasis amongst pa-tients attending the main hospitals in the Vhembe district of Limpopo Province, South Africa, between 1998 and 2004. b) Occurrence of S. haematobium in urine samples submitted for urinary tract infections to the laboratory of the Vhembe district hospitals between 2001 and 2003. Data elaborated from Samie et al [9].
2.2
Life cycle and schistosomiasis transmissionThe infectious form of the parasite for humans is a freely swimming, short-lived larval stage of the parasites known as cercaria and periodically shed by infected snails (see Fig. 2.4 for a sample data collection of cercarial pro-duction [11]). People become infected when cercariae penetrate their skin during contact with infested water. Within the human body, cercariae shed their tails and become schistosomula, the maturing larvae of the parasite. These larvae migrate through the body and need about 5-7 weeks before becoming sexually mature adults [7]. The adult male and female worms colonize human blood vessels, where they can live for years, mating and producing hundreds to thousands of fertilized eggs daily. The eggs can ei-ther leave the body of the host by being shed in the environment through faeces (S. mansoni, S. japonicum) or urine (S. haematobium), or become trapped inside the human host tissues. The severity and complexity of the pathology of schistosomiasis are related to the quantity of these encysted
1999 00 01 02 03 04 05 06 07 2008 0 150 300 450 600 J F M A M J J A S O N D 0 30 60 90 120 N u mb e r o f p a ti e n ts (a) (b) 1999 2000 2001
Figure 2.3: Sample patterns of yearly and monthly occurrence of schistosomiasis in Ethiopia. a) Overall yearly number of S. mansoni patients between 1999 and 2008 from the only hospital in Wonji, Ethiopia. b) Monthly data of the number of S. mansoni patients in Wonji between 1999 and 2001. Data elaborated from Xue et al [10].
eggs [2]. The eggs released out of the human body that reach freshwater can hatch into larvae called miracidia, the parasite larval form that is in-fectious for snail hosts. In the snail, miracidia undergo asexual replication for 4-6 weeks [7], then the snail becomes infective and starts releasing tens of thousands of cercariae into the water. The time between initial snail infection and onset of infectiousness is the so-called prepatent period [7]. Cercariae can survive in freshwater for 1-3 days [7], then they need to find a human host to complete the parasite’s life cycle. A sketched scheme of the parasite life cycle is shown in Fig. 2.5.
2.3
Schistosomiasis models: state-of-the-artMathematical modelling of disease dynamics has proved to be a useful tool in many human infections, in particular to understand the transmission characteristics of parasitic diseases in order to develop and evaluate the ef-fects of control programs [13–15], and to make predictions on the efef-fects of different intervention options [16–18], also exploring control strategies targeted at high-risk behavioural groups [19, 20]. Transmission models of schistosomiasis have been in existence since the 1960s [21,22]. These mod-els are generally based on some limiting assumptions (e.g. homogeneous population [22], single host population and one parasite stage [23]).
30 40 50 60 70 80 90 100 110 120 130 140 Days post-infection 0 100 200 300 400 500 600 700 800 C e rca ri a e p e r sn a il, ς
Figure 2.4: Varying cercarial release by infected snails. The number of cercariae re-leased by one snail changes with time since infection. Data elaborated from Feng et al. [12]. Infected human Water contamination Schistosome egg Miracidium Snail Cercaria Exposure to infested water Adult parasites
Figure 2.5: Schistosoma life cycle. Adult schistosomes within infected human hosts pro-duce eggs, which are shed in the environment through excreta. The eggs that reach freshwater can hatch into miracidia and infect intermediate species-specific snail hosts. Infective snails shed free-swimming cercariae that can penetrate human skin and eventually develop into reproductive worms. See text for further details about transmission.
in 1965. It describes transmission dynamics through two state variables, namely the average parasite burden in the human population (W ) and the prevalence of infection in snails (Y ). Population dynamics are neglected in both human and snail hosts, assuming instead demographic equilibrium. Also, the model does not include the dynamics of cercariae and miracidia, whose abundances are considered to be proportional to Y and W , respec-tively. The model thus reads as follows:
dW dt = βSY − µPW dY dt = χHW (1 − Y ) − µSY, (2.1)
where β is the snail-to-human transmission rate, S is the abundance of the snail population, µP is the mortality rate of adult parasites in human hosts,
χ is the human-to-snail transmission rate, H is the abundance of the human population and µS is the mortality rate of infected snails. The parameters
µP and µS can be evaluated as the inverse of the average lifespans of adult
worms and infected snails (around five years and two months, respectively; see e.g. [11]), whereas β and χ represent aggregated parameters accounting for several epidemiological and socioeconomical processes.
Model 2.1 has two steady-state solutions. The first one is the so-called disease-free equilibrium (DFE), i.e. a state of the system in which the para-site is not present (thus W = 0 and Y = 0). The second one is the endemic equilibrium (EE), i.e a state of the system in which parasite transmission is permanent, namely
W = βSχH − µPµS µPχH
, Y = βSχH − µPµS
βSχH . (2.2)
By standard linear stability analysis arguments, it can be shown that the DFE is stable if µPµS− βSχH > 0, also corresponding to the parameter
region in which the EE is not feasible (i.e. characterized by negative com-ponents). This stability condition can be equivalently stated in terms of the so-called basic reproduction number
r0 =
βSχH µPµS
, (2.3)
i.e. the average number of secondary infections produced by one infected individual in a community with no pre-existing immunity to the disease [13]. Specifically, the DFE is asymptotically stable if r0 < 1, unstable if
r0 > 1. Conversely, the EE is found to be stable if r0 > 1. The condition
r0 = 1 thus marks an exchange of stability between the DFE and the EE: for
r0 < 1 the DFE is stable, while the EE is unfeasible and unstable; at r0 = 1
the two equilibria collide; and for r0 > 1 the DFE is unstable, while the
EE is positive and stable (transcritical bifurcation; see e.g. [24]). Fig. 2.6 illustrates the basic properties of the equilibria of model 2.1. From a disease control perspective, it also explains why preventing water contamination (i.e. decreasing the overall human-to-snail transmission rate χH) might be less an effective control measure than preventing human water contact (i.e. decreasing the overall snail-to-human transmission rate βS), as suggested by Macdonald (1965) in his seminal work (see also [25] for discussion). In fact, the burden W is more sensitive to βS than to χH.
10-5 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 100
Snail-to-human transmission rate βS
H u ma n -t o -sn a il t ra n smi ssi o n ra te χ H W = 1 Y = 0.9 Y = 0.5 r 0 < 1 r 0 > 1 W = 10 W = 100 r 0 = 1
Figure 2.6: Equilibria of Macdonald’s (1965) schistosomiasis model. The DFE is stable if r0 < 1, while the EE is feasible and stable if r0 > 1. The black line (r0 = 1) indicates the transcritical bifurcation through which the two equilibria collide and ex-change stability (endemicity boundary). The dashed [dash-dotted] curves are contour lines for the prevalence of infected snails [average worm burden in human hosts] at the EE. Parameter values:µP=5.5· 10−4,µS= 1.7· 10−2. All rates are expressed in [day−1].
More recent studies have tried to relax the assumptions of Macdonald’s model, e.g. by accounting for the demographic dynamics of human and snail populations, or by including some additional aspects (e.g. infection
age of snails [11, 12], parasite’s mating structure and multiple resistant schistosome strains [26]). They usually neglect the dynamics of the in-termediate life stages of the parasite, thus reducing model complexity.
Previous studies have already shown that disease dynamics not only de-pend upon interactions between infectious agents and the hosts, but they are also strongly affected by environmental and socioeconomic factors [27,28]. In this respect, transmission heterogeneity represents a central issue in the study of infectious disease dynamics; in fact, it has been receiving con-siderable attention for more than thirty years, with research focusing on the role played by heterogeneity in both short-term epidemic dynamics and long-term transmission maintenance, as well as on the challenges and op-portunities that heterogeneity poses to disease control (see e.g. [29–38]). A full understanding of the drivers and the consequences of heterogeneity still represents a major challenge for epidemiology, especially for diseases characterized by indirect transmission, such as vector-borne, water-related or macroparasitic infections [39]. In this respect, heterogeneity has been accounted for in several mathematical models for schistosomiasis transmis-sion. A seminal contribution was given in 1978 by Barbour, who elaborated on Macdonald’s (1965) basic model, and extended it to account for individ-ual variations in water contact patterns and source heterogeneity. Later works focused on the integration of field data and the implications of het-erogeneity for disease control [40, 41], as well as on the effects of site-specific environmental features on schistosomiasis transmission [17, 42].
Afterwards, the interplay between local-scale transmission heterogene-ities and spatial coupling mechanisms was investigated, together with its implications for long-term disease control [19]. Metapopulation models have proved to be a powerful tool in order to understand disease persistence and infection intensity in human societies [43, 44]. In the case of schistoso-miasis, the movement of infectious agents can occur via various transport processes involving hosts and pathogens, including human mobility, larval transport along canals and streams, and snails dispersal through hydrologi-cal interconnections. The spread of schistosomiasis is thus the result of the interplay between various mechanisms acting at different spatial and tem-poral scales. On the human host side, social connections provide a pathway for adult parasite transport while people travel between endemic and non-endemic areas. This movement can involve very large spatial scales in ways that are often difficult to predict [45], and constitutes an effective transmis-sion mechanism provided that disease-transmitting snails live in the visited areas. Despite recent advances in the modelling of human mobility [46–48], there still remain fundamental limits to our understanding of where, when,
why and how people move [49, 50]. Standard mobility models have been found to perform poorly in the African context [51]. Therefore, proxies of human mobility that can be remotely acquired, properly anonymized and quantitatively elaborated represent an invaluable tool to inform epidemio-logical models. In this respect, the analysis of CDRs represents one of the most promising tools to infer human mobility patterns [52, 53] – also in an epidemiological context, as shown by the increasing number of studies that make use of CDR analysis [54–60]. On the snail and parasite side, con-nectivity via hydrological transport and animal dispersal increases the risk of larval and snail propagation over shorter spatial scales. As an example, all over the world, an estimated 63 million people at risk for schistosomia-sis live in irrigated environments, with an increased relative risk of urinary and intestinal schistosomiasis of 1.1 and 4.7, respectively, compared with non-irrigated environments [61].
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THEORETICAL
INVESTIGATIONS
CHAPTER
3
Temporal patterns of schistosomiasis
dynamics
In this chapter, different sources of seasonality in schistosomiasis preva-lence patterns are investigated via local disease models. In fact, the oc-currence of schistosomiasis usually displays both inter- and intra-annual variability: while intra-annual oscillations can be explained by the strong seasonal fluctuations of snail demography, interannual fluctuations, which can be more or less wide, may be caused by nonlinearities in demographic and epidemiological mechanisms. In the next sections, the mechanisms that drive the temporal variability of schistosomiasis severity and prevalence are explored using different mathematical models. The main innovation con-cerns specifically snail dynamics. In fact, the presence of a prepatent pe-riod in snails [1–3], during which the infection is ongoing but release of cercariae cannot be detected, suggests the division of snails into different classes, according to their infection age or epidemiological status.
3.1
Age-of-infection structured modelA macroparasite model of schistosomiasis that includes age-of-infection in snail hosts (i.e. time since infection) and intensity of human infection is first analyzed. Sensitivity and preliminary bifurcation analyses are used to understand the effects of some ecological and control parameters on the mean worm burden within human hosts. In particular, different recruitment functions are tested for snails, proving that the introduction of a density-dependent recruitment function (e.g. logistic equation) leads to more com-plex and realistic prevalence patterns.
3.1.1 Model formulation
The analysis is based on a system of differential equations that describe the dynamics of humans, schistosomes (adult parasites, miracidia, and cer-cariae), and snails, according to the ecological and epidemiological mech-anisms that characterize human schistosomiasis transmission cycle (see Chapter 2). A diagram sketch of the model is shown in Fig. 3.1, while mathematical details are presented in the following.
Human hosts N Adult parasites P Uninfected snails S Miracidia M X 1 Cercariae C ... X τ +1 Infected snails Infected snails max
Figure 3.1: Schematic diagram of the age-of-infection structured model. The model is characterized by (τmax+ 4) state variables: human hosts, adult parasites, uninfected snails and infected snails, classified according to their infection age (grey blocks). The larval forms of the parasite are assumed to be proportional to other relevant variables (white blocks). The arrows describe the fluxes, the dashed arrows represent indirect interactions. See details about the model in the text.
The model can be considered as the combination of two sub-systems describing, respectively, the dynamics of humans and adult worms in their tissues, and snails dynamics. The first sub-system is governed by two dif-ferential equations that describe the dynamics of human hosts N and of the adult form of the parasite P . The man-schistosome interaction is modeled as a macroparasitic infection [4]: ˙ N = µH(H − N ) − αP ˙ P = βCN − (µH + µP + α)P − α k + 1 k P2 N . (3.1)
Human hosts are assumed to be characterized by a constant recruitment µHH (with H being the community size in absence of the disease), and a
natural death rate µH. Parasites are assumed to induce mortality
proportion-ally to their number within each human host, i.e. proportionproportion-ally to the mean worm burden P/N . With α being a constant determining the pathogenicity of the parasite to the human host [5], the total losses due to disease-induced mortality of humans thus are:
αP
NN = αP. (3.2)
Parasites’ recruitment is proportional to the number of cercariae C in the environment and to an infection rate β that includes the contact rate as well as several aspects of survival and maturation of cercariae inside and outside the human hosts. Since the timescale of the free-living larval stages is much shorter than that of the mature worm in the human host and the developing stage in snails, cercariae are assumed to be proportional to the number Iτ of
infectious shedding snails after τ days since first infection [2, 4], according to the following function:
C = X
τ
ζ(τ )Iτ, (3.3)
where ζ(τ ) represents the number of cercariae released by one infected snail after τ days from first infection multiplied by the residence time of cercariae in water. Similarly to other studies [3], the cercarial release func-tion is assumed to be oscillating, i.e.:
ζ(τ ) = ζ0 1 + A sin2πτ T if τmin ≤ τ ≤ τmax, 0 elsewhere, (3.4)
where ζ0 is a baseline reference value for cercarial production, A and T
are the amplitude and the period of their oscillations, respectively, and τmin
[τmax] is the first [last] day of cercarial release by snails after infection.
The death rate of parasites within the human host population has three components. First, there are losses due to natural host mortality (µHP ).
Second, there is a component linked to parasite mortality within the host. Assuming an intrinsic death rate µP, these losses make a contribution of
µPP . Third, there are losses due to disease-induced mortality of humans.
Since macroparasites are almost always unevenly distributed across their host populations, negative binomial is often used for modelling parasite differences in load among different individuals [2]. Thus, these losses can be written as a function of the clumping parameter k, which gives an inverse measure of the degree of aggregation of the parasites within the hosts [5]:
αP NN (1 + k + 1 k P N) = αP + α k + 1 k P2 N . (3.5)
The second sub-system describes the dynamics of uninfected (S) and infected snails (Iτ), which are assumed to be age-structured. The density
of infected snails is modelled through τmax + 1 variables describing the
daily flow after first infection, with Iτmax+1 including all snails infected for
more than τmaxdays. The system that describes the snails dynamics is thus
defined as ˙ S = b(S, I1, I2, ..., Iτmax+1) − µSS − ρM S ˙ I1 = ρM S − (µS+ η)I1− εI1 ˙
Ii = ε(Ii−1− Ii) − (µS + η)Ii i ∈ [2, τmax]
˙
Iτmax+1 = εIτmax− (µS + η)Iτmax+1.
(3.6)
The recruitment function of susceptible snails, b, will be discussed in the following. In addition to a natural death rate µS, infected snails are also
subject to a disease-induced death rate η. Recruitment of infected snails de-pends on the infection rate ρ and the number of miracidia M in the aquatic environment. Miracidia, being short-lived, are assumed to be proportional to the number of adult parasites through the per capita egg laying rate lP
of adult parasites and the residence time of miracidia in water τM, so that
M = lPτMP . For simplicity of notation, a new parameter χ = ρlPτM
is introduced to aggregate the various intermediate steps involved in snail infection. Infected snails move from the first class I1 to the following one
after a period of average duration 1/ε, set to 1 day to account for days post infection in cercarial production.
In the following, the range of possible behaviours of the system is ex-plored by considering two different recruitment functions for snails demog-raphy b. A simple case is first considered in which snails recruitment is assumed to be constant, i.e. b = ΛS. Combining (3.1)–(3.6), in this simple
case the full model that describes schistosomiasis dynamics is represented by the following system of (τmax+ 4) nonlinear differential equations:
˙ N = µH(H − N ) − αP ˙ P = βN P τζ(τ )Iτ − (µH + µP + α)P − α k + 1 k P2 N ˙ S = ΛS− µSS − χP S ˙ I1 = χP S − (µS+ η)I1 − εI1 ˙
Ii = ε(Ii−1− Ii) − (µS+ η)Ii i ∈ [2, τmax]
˙
Iτmax+1 = εIτmax− (µS+ η)Iτmax+1.
(3.7)
However, other studies [6] have shown that snail populations are regu-lated by density-dependent mechanisms and that the introduction of a non-linear recruitment function seems to play an important role in generating realistic patterns of schistosome infections [2]. Therefore, a more complex case is also considered in which snails are assumed to be born uninfected according to a logistic recruitment function and that infected snails (both exposed and infectious) are unable to reproduce, thus:
b = νS " 1 − γ S +X τ Iτ !# , (3.8)
where ν is the intrinsic natality rate and γ captures the effect of density de-pendence among snails. The full model is thus represented by the following system: ˙ N = µH(H − N ) − αP ˙ P = βN P τζ(τ )Iτ − (µH + µP + α)P − α k + 1 k P2 N ˙ S = νS[1 − γ(S +P τIτ)] − µSS − χP S ˙ I1 = χP S − (µS+ η)I1 − εI1 ˙
Ii = ε(Ii−1− Ii) − (µS+ η)Ii i ∈ [2, τmax]
˙
Iτmax+1 = εIτmax− (µS+ η)Iτmax+1.
3.1.2 Parameter estimation
Parameters associated with infection of human hosts are usually quite dif-ficult to estimate. Here, part of the parameters is derived from literature, and part is allowed to vary because their values are uncertain or dependent upon exogenous conditions. In these latter cases, sensitivity and bifurcation analyses are usually the best approach to identify parameters able to pro-duce realistic results [7] and to evaluate the effects of combined changes in the values of specific parameters [2]. This approach has been used also in previous studies on other schistosomiasis models [8, 9].
The analysis is conducted on a human population of size H = 1, 000 individuals with a life expectancy of 70 years (µH = 4 · 10−5/day). The
life expectancy of adult parasites is 5 years [3, 10] (µP = 5.5 · 10−4/day).
Parasite-induced mortality in the human host is considered to be negligible (α = 0). As for snails, the average lifetime is considered to be 1 year [3] for uninfected snails (µS = 2.7 · 10−3/day) and 2 months [3] for infected snails
(η = 1.37 · 10−2/day). The average residence time in each age class is 1 day (ε = 1/day), and the cercarial release function is assumed to be periodic of period 30 days, with a prepatent period of one month (τmin = 31), and
having a maximum support of 140 days (τmax = 140) [3]. The baseline
ref-erence number of cercariae released by one infected snail is assumed to be 350/day [3] (thus ζ0 = 700 cercariae/snail, because the average residence
time of cercariae in the water environment is assumed to be 2 days [1]), with amplitude of oscillations A = 0.2. All parameters are listed in Tab. 3.1.
The rates of human and snail infection (β and χ) are the most difficult to be quantified. Note that both parameters are the product of several fac-tors. The first includes the probability of contact with cercariae, cercarial survival inside the human host and maturation into the adult form of the parasite; the latter includes the number of produced eggs and their proba-bility of reaching the aquatic environment, developing into miracidia and successfully infecting a susceptible snail. Due to the complexity associated with the definition of these parameters, sensitivity and bifurcation analyses are conducted over a large parametric range.
T able 3.1: State variables and parameters of the age-of-inf ection structured model : description, value , units and refer ences. Name Description V alue Units References N Number of human hosts indi vidual P Number of adult parasites parasite S Density of susceptible snails snail · m − 2 Iτ Density of infected snails after τ days post infection snail · m − 2 H Human community size 1,000 indi vidual fix ed µH Per capita natural death rate of humans 4 · 10 − 5 day − 1 fix ed µP Per capita natural death rate of adult parasites 5.5 · 10 − 4 day − 1 [3, 10] µS Per capita natural death rate of snails 2.7 · 10 − 3 day − 1 [3] η Disease-induced death rate of snails 1.37 · 10 − 2 day − 1 [3] α Disease-induced death rate of humans per unit of parasite b urden 0 day − 1 [11] k Clumping parameter of parasite distrib ution -not used ζ0 Reference releasing rate of cercariae by one snail 700 cercariae · snail − 1 [1, 3] A Amplitude of oscillations in cercarial release 0.2 -fix ed T Period of oscillations in cercarial release 30 days [2] τmin Onset of infectiousness in snails 31 days [2] τmax Maximum support for cercarial release 140 days [2] ΛS Constant snail recruitment 2.7 · 10 − 1 snails · day − 1 m − 2 [12] γ Rate of competition for resources among snails 10 − 2 m 2· snail − 1 fix ed ν Intrinsic natality rate of snails 0.7 day − 1 fix ed χ Per capita rate of infection of snails V ariable parasite − 1· day − 1 β Per capita rate of infection of humans V ariable parasite · m 2 · cercaria − 1 · indi vidual − 1 · day − 1
3.1.3 Results
Results show that, given a constant recruitment for snails, the mean worm burden at the endemic equilibrium P /H increases with both the human and snail infection rates, β and χ, with larger effects induced by rate β (Fig. 3.2).
Rate of human infection, β (∙10-8)
R a te o f sn a il in fe ct io n , χ (∙ 1 0 -4) 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 P/H=4 7 10 13 16 19 22 ¯
Figure 3.2: Endemic equilibrium of the age-of-infection structured model with constant snail recruitment. Black lines represent the level curves of the mean worm burden P /H at the endemic equilibrium for different combinations of human and snail infec-tion ratesβ and χ. Constant snail recruitment ΛS is set to2.7 · 10−1 snails/day/m2 (≈ 100 snails/m2in absence of the parasite [12]). All other parameters as in Tab. 3.1.
On the contrary, numerical simulations of the model with density-de-pendent snails recruitment show that a bifurcation occurs for some critical values of the infection rates, in which case the endemic equilibrium is re-placed by stable periodic solutions (Fig. 3.3). After the bifurcation point is reached, the periodic solution displays larger amplitudes of oscillation for higher values of β and χ (see some numerical examples for increasing β in Fig. 3.4).
This preliminary analysis conducted on the age-of-infection structured model shows that varying cercarial release is not responsible for periodicity in disease patterns, because only the model with density-dependent recruit-ment function is able to reproduce inter-annual variability in schistosomi-asis prevalence patterns. Infection age structure introduces complexity in the model that makes difficult a thorough mathematical analysis, other than
Rate of human infection, β (∙10-7) Me a n w o rm b u rd e n , P/ H 0 50 100 150 200
Rate of snail infection, χ (∙10-3) 300 250 200 150 100 50 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.4 0.8 1.2 1.6 2 ¯ (a) (b) min-max
Figure 3.3: 1-D bifurcation diagrams of the age-of-infection structured model with lo-gistic snail recruitment. A bifurcation occurs at some critical point. In panel (a), the snail infection rate is set toχ = 2 · 10−4parasite−1· day−1. In panel (b), the human infection rate is set toβ = 4 · 10−8parasite· m2· cercaria−1· individual−1· day−1. Numerical examples are computed forν = 0.7 day−1andγ = 10−2m2/snail (≈ 100 snails/m2). All other parameters as in Tab. 3.1.
Me a n w o rm b u rd e n , P(t )/ H Time (years) 15 30 45 60 75 90 (b) (c) (a) 0 100 0 200 300 100 0 200 300 100 0 200 300
Figure 3.4: Temporal patterns of adult parasites in the age-of-infection structured model with logistic snail recruitment. Temporal patterns vary from endemic equi-librium (a) to periodic solutions (b-c). Simulations are obtained for χ = 2 · 10−4 parasite−1 · day−1. The human infection rates are set as: a) β = 4 · 10−8, b)β = 1.2 · 10−7, c)β = 2 · 10−7parasite· m2· cercaria−1· individual−1· day−1. All other parameters as in Tab. 3.1.
numerical simulations. In absence of consolidated evidence of its ecologi-cal and epidemiologiecologi-cal significance, in the next sections the analysis will be focused on a simpler model formulation, which is amenable to analytical investigations.
3.2
SEI-like compartmental modelIn next sections the mechanisms that drive the temporal variability of schis-tosomiasis severity and prevalence are explored using a new model of in-termediate complexity, which allows a thorough bifurcation analysis. This model is less complex than those including the snails’ infection age, yet it is fairly realistic and simple enough to allow an analytical investigation of the link between the ecology of the snails and the variability of typical disease patterns observed in many endemic regions.
3.2.1 Model formulation
The new model can be still considered as the combination of two sub-systems. The interaction between humans and parasites is described, again, by system 3.1. In this case, cercariae are assumed to be proportional to the total number I of infectious snails, C = ζI, where ζ represents the con-stant number of cercariae released by one infected snail multiplied by the residence time of cercariae in water.
In order to consider, in a less complex way, the prepatent period after initial infection, snail dynamics is now described via a compartmental SEI-like model, that introduces a delay between infection and onset of infec-tiousness. The snail population is thus divided into Susceptible, Exposed, and Infectious individuals, according to their ability to be infected and to infect (i.e. their ability to release cercariae after initial infection). The sys-tem has the following form:
˙ S = νS [1 − γ(S + E + I)] − µSS − ρM S ˙ E = ρM S − (µS+ η)E − δE ˙ I = δE − (µS+ η)I, (3.10)
where all the parameters have the same meaning as before. After infection, snails enter the Exposed compartment (they are infected but not yet infec-tious, i.e. they do not release cercariae). After a prepatent period of average duration 1/δ, they complete the cycle entering into the Infectious class and starting to release cercariae. A sketched diagram of the model is shown in Fig. 3.5.
Human hosts N Adult parasites P Susceptible snails S Miracidia M E Cercariae C I Exposed snails Infectious snails
Figure 3.5: Schematic diagram of the compartmental SEI-like model. The model is characterized by five state variables: human hosts, adult parasites, and uninfected/-exposed/infected snails (grey blocks). The larval forms of the parasites are assumed to be proportional to other relevant variables (white blocks). The arrows describe the fluxes, the dashed arrows represent indirect interactions. See details about the model in the text.
The full model that describes schistosomiasis dynamics is thus repre-sented by the following system of five nonlinear differential equations:
˙ N = µH(H − N ) − αP ˙ P = βζIN − (µH + µP + α)P − α k + 1 k P2 N ˙ S = νS [1 − γ(S + E + I)] − µSS − χP S ˙ E = χP S − (µS+ η)E − δE ˙ I = δE − (µS+ η)I. (3.11)
All variables and parameters of the model are listed in Tab. 3.2. 3.2.2 Parameter estimation
The model is parameterized as in Section 3.1.2, while, again, sensitivity and bifurcation analyses are used to explore uncertain parameters effects.
The parasite-induced mortality in the human host is first considered as negligible (i.e. α = 0), in order to make the model amenable to analytical investigation. This hypothesis is later relaxed, by allowing α to assume values > 0 (for example, the value estimated for an endemic area in Sudan was α = 1.1 · 10−7/day [11]). In this latter case, the clumping parameter of
the negative binomial distribution of parasites within human hosts is set to the value estimated in a previous study of a similar model [3], k = 0.243 (see Fig. 3.6 for the distribution of S. mansoni eggs in a study sample [13]).
0 100 200 300 400 500 600
Eggs per gram of faeces
0 50 100 150 200 250 F re q u e n cy (# o f in d ivi d u a ls)
Figure 3.6: Data and theoretical distribution of parasites among human hosts. The bars represent the data for a Brazilian population of 597 individuals (data extracted from Bethony et al. [13]). The red points represent a negative binomial distribution with clumping parameterk = 0.243 estimated from the same data in Feng et al. [3].
In addition to all the parameters already introduced in previous sections, the average duration of the prepatent period is assumed to be about 2 weeks [3, 10] (δ = 6.7 · 10−2/day). As for the logistic function, carrying capacity is assumed to be ≈ 100 snails/m2 in absence of the parasite [12] (γ = 10−2 m2/snail). The intrinsic natality rate of snails is strongly dependent on the environmental conditions [14] and exposition to schistosomes [6]. Each snail is assumed to produce at most 180 eggs in 10 weeks [6] and, with a hatching rate of about 30%, ν is set to 0.7/day. All parameters are listed in Tab. 3.2.
A thorough stability and bifurcation analysis of the model is given in the following.
T able 3.2: State variables and parameters of the SEI-lik e compartmental model : description, value , units and refer ences. Name Description V alue Units References N Number of human hosts indi vidual P Number of adult parasites parasite S Density of susceptible snails snail · m − 2 E Density of exposed snails snail · m − 2 I Density of infectious snails snail · m − 2 H Human community size 1,000 indi vidual fix ed µH Per capita natural death rate of humans 4 · 10 − 5 day − 1 fix ed µP Per capita natural death rate of adult parasites 5.5 · 10 − 4 day − 1 [3, 10] µS Per capita natural death rate of snails 2.7 · 10 − 3 day − 1 [3] η Disease-induced death rate of snails 1.37 · 10 − 2 day − 1 [3] δ Exit-from-prepatenc y rate in snails 6.7 · 10 − 2 day − 1 [3, 10] ζ Cercariae releasing rate by one snail 700 cercaria · snail − 1 [1, 3] α Disease-induced death rate of humans per unit of parasite b urden 0, 1.1 · 10 − 7 day − 1 [11] k Clumping parameter of parasite distrib ution 0.243 -[3] γ Rate of competition for resources among snails 10 − 2 m 2 · snail − 1 [12] ν Intrinsic natality rate of snails 0.7 day − 1 [6] χ Per capita rate of infection of snails V ariable parasite − 1 · day − 1 β Per capita rate of infection of humans V ariable parasite · m 2 · cercaria − 1 · indi vidual − 1 · day − 1
3.2.3 Equilibria and stability
The nonlinear analysis of the system is performed under the assumption that the extra human mortality rate induced by one adult parasite is much smaller than the other parameters, so that it can be set to zero. Thus the following simplified system is studied:
˙ N = µH(H − N ) ˙ P = βζIN − (µH + µP)P ˙ S = νS [1 − γ(S + E + I)] − µsS − χP S ˙ E = χP S − (µS+ η)E − δE ˙ I = δE − (µS+ η)I. (3.12)
Because the first equation is not coupled to the rest of system (3.12), it is sufficient to analyze the sub-system with state vector X = [P, S, E, I]. Setting ˙X = 0 and N = H provides two equilibria:
• A parasite-free equilibrium X0 = (P0, S0, E0, I0), given by:
X0 = S0 = ν − µS νγ P0 = 0, E0 = I0 = 0 (3.13)
• An endemic equilibrium X+= (P+, S+, E+, I+), given by:
X+= P+= βζH µH + µP I+ S+ = (µH + µP)(µS+ η + δ)(µS+ η) χβζHδ E+ = µS+ η δ I+ I+ = ν − µS νγ − S+ µS+ η δ + χβζH νγ(µH + µP) + 1 (3.14)
