Christophe Diagne

Emmanuelle Gilot-Fromont

Stéphane Cornet

Laëtitiahusse

Ambroise Souleymanedoucouré

Dalecky

Mamadou Khaliloubâ

YoussouphaNiang Mamoudou Kane

Aliou Diallo

Odile Sow

Sylvain Fossati-Gaschignard

Emmanuelle Piry

MbackéSembène Carine Artige

Nathalie Brouat

C Charbonnel

Diagne

A Dalecky

O Fossati-Gaschignard

L Husse

K Bâ

Y Niang

M Diallo

A Sow

IRD, CBGP M Sembène

S Piry

E Artige

N Charbonnel

Contemporary variations of immune responsiveness during range expansion of twoinvasive rodentsin Senegal

Biological invasions provide unique opportunities for studying life history trait changes over contemporary time scales. As spatial spread may be related to changes in parasite communities, several hypotheses (such as the evolution of increased competitive ability (EICA) or EICA-refined hypotheses) suggest immune changes in invasive species along invasion gradients. Although native hosts may be subject to similar changes in parasite selection pressures, their immune responses have been rarely investigated in invasion contexts. In this study, we evaluated immunevariations for invasive house mice Mus musculusdomesticus, invasive black ratsRattusrattusand native rodents Mastomyserythroleucusand Ma. natalensis along well-characterised invasion gradients in Senegal. We focused on antibody-mediated (natural antibodies and complement) and inflammatory (haptoglobin) responses. One invasion route was considered for each invasive species, and environmental conditions were recorded. Natural-antibody mediated responses increased between sites of long-established invasion and recently invaded sites only in house mice. Both invasive speciesexhibited higher inflammatory responses at the invasion front than in sites of long-established invasion. The immune responses of native species did not change with the presence of invasive species. These patterns of immune variations do not support the EICA and EICA refined hypotheses, andthey rather suggest a higher risk of exposure to parasites on the invasion front. Altogether, these results provide a first basis to further assess the role of immune changes in invasion success.

INTRODUCTION

Biological invasions, i.e., the successful establishment and spread of species outside their native range, are increasingly frequent worldwide mostly due to human activities. They often have detrimental consequences for the communities invaded [START_REF] Kolar | Progress in invasion biology: predicting invaders[END_REF]. Despite the increasing burgeoning interest in invasion science these last decades, several gaps exist in our knowledge and understanding of the factors and forces driving invasion success of nonnative species in new areas [START_REF] Facon | A general eco-evolutionary framework for understanding bioinvasions[END_REF][START_REF] Lowry | Biological invasions: a field synopsis, systematic review, and database of the literature[END_REF].

Parasitism is one factor likely to promote invasion success by influencing a range of host community interactions [START_REF] Dunn | Parasites and biological invasions: parallels, interactions, and control[END_REF]. As such, invasive species may lose their natural enemies, including micro-and macro-parasites, in their non-native ranges, with positive outcomes on invader fitness and demography (enemy release hypothesis, [START_REF] Colautti | Is invasion success explained by the enemy release hypothesis?[END_REF]). Alternatively, invaders may introduce exotic parasites with detrimental effects on the survival, fecundity and/or regulation of indigenous host populations and provide opportunities for diseases to emerge (spill-over or novel weapon hypothesis, [START_REF] Strauss | Invading with biological weapons: the importance of diseasemediated invasions[END_REF]. They may also acquire local parasites from their new environment, amplifying the impact of some of them at the expense of native species, with effects at both the individual and population scales (spillback hypothesis, [START_REF] Kelly | Parasite spillback: a neglected concept in invasion ecology?[END_REF]).As immunity is costly in terms of energy and immunopathology [START_REF] Klasing | The costs of immunity[END_REF][START_REF] Raberg | Basal metabolic rate and the evolution of the adaptive immune system[END_REF], these changes in parasite pressure have led to some predictions regarding potential variations of invaders" immune defences in the course of invasion. With regards tothe enemy release hypothesis, the invaders are expected to reallocate energetic resources from unnecessary defence mechanisms to life history traits favouring invasion success, such as dispersal or enhanced reproductive output. This prediction, known as the Evolution of Increased Competitive Ability (EICA) hypothesis [START_REF] Blossey | Evolution of increased competitive ability in invasive nonindigenous plants -a hypothesis[END_REF], has been refined to take into account the widespread occurrence of parasitism [START_REF] Lee | A role for immunology in invasion biology[END_REF]). In addition to parasites that would not be lost during invasion, invaders could be infected by local parasites (including the possibility of spill-back of parasites carried by native hosts). A drastic reduction of immune responses would make invaders highly susceptible to generalist pathogens encountered in recently invaded host communities [START_REF] Roberts | Foundations of Parasitology[END_REF]. Then, successful invaders should be those that dampen the most expensive and/or least effective immune defences instead favouring less costly and more efficient immune strategies with respect to parasites that are kept from the source area or those newly encountered on the invasion front (EICA-refined hypothesis, [START_REF] Lee | A role for immunology in invasion biology[END_REF]. Such trade-off would allow promoting other life history traits without dampening invader defences.

The mechanisms potentially mediating these phenotypic variationsinclude various processes:evolutionary ones as suggested in the name of this hypothesis, but also ecological changes. A major part of immune variation relies on differences in genetic background (reviewed in [START_REF] Ardia | The role of constraints and limitation in driving individual variation in immune response[END_REF]. Genetic drift and natural selection may therefore be key evolutionary processes by which invaders will evolve on invasion front after few generations (review in [START_REF] Charbonnel | Molecular epidemiology of disease resistance genes with perspectives for researches on biological invasions and hybrid zones[END_REF].Recently, it was also assessed that epigenetic modifications -defined as changes in phenotypes that persist through mitosis and even meiosis, but occur independently of changes in underlying DNA sequence -may also play pivotal roles in immune changes related to invasion process [START_REF] Brown | Invader immunology: invasion history alters immune system function in cane toads (Rhinella marina) in tropical Australia[END_REF][START_REF] Na | Escaping Host Immunity: New Tricks for Plant Pathogens[END_REF]. But alternatively, another important source of immune variation during invasion process could be related to phenotypic plasticity, which is the tendency/ability for phenotypes to change across different environments within generations (reviewed in [START_REF] Gervasi | The context of host competence: a role for plasticity in host-parasite dynamics[END_REF].

This mechanism has been widely reported using experimental works modifying environmental conditions (resource quality and availability, cross-fostering) or individual surveys throughout their life, in both vertebrates and invertebrates [START_REF] Schulenburg | Introduction. Ecological immunology[END_REF].

Another complexity resides in the immune system itself.In vertebrates, it is a highly diverse network of organs, cells and molecules that are generally classified into innate and adaptive compartments, both including cellular (e.g., macrophages, lymphocytes) and humoral components (e.g., antibodies, peptides), which interact together but may have different costs and benefits for the organism [START_REF] Klasing | The costs of immunity[END_REF]). It has been assessed that innate responses associated with local and/or systemic inflammation incur high-energy expenditure and major physiological, behavioural and pathological costs [START_REF] Sorci | Inflammation and oxidative stress in vertebrate host-parasite systems[END_REF] compared to responses mediated by other effectors, such as both natural (innate) and specific (acquired) antibodies [START_REF] Raberg | Basal metabolic rate and the evolution of the adaptive immune system[END_REF][START_REF] Klasing | The costs of immunity[END_REF]). This has led to suggest that invasion success would therefore be associated with dampened costly systemic inflammatory response and stronger less costly antibody-mediated immunity [START_REF] Lee | A role for immunology in invasion biology[END_REF]. Alternatively, because of potential high infection risks encountered on invasion fronts, the invaders" capacity to mount rapid, non-specific immune responses could be essential for invasion success. It would be therefore possible to make different predictions than those of [START_REF] Lee | A role for immunology in invasion biology[END_REF] for immune changes along invasion gradients, such as an increase of immune investment, or the absence of immune trade-offs [START_REF] Phillips | Life-history evolution in range-shifting populations[END_REF].

The impacts that bioinvasions may have on immune responses in native species remain scarcely described and investigated. We can first consider that native species might be exposed to a lower infection risk as a result of dilution effects (the presence of a less competent host decreases the infection prevalence in the native host, Dunn 2009) or of density-dependent effects (the density of native hosts may decrease due to competition with invaders, and infection prevalence of parasites transmitted through contacts will consequently decreased too, [START_REF] Keesing | Impacts of biodiversity on the emergence and transmission of infectious diseases[END_REF]. In this context, a decrease in immune investment could be selected expected (EICA hypothesis). Nevertheless, parasite spill-over from invasive to native species is frequent [START_REF] Harris | Review of negative effects of introduced rodents on small mammals on islands[END_REF]) and can strongly affect native host communities, even leading to local extinction, which favorsthe invasive host [START_REF] Daszak | Emerging infectious diseases of wildlife -threats to biodiversity and human health[END_REF][START_REF] Prenter | Roles of parasites in animal invasions[END_REF]). Such a strong impact of exotic infections on native species could result from the lack of efficient immune responses (either in terms of immune effectors or amounts of responsiveness). A dampened immune defense could result from direct competition with invaders through a lowered access to resources or from stress hormones such as glucocorticoids that may ultimately compromise immunity (Martin et al. 2010a). Strong immunopathologic effects affecting natives" fitness may also result from infections with exotic parasites (spill-over) or from amplified risk of infections with native parasites (spillback) (Martin et al. 2010b).

Limitedempirical data are currently available concerning the ecoimmunology of invasions [START_REF] Pedersen | Wild immunology[END_REF]Babayan 2011, White and[START_REF] White | The ecoimmunology of invasive species[END_REF]. Indeed, few studies have investigated differences in immune phenotypes along invasion gradients for a given invasive species, and even fewer have considered several immune pathways simultaneously [START_REF] Cornet | Eco-immunology and bioinvasion: revisiting the evolution of increased competitive ability hypotheses[END_REF] see [START_REF] Llewellyn | Reduced investment in immune function in invasion-front populations of the cane toad (Rhinella marina) in Australia[END_REF][START_REF] Brown | Invader immunology: invasion history alters immune system function in cane toads (Rhinella marina) in tropical Australia[END_REF]. Therefore, it remains important to improve our knowledge ofsuch immune variations associated with invasion. It is a necessary pre-requisite before addressing the eco-evolutionary processes dictating invasion success.In this study, we focused on two invasive rodent species, the domestic mouse (Mus musculusdomesticus) and the black rat (Rattusrattus).Weanalyzed phenotypic variations in immune responsiveness occurring along their well-known invasion routes in Senegal. These murid rodents are exclusively commensal in the study area (i.e., living in/around human dwellings/man-made structures) and are worldwide significant invasive species (Global Invasive Species Database -http://www.issg.org/database/). Complete syntheses reporting description of their current invasion histories in Senegal (including data of historical inventories, molecular analyses and ecological longitudinal surveys) are provided elsewhere [START_REF] Dalecky | Range expansion of the invasive house mouse Mus musculus domesticus in Senegal, West Africa: a synthesis of trapping data over three decades[END_REF][START_REF] Konecny | Invasion genetics of the introduced black rat (Rattus rattus) in Senegal, West Africa[END_REF]. Briefly, both species originated from Asia and have expanded their distribution range worldwide, making use of human migration to colonize all the continents. They were first brought to West Africa by European explorers and settlers from the 15th century. Large and likely standing populations of rats and mice are reported in coastal colonial cities from the middle of the 19 th century. At the beginning of the 20 th century, they began to expand eastwards with the development of inland commercial transport. Their distribution currently covers much of North and Central Senegal for M. m. domesticus and South and Central Senegal for R. rattus. Recent longitudinal surveys [START_REF] Granjon | Les rongeurs de l'Afrique sahélo-soudanienne[END_REF]Duplantier 2009, Dalecky et al. 2015) have documented how the ongoing expansion of both species has resulted in the extirpation of native rodents (mostly Mastomyserythroleucus and Ma. natalensis) from commensal habitats, these latter species being now found almost only in villages at the invasion front and in non-invaded areas.

Immune phenotypes were described using effectors involved in natural antibody (NAb)mediated and inflammatory immune pathways. They are main components of innate constitutive immunity acting in the earliest phase of immune defence against general challenges and new parasites [START_REF] Rossi | Innate immunity correlates with host fitness in wild boar (Sus scrofa) exposed to classical swine fever[END_REF]. First, NAbs are humoral components of constitutive immunity, providing the first-line of non-cellular protection against antigens [START_REF] Matson | A hemolysis-hemagglutination assay for characterizing constitutive innate humoral immunity in wild and domestic birds[END_REF]. They serve as recognition molecules capable of opsonising invading micro-organisms and initialising the Complement (Cp) enzyme cascade, which ends in cell lysis [START_REF] Carroll | Linkages of innate and adaptive immunity[END_REF]. They are unique among immunoglobulin molecules because their presence does not require prior exposure to exogenous antigens. The assessment of this immune pathway is particularly appealing because NAbs should be less sensitive than acquired antibody responses to short-term variations in environmental conditions, nutritional status, or stress levels [START_REF] Baumgarth | Innate and acquired humoral immunities to influenza virus are mediated by distinct arms of the immune system[END_REF]. Moreover, Nab levels in the serum could be positively correlated with the ability to produce antibodies after a challenge [START_REF] Matson | A hemolysis-hemagglutination assay for characterizing constitutive innate humoral immunity in wild and domestic birds[END_REF]. Second, we used haptoglobin (Hp) concentration to assess the inflammatory state of the rodents, as previously performed in wild birds (Martin et al. 2010a). Hp is a multifunctional hepatic acute-phase protein highly released in the blood during inflammation, with strong anti-inflammatory and anti-oxidant properties [START_REF] Huntoon | The acute phase protein haptoglobin regulates host immunity[END_REF]. It circulates at low concentrations in the blood of a range of animal species and its concentration is dependent on health status [START_REF] Dobryszycka | Biological functions of haptoglobin-new pieces to an old puzzle[END_REF]. Hp has been shown to be a distinctive trait of individuals [START_REF] Matson | Baseline haptoglobin concentrations are repeatable and predictive of certain aspects of a subsequent experimentally-induced inflammatory response[END_REF]. Both immune tests used here are simple, highly repeatable and non-specific. Moreover they do not require the recapture of animals or their maintenance in animal facilities. For these reasons, they enable to cope with the specific constraints of comparative immunological studies dealing with numerous samples of different non-model species caught in natural populations and sacrificed at the time of capture. All assays were performed at both long-established and recently invaded sites, for both invasive species, and for Mastomys species found either in sympatry with the invaders at the invasion frontor alone at sites not yet invaded.

We specifically addressed the two following questions: 1)do the immune responsiveness patterns observed in rats and mice along invasion routes support the EICA or EICA refined hypotheses? We expected a decrease of immune responsiveness (Nabs and Hp) in recently invaded localities compared to long established ones under the EICA hypothesis, or a decrease of costly immune responsiveness (Hp) at the expense of less costly ones (NAbs) under the EICA-refined hypotheses;2) do native species exhibit variations in immune responsiveness associated with the presence of invasive ones ?Up to now, no framework has been developed with regard to this question, what prevents us to make any predictions.Altogether, these results enabled to discuss some of the general predictions of Lee and Klasing"s paper (2004), which is one of the reference work when studying the potential links between immunology and invasion. Nevertheless, this study is descriptive and as such, it was not designed todisentangle the ecological and evolutionary processes potentially underlying the immune patterns observed.

MATERIALS AND METHODS

Ethical Statement

Trapping campaigns within private properties were systematically realized with prior explicit agreement from relevant institutional (CBGP: 34 169 003) and local authorities. All animalrelated procedures were carried out in accordance with relevant requirements of Senegalese and Frenchlegislation and following official ethical guidelines [START_REF] Sikes | Guidelines of the American Society of Mammalogists for the use of wild mammals in research[END_REF].

Rodent sampling and blood collection

Field sampling was conducted separately along an invasion route for each invasive species (Figure 1). It was carried out during the dry season, in March-April 2013 for the "mouse" invasion route and from November 2013 to February 2014 for the "rat" invasion route. The sampling sites belonged to one of three invasion status categories, defined on the basis of historical records and longitudinal surveys (see references in [START_REF] Dalecky | Range expansion of the invasive house mouse Mus musculus domesticus in Senegal, West Africa: a synthesis of trapping data over three decades[END_REF][START_REF] Konecny | Invasion genetics of the introduced black rat (Rattus rattus) in Senegal, West Africa[END_REF]): (i) sites of long-established invasion, in which invasive populations have been recorded for centuries; (ii) sites at the invasion front (= sites recently invaded) in which invasive populations have been established for less than 30 years; and (iii) non-invaded sites in which invasive species have never yet been recorded or trapped. For each category, we sampled three to four sites (Figure 1). We used a standardised sampling protocol for all localities of the three categories of sites considered. It enabledto standardize the potential impacts of stress on immune responses due to animal capture and handling. The trapping procedures are described in detail elsewhere [START_REF] Dalecky | Range expansion of the invasive house mouse Mus musculus domesticus in Senegal, West Africa: a synthesis of trapping data over three decades[END_REF]. At each site, we set at least 120 traps within houses during one to three successive nights, with the aim to capture 20 adult rodents per species. Traps were checked and baited once a day with peanut butter supplemented with fresh onions. Rodents captured at nightwere retrieved the following morning and then sacrificed by cervical dislocation within the following four hours. They wereweighed to the nearest 0.5 g, sexed and dissected. Identification was based on morphometrics (head-body, tail, hind foot and ear lengths) and genetics (ZFY2 gene-based RFLP for identification to the subspecies level for M. musculus; cytochrome b gene-based RFLP for species characterisation in the genus Mastomys). As suggested by [START_REF] Granjon | Les rongeurs de l'Afrique sahélo-soudanienne[END_REF], rodents were considered to be adults on the basis of body weight and reproductive status. Blood samples were collected by cardiac puncture after the animals were euthanatized. They were kept on ice for 24 hours and then centrifuged. The floating serum supernatant fraction was removed, frozen in liquid nitrogen, and then stored in a freezer (-20°C) in Dakar (Senegal). Samples were transferred in dry ice from Dakar to Montpellier (France) in accordance with the regulations enforced in Senegal and France.

Environmental data

Because environmental variations may drive differences in immune responses between rodent populations independently of the invasion status categories, we described relevant climatic and commensal habitat parameters for all sampling sites and included them in further statistical analyses. We focused on these factors and did not include vegetation information because house mice and black rats are strictly commensal in Senegal, and because both invasive species expand their range through human trade and transport rather than by individual dispersal in the wild [START_REF] Dalecky | Range expansion of the invasive house mouse Mus musculus domesticus in Senegal, West Africa: a synthesis of trapping data over three decades[END_REF]. Means and standard deviations of climatic data collected between 1997 and 2012 were used (data available on http://www.ncdc.noaa.gov/cdo-web/datasets for temperatures, and http://richardis.univ-paris1.fr/precip/rainser1.html for rainfall with GPCP-1DD as the source of data). We considered rainfall data in mm (for each year:cumulated annual rainfall, cumulated rainfall during the rainy season, minimum and maximum monthly rainfall during the rainy season) and temperature data in degrees Celsius (for each year: monthly mean, monthly mean minimum and maximum, minimum of monthly mean minimum and maximum of monthly mean maximum). Commensal habitat characteristics were recorded during trapping sessions.

In particular, we recorded for each sampled room the material used (sand, banco, cement, sheet metal, fibers) for each part of the building (floor, wall, ceiling), the type of room (dwelling house, shop, storehouse, kitchen) and for each site the inhabited area surface estimated using Google Earth Pro 7.1.

Hemagglutination-hemolysis (HA-HL) assay

The goal of this assay was to evaluate components of humoral innate immunity through the ability of plasma to agglutinate and lyse foreign cells through NAbs-Cp system. We characterised the serum agglutination of heterologous red blood cells (RBC) due to NAbs (HA assay) and RBC hemolysis due to NAbs-mediated Cp activation (HL assay), with a slightly modified version of the protocol described by [START_REF] Matson | A hemolysis-hemagglutination assay for characterizing constitutive innate humoral immunity in wild and domestic birds[END_REF] for mammal species. Briefly, we used chicken RBCs as target cells, and serum samples from a rabbit immunised against chicken RBCs as positive controls. All the HA/HL assays from an invasion route were carried out using the same RBC suspension. We diluted twofold serially (from 1/2 to 1/128) ten rodent serum samples per plate with 10µl in every well. The plates were read blindly following [START_REF] Rossi | Innate immunity correlates with host fitness in wild boar (Sus scrofa) exposed to classical swine fever[END_REF] and scores were attributed for each sample (column) as the log2 of the last dilution exhibiting each phenomenon (HA, HL). To exclude potential observer effects, all images were scored the same day by the same trained observer; thus correction for observer effect was not necessary in the subsequent analyses. It was possible to assign a half-point score in cases of ambiguity. High scores of HA reflected high concentrations of NAbs in the blood. Levels of HL reflected both the NAbs and complement activities.

Haptoglobin (Hp) assay

Hp was quantified from 10µl of serum with a commercially available colorimetric immunoassay kit ("PHASE" Haptogloblin Assay, TP-801, Tridelta), according to the manufacturer"s instructions. Absorbance at 650 nm was determined with a spectrophotometer, both before and after the addition of the final reagent triggering the colorimetric reaction. We used the pre-scan absorbance to correct for differences in plasma colour and cloudiness, including the initial redness of the serum(i.e., the initial serum hemolysis) that can hamper the assay if not taken into account [START_REF] Matson | Baseline haptoglobin concentrations are repeatable and predictive of certain aspects of a subsequent experimentally-induced inflammatory response[END_REF]. Serum Hp concentration (mg/ml) was estimated by comparing the difference in absorbance (finalpre-scan) to a calibration curve.

Statistical analyses

In order to assess whether environmental features differed between the three categories of sampled sites, we carried out a two-stage Principal Component Analysis (PCA) with sites as observations, independently for climatic features (using 16-years mean values and their respective standard deviations) and commensal habitat characteristics (proportion data). As many indicators appeared highly correlated (r> 0.75), we kept only one variable in each set of highly correlated variables, based on both contributions to the axis construction and quality of representation. We then performed a final PCA with remaining variables. We tested statistically whether data were structured according to invasion status categories using a Between/Within-groups Analysis (BWA). Monte-Carlo tests of permutations (999 permutations) were applied to analyse the statistical significance of the groups graphically observed on the PCAs.

Variations in Hp concentration (log-transformed), HA and HL values were analysed independently, using linear models, in R software v.3.1.0 (R Core Team 2015). The "mouse" and "rat" invasion routes were investigated separately. The starting models included a factor combining the species and site status on the invasion route, hereafter referred to as "specific invasion status" and classified into four categories : A) invasive species alone in sites of long-established invasion; B) invasive species at recently invaded sites; C) native species at recently invaded sites and D) native species at non-invaded sites. Combining the two factors "species" and "site category" into a single one (called "specific invasion status" hereafter) enabled to avoid confounding effects between host species and invasion on immune responses. Individual factors (sex, body mass and age class) may greatly affect immune system parameters. We therefore included these factors and their 2-ways interactions with specific invasion status as fixed effects. We also added the descriptors of environmental variations as potential predictors in the full models. Therefore, coordinates of each site on the two first PCA axes for both climatic and commensal habitat data were included. Finally, we had to include several additional factors specific to the different experimental protocols used:

(i) a "plate" factor in the HA model to take into account both the chronology in which the plates were analysed and the involvement of two different experimenters, and (ii) HA scores in the HL model as HL directly depends on the presence of antibody-antigen complex revealed by HA. For both the HA and HL models, we also included the initial level of serum hemolysis as initial hemolysisis expected to interfere with the interpretation of HA and HL results.A value was then systematically given to the serum prior dilution, ranging from 1 (not red) to 8 (dark red) according to the intensity of the initial serum redness in order to correct for non-visibility and over-interpretation of the phenomena during the plate reading. The Akaike information criterion with correction for samples of finite size (AICc) was used for model selection. Models with all possible combinations of the terms included in the starting model were generated with the MuMIn v.1.10.5 R package (dredge function, [START_REF] Barton | MuMIn: Multi-Model Inference[END_REF]).

Models with a ΔAICc< 2 with respect to the model with the lowest AICc were selected and the most parsimonious of these models was chosen. The significance of explanatory variables and their interactions was determined by deletion testing and log-likelihood ratio tests. The assumptions of each final model were checked graphically, by an analysis of their residuals.

Post-hoc tests for multiple comparisons were carried out with Tukey"s test (95% family-wise confidence level).

RESULTS

We analysed serum samples from 646 individuals belonging to four rodent species captured at 23 sites (Table 1, Figure 1). The dominant species in the native rodent communities were Ma. erythroleucus along the "mouse" invasion route and Ma. erythroleucus or Ma. natalensis along the "rat" invasion route. Mastomysnatalensis was found specifically at the invasion front of R. rattus, coinciding geographically with the limited distribution area of this native species in Senegal. Mastomyserythroleucus was occasionally captured in sites of long-established invasion (n = 15) and in non-invaded sites beyond the invasion front in western Senegal (n = 3) along the "rat" invasion route. These individuals were too few in number for more detailed analysis. No significant difference was detected in HA (F 1,49 = 0.18, p = 0.71), HL (F 1,49 = 2.14, p = 0.28) or Hp(F 1,53 = 1.27, p = 0.38) levels between Mastomys species at sites recently invaded by R. rattus. Mastomysnatalensis and Ma.erythroleucuswere therefore considered as a single taxon, native Mastomys sp., in further statistical analyses on the "rat" invasion route.

Environmental data

We found significant climatic differences along "mouse"and "rat"(Monte-Carlo tests, p < 0.05) invasion routes. Along the "mouse" invasion route, climate seemed cooler and drier in sites of long-established invasion compared to the other sites. Along the "rat" invasion route, climate seemed cooler and wetter in sites of long-established invasion compared to the other sites.

These results are well represented on the second axis of mouse and rat PCAs (Figure S1a, Figure S2a, supplementary materials). The Between/Within analysis on commensal habitat characteristics revealed a significant segregation between invaded and non-invaded sites along the "mouse" invasion route (Figure S1b, supplementary materials), suggesting more traditional buildings in non-invaded sites. Along the "rat" invasion route, no significant difference in commensal habitat characteristics was detected (Monte-Carlo test, p > 0.05) with regard to site invasion status (Figure S2b, supplementary materials). However, these results must be taken cautiously as this study was not designed to investigate the influence of environmental characteristics on immune responsiveness.

Variation in HA/HL estimates

"Mouse" invasion route -The most parsimonious model explaining variation in HA included the climate (component 1: F 1,290 = 5.83, p< 0.0001; component 2: F 1,290 = 16.81, p< 0.0001), plate factor (F 1,290 = 11.27, p = 0.0163), specific invasion status (F 3,290 = 12.48, p< 0.0001, Figure 2) and its interaction with sex (F 3,290 = 4.75, p = 0.0029) (Table S1, supplementary materials). Sex accounted for differences in HA levels only at non-invaded sites, with higher values recorded for males than females. Mice from sites of long-established invasion had the lowest HA values. No significant difference in mean HA was detected in Ma. erythroleucus between recently invaded and non-invaded sites (post-hoc Tukey"s test, p = 0.9999), nor between Ma. erythroleucus and M. m. domesticus at the invasion front (post-hoc Tukey"s test, p = 0.9999).

The most parsimonious model explaining variation in HL included the initial degree of serum hemolysis (F 1,293 = 99.82, p< 0.0001), HA score (F 1,293 = 50.72, p< 0.0001), sex(F 1,293 = 9.87, p = 0.0018) and specific invasion status (F 3,293 = 26.56, p< 0.0001, Figure 2) (Table S1, supplementary materials). High values of HL were recorded for high values of HA and initial serum hemolysis, and for males compared to females. Mice from sites of long-established invasion had lower HL values compared with other invasion statuses. On invasion front, M. m. domesticus had lower values of HL than Ma. erythroleucus from recently invaded (posthoc Tukey"s test, p< 0.0001) and non-invaded sites (post-hoc Tukey"s test, p = 0.0006). No significant difference in mean HL between recently invaded and non-invaded sites was detected in Ma. erythroleucus (post-hoc Tukey"s test, p = 0.8723).

'Rat' invasion route -The most parsimonious model explaining variation in HA included the initial degree of serum hemolysis (F 1,299 = 12.74, p = 0.0004), age (F 1,299 = 9.40, p = 0.0024), body mass (F 1,299 = 4.93, p = 0.0270) and climate (F 1,299 = 12.91, p = 0.0004) (Table S2, supplementary materials). Heavier and adult rodents had higher HA levels than lighter and juveniles, respectively. Serum hemolysis was negatively correlated with HA. No significant difference was found according to specific invasion status (F 3,299 = 1.41, p = 0.2399).

The most parsimonious model explaining variations in HL included the initial degree of serum hemolysis (F 1,300 = 57.76, p< 0.0001), HA score (F 1,300 = 355.92, p< 0.0001) and commensal habitat characteristics (component 1: F 1,300 = 21.82, p< 0.0001) (Table S2, supplementary materials). HL values were positively correlated with both HA scores and with initial serum hemolysis. Specific invasion status had no significant effect on HL values (F 3,297 = 0.82, p = 0.4862).

Variation in Hp estimates

'Mouse' invasion route -The most parsimonious model explaining variation in Hp concentration included climate (component 1: F 1,183 = 13.31, p = 0.0003; component 2: F 1,183 = 22.74, p< 0.0001) and specific invasion status (F 3,183 = 5.22, p = 0.0017) (Table S1, supplementary materials; Figure 2). Mice sampled from the invasion front had Hp levels twice those of mice from sites of long-established invasion (post-hoc Tukey"s test, p = 0.0039). No significant difference in mean Hp concentration was found between Ma. erythroleucus from recently invaded and non-invaded sites (post-hoc Tukey"s test, p = 0.9307). Furthermore, Hp levels were similar in Ma. erythroleucus and M. m. domesticus sampled in recently invaded sites (post-hoc Tukey"s test, p = 0.4349).

'Rat' invasion route -The most parsimonious model best explaining variation in Hp included climate (F 3,308 = 12.62, p = 0.0004; Table S2, supplementary materials ) only. However, the specific invasion status was included in about half of the models within ΔAICc< 2 (in seven of the 16 selected) with R. rattus from sites of long-established invasion having lower Hp concentrations than rodents trapped in recently invaded sites (R. rattus: post-hoc Tukey"s test, p = 0.0362; Mastomys sp.: post-hoc Tukey"s test, p = 0.0058; Figure 3). At the invasion front, Hp levels were similar in invasive and native species (post-hoc Tukey"s test, p = 0.74). No significant difference was detected in mean Hp between Mastomys sp. from recently invaded and non-invaded sites (post-hoc Tukey"s test, p = 0.2163).

DISCUSSION

Most studies investigating the role of immunity in the context of biological invasions have focused on comparisons between sympatric invasive and native species or between phylogenetically related species with contrasted levels of invasion success [START_REF] Lee | Contrasting adaptive immune defenses and blood parasite prevalence in closely related Passer sparrows[END_REF], Martin et al. 2010a). Because differences in immune investment may be due to intrinsic species characteristics, sampling designs focusing on interspecific comparisons may not be entirely appropriate for evaluating immune defencesinvasion issues. Comparing several invasive (areas in which invaders are well established vs newly colonised areas) and native (non-invaded vs recently invaded areas) populations along a well-defined invasion route appears to be a more relevant approach, as it allows overcoming specific differences and taking into account the invasion history as a spatio-temporal continuum. Moreover, potential immune changes in native communities should be considered as a full issue that may ultimately impact the invasion process. Several predictions have been proposed with regard to immunity and invasion [START_REF] Lee | Contrasting adaptive immune defenses and blood parasite prevalence in closely related Passer sparrows[END_REF]Klasing 2004, Phillips et al. 2010). Our results did not fully support the major one, which is the EICA-refined hypothesis. Antibody-mediated defences were found to increase along the invasion route, but for M. m. domesticus only (Fig. 2; Table S1, Supplementary materials). Moreover, the inflammatory response was found to be stronger in invasive populations at the invasion front than in populations from sites of long-established invasion, for both invasive species (Fig. 2; Fig. 3; Table S1, Table S2, Supplementary materials). Finally, we did not observe any difference in the immune effectors surveyed in native species between invaded and non-invaded localities.

Effects of methodological biasesand individual characteristics on immune variations

We carefully considered experimental factors in our statistical models as they could have biased our results. The initial levels of serum hemolysis and HA scores were found to significantly influence HA/HL results irrespective of the invasion route (Table S1, Table S2, Supplementary materials). Higher values lead to higher HA/HL scores, regardless to the species and the invasion status of sites. These results were expected as Cp lysis is activated by the formation of antigen-NAb complexes [START_REF] Matson | A hemolysis-hemagglutination assay for characterizing constitutive innate humoral immunity in wild and domestic birds[END_REF]. Furthermore, the initial serum hemolysis is part of the HL score and may represent the total final HL in extreme cases where NAb-mediated lysis did not occur in a serum. These results highlight the crucial importance to incorporate such "experimental" factors as potential predictors in the explanatory models.

Immune responses could also be biased due tostress resulting from capture and handling procedures. However,the immune effectors studied here are not strongly affected bystressors.

In particular, it has recently been shown that Nabs and Cpare insensitive to capture and handling stress (Buehler et al. 2008). Besides, Nab production seems to be largely independent of internal or external stimuli [START_REF] Ochsenbein | Natural antibodies and complement link innate and acquired immunity[END_REF]. In addition, we minimized the potential biases due to different effects of stress between localities of colonization statuses by using a consistent standardized capture protocol among sampling sites. This design aimed at preventing any differences in methodology-related stressors between rodent populations. Finally, a strong argument showing that capture and handling procedures had a very few impact on the immune patterns observed is that native rodents, which are also subject to these methodological stresses, exhibited no immune variation between non-invaded sites and invasion fronts. This result suggests that the invasion front itself is not an area generating specific stress reactions. Altogether, these elements argue in favour of a limited impact of stress on the immune variations observed here.

We found a significant influence of age, body mass and sex on some immune responses that were higher in adults, heavier individuals and males, respectively (Table S1, Table S2, Supplementary materials). Age and body mass influenced significantly NAb levels for both native and invasive rodents on the "rat" invasion route. These findings were congruent with other works [START_REF] Rossi | Innate immunity correlates with host fitness in wild boar (Sus scrofa) exposed to classical swine fever[END_REF] and corroborated the common trend of an increased immune capacity with host growth and condition. Our result of male mice showing higher HL levels than females corroboratesthe life-history theory predictingthat females should invest more in specific immune pathways than in constitutive immunity, which would thus be expected to be downregulated [START_REF] Lee | Contrasting adaptive immune defenses and blood parasite prevalence in closely related Passer sparrows[END_REF]. Moreover,other factors including ecological, physiological, social and behavioural ones, probably interacting in a complex way, may contribute to the observed patterns of sexual differences observed in rodent immunity. For instance, it has been shown in voles that social environment and steroid hormones affect in a complex way sex differences in immune function [START_REF] Klein | Social environment and steroid hormones affect species and sex differences in immune function among voles[END_REF]. This question therefore remains a challenging area of research in evolutionary ecology.

Impact of environmental conditions on immune patterns

As M. m. domesticus and R. rattusare likelyto encounternew environments along their respective invasion route, immune variations observed in these invasive species may be considered as responses to environmental factors(reviewed in [START_REF] Colautti | Contemporary evolution during invasion: evidence for differentiation, natural selection, and local adaptation[END_REF]. It has been previously shown that environmental parameters can affect the immune functions of rodents (see Beldomenico et al. 2008). Climate and commensal habitat structure may influence rodent immune responses, for instance through resource availability or community composition, and were thus considered in our statistical models. Nevertheless, these environmental featurescould not explain alone the patterns observed. Climatic data differed between sites of long-and recently-established invasion for both invasive mice and rats but no variation was observed in HA and HL levels for R. rattus, and climate was not a relevant predictor to explain variations of HL in M. m. domesticus. Likewise, habitat structure did not differ between sites of long-and recently-established invasion along the "mouse" invasion route, although immune responses significantly varied.Conversely, habitat structure differed between non-invaded sites and those recently invaded by the house mouse, but no immune variationwas observed in native populations along this invasion transect.

Absence of immune patterns supporting the EICA and EICA refined hypotheses

Variations in immune responses between long-and recently-established invasive populations did not support the predictions expected under the EICAand EICA refined hypotheses, which are respectively a general decrease of immune responsiveness or opposite changes between energetically costly immune pathways (decrease of responses) and less costly ones (increase of responses). Instead, this study revealed increasedinflammatory and/or humoral responses in expanding invasive populations of mice and rats compared to long established populations (Fig. 2; Fig. 3).

These results questionedthe relevance of thephysiological and ecological interactionsthat are at the basis of EICA expectations, namely parasitism and immune costs. The risks of exposure to parasites and subsequent infections are supposed to be an important component of the novel environment encountered during invasion, which may explain these immune variations along invasion routes. Nevertheless, by contrast with the enemy release emphasized in the EICA hypothesis, invaders may experience novel parasite pressure as they may be exposed and/or infected with novel local parasites when pathobiome communities differ between native and invasive species (see Diagne et al. in press;Galan et al. 2016 for examples). The overall infection risk in recently invaded areas could thus increase on invasion front. In particular, the ability to mount strong inflammatory responses may prevent "naïve host syndrome" in invasive species or within the invasion front, a severe adverse effect of parasites on hosts with which they have no co-evolutionary history [START_REF] Mastitsky | Parasites of exotic species in invaded areas: does lower diversity mean lower epizootic impact?[END_REF]. Besides, larger responses for Hp in invaders would be consistent with efforts to attenuate inflammation. Indeed, although the secretion of Hp is enhanced in inflammatory states, Hp has strong anti-inflammatory and anti-oxidative properties. During infection, Hp is best characterized as a protein that protects hosts against "all the dangers of the acute phase response" [START_REF] Dobryszycka | Biological functions of haptoglobin-new pieces to an old puzzle[END_REF]. Within 24h of infection, circulating Hp concentration increases considerably, to replenish hemoglobin stores and damp down inflammatory responses and thus, their detrimental effects. High Hp levels may therefore be of benefit to invaders due to a lessening of the negative impacts of immunity.

The extent to which the presumed costs of immune defences (high for inflammation, low for antibody-mediated responses, [START_REF] Lee | A role for immunology in invasion biology[END_REF]) can be generalized to all biological models and ecological conditions must also be questioned. Immune strategies have been shown to differ significantly between species, or even between races of domesticated species [START_REF] Mendes | Variation in the innate and acquired arms of the immune system among five shorebird species[END_REF]. Moreover, the estimation of these costs may be specifically dependent on the response examined for a given pathway, and/or the condition/energetics parameters chosen. However trade-offs between inflammation and humoral responses involving immune effectors other than those investigated here cannot be excluded.We must keep in mind that the indicators studied here only reflect one part of the immune capacities to fight infections and may not be generalizable to other effectors. Although it is very difficult to measure immune costs in the wild, a full study dedicated to this issue should provide crucial information on immune trade-offs during invasion process.

Stronger patterns of immune variationsin mouse compared to rat invasive species

Although environmental parameters and interspecific differences might beobvious reasons to explain contrasted immune variationsobserved between rats and mice along their respective invasion routes, different abilities in direct competition with native communities may also be proposed. Direct competition is often put forward as an explanation for the replacement of native species by exotic rodents [START_REF] Drake | Invasive rodents on islands: integrating historical and contemporary ecology[END_REF]. The black rat and house mouse strongly differ in their competitive interactions with native rodents. R.rattus has been shown to be aggressive to intruders, physically eliminating experimentally introduced conspecific individuals from an insular population [START_REF] Smith | Disease and competition, not just predation, as drivers of impacts of the black rat (Rattus rattus) on island mammals[END_REF]. They are much larger than Mastomys species and would therefore be at an advantage in direct competition.On the contrary, M. m. domesticus is known to perform poorly in direct competition with several native rodents [START_REF] Gomez Villafane | Differences in population parameters of Rattus norvegicus in urban and rural habitats of central Argentina[END_REF]).Variation in immune phenotype may therefore be a less important strategy for the invasion success of R. rattusthan for M. m. domesticus.

Invasion success mediated by immune variations in house mice would account for observations that this species is less parasitized than rats, for zoonoses for instance [START_REF] Blackwell | The role of the house mouse in diseases and zoonoses[END_REF][START_REF] Meerburg | Rodent-borne diseases and their risks for public health[END_REF]. This hypothesis needs to be tested, and it would also be interesting to investigate whether it is observed in other cases of biological invasions.

No variation of immune response in native species

The questions of what happens in native species and whether immune responses change in invaded compared to non-invaded areas remain scarcely explored. In this work, native Mastomys species exhibited similar immune responses in non-invaded and recently invaded sites, along both the "mouse" and "rat" invasion routes (Fig. 2; Fig. 3). This pattern could result from the matching between environmental conditions (especially parasite pressure, resource availability, etc.) and the development of adequate immune responses by native rodents that have co-evolved with their natural environment. However, we found significant environmental differences between non-invaded sites and invasion fronts for both invasion routes. It is also likely that novel epidemiological conditions are established during invasion as the introduction of invaders directly affects host community composition and may also modify the infection risk of native species (Dunn and Hatcher 2015, Diagne et al. in revision, Galan et al. unpublished data). As such, the infection risk may be either amplified by spillover and/or spill-back mechanisms, or decreased when invaders are not competent to maintain and transmit local parasites (dilution effect). Thus, the absence of immune variations in native species could reflect their inability to adapt their immune phenotypes.

In summary, our data provide no support for (i) the predictions of the EICA-refined hypothesis that antibody-mediated immunity should be favoured over inflammation during invader range expansion, and (ii) our expectations on immune changes experienced by native species when co-occurring with invaders. Whether immune changes could occur due to a higher risk of exposition to/infection with novel parasites at the invasion front and a greater ability of invasive species, such as M. m. domesticus in particular, to adjust their immune phenotypes during invasion are hypotheses that need to be confirmed experimentally in the future. Such studies could also enable to assess the respective roles of evolutionary and ecological processes in driving these phenotypic immune variations. Table S2: Results of the linear mixed-effect models for hemagglutination (HA),hemolysis (HL) and haptoglobin (Hp) variations along the Rattus. rattusinvasion route. "i:j" indicates the interaction between the factors "i" and "j". Significant effects are highlighted in bold. AICc:

Akaike"s information criterion with correction for finite sample size. Δ: difference between the model chosen and the model with the lowest AICc. N: total number of models selectedasΔAICc< 2 with respect to the model with the lowest AICc.S: number of models selected in which the factor was significant.

Figure 1 .

 1 Figure 1.Sampling sites. Triangles, squares and circles correspond respectively to sites of

Figure 2 .

 2 Figure 2. Effects of specific invasion status on the variation in (i) hemagglutination (HA), (ii)

Figure 3 .

 3 Figure 3.Effects of specific invasion status on the variation of serum haptoglobin (Hp)

Fig. S2 .

 S2 Fig. S2.Principal component analysis (PCA) of climatic (a) and commensal habitat (b) data for

Fig. S3 .

 S3 Fig. S3.Principal component analysis (PCA) of climatic (a) and commensal habitat (b) data

Table 1 :

 1 Sample size for each assay, by invasion status (LI = sites of long-established invasion; IF = invasion front; NI = non-invaded sites), sampled site (code in parentheses) and host species for a) the "mouse"invasion route (166 Mus musculusdomesticus, 145Mastomyserythroleucus) and b) the "rat" invasion route(196 Rattusrattus, 88Mastomysnatalensis, 50 Mastomyserythroleucus). Numbers in parentheses indicate sample size for males/females, respectively. "-" indicates that no rodent was trapped or analysed.Sample sizes differ between immune assays because of the limited volume of some serum samples prevented to perform both assays.

	a)
	Invasion status	Sites	Hemagglutination-Hemolysis M. m. Ma. domesticus erythroleucus	Haptoglobin M. m. Ma. domesticus erythroleucus
		Dagathie (DAG)	19(10/9)	-		12(6/6)	-
	LI	Mbakhana (MBA) Thilene (THL)	23(13/10) 20(8/12)	--		12(8/4) 13(3/10)	--
		Ndombo (NDB)	21(10/11)	-		15(7/8)	-
		Dodel (DOD)	23(9/14)	19(11/8)	10(3/7)	12(9/3)
	IF	Aere Lao (AEL) Dendoudi (DEN)	21(10/11) 17(12/5)	19(12/7) 23(16/7)	17(8/9) 10(6/4)	11(7/4) 16(12/4)
		Lougue (LOU)	22(10/12)	22(15/7)	10(3/7)	14(11/3)
		Doumga Lao (DOL)	-	20(10/10)		-	14(7/7)
	NI	Lambango (LAM) SaréMaoundé (SAM)	--	20(9/11) 9(5/4)		--	16(7/9) 5(3/2)
		DiomandouWalo (DIW)	-	13(9/4)			-	11(7/4)
	b)
	Invasion status	Sites	Hemagglutination-Hemolysis R. rattus Ma. erythroleucus Ma. natalensis	R. rattus	Haptoglobin Ma. erythroleucus	Ma. natalensis
		Diakene Wolof (DIK)	24(9/15)	-	-		24(9/15)	-	-
	LI	Diattacounda (DIT) Marsassoum (MAR)	27(13/14) 26(13/13)	--	--		27(13/14) 26(13/13)	--	--
		Tobor (TOB)	20(6/14)	-	-		20(6/14)	-	-
		BadiNieriko (BAN)	21(8/13)	11(5/6)	-		23(10/13)	12(6/6)	-
	IF	Bountougoufara (BOU) Kedougou (KED)	31(9/22) 22(9/13)	12(7/5) -	-22(9/13)	31(9/22) 22(9/13)	13(8/5) -	-22(9/13)
		Soutouta (SOU)	22(11/11)	9(5/4)	-		23(12/11)	9(5/4)	-
		Bransan (BRA)	-	-	23(10/13)	-	-	23(10/13)
	NI	Mako (MAK)	-	-	23(11/12)	-	-	26(13/13)
		Segou (SEG)	-	-	20(7/13)	-	-	21(8/13)

  Temperature data were recorded from local weather stations closest to sampled localities and available on http://www.ncdc.noaa.gov/cdo-web/datasets; rainfall data were recorded from satellite products available on http://richardis.univparis1.fr/precip/rainser1.html with GPCP-1DD as date source. Commensal habitat data were recorded directly on the field during rodent sampling.

  Temperature data were recorded from local weather stations closest to sampled localities and available on http://www.ncdc.noaa.gov/cdo-web/datasets; rainfall data were recorded from satellite products available on http://richardis.univparis1.fr/precip/rainser1.html with GPCP-1DD as date source.Commensal habitat data were recorded directly on the field during rodent sampling.

Acknowledgments.We thank Laurent Granjon and Jean-Marc Duplantier for their precious help and advice during this work. Jean-Claude Berges and Natacha Volto are thanked for their help with climatic variables. This work was supported by the ANR ENEMI project (ANR-11-JSV7-0006). We acknowledge support from the Service de Coopérationetd'ActionCulturelle (French Embassy, Senegal) and the AgenceUniversitaire de la Francophonie through Ph.D. scholarships. We are particularly indebted to all the people in Senegal who allowed us to carry out rodent trapping in their homes.

Data archiving.Data for this study will be available onDryad (DataDryad.Org) after manuscript is accepted for publication.

Supplementary materials

Table S1: Results of the linear mixed-effect models for hemagglutination (HA),hemolysis (HL) and haptoglobin (Hp) variations along the Mus musculusdomesticusinvasion route. "i:j" indicates the interaction between the factors "i" and "j". Significant effects are highlighted in bold. AICc: Akaike"s information criterion with correction for finite sample size. Δ: difference between the model chosen and the model with the lowest AICc. N: total number of models selectedasΔAICc< 2 with respect to the model with the lowest AICc.S: number of models selected in which the factor was significant.


Theodoros Spyridopoulos
email: th.spyridopoulos@bristol.ac.uk

George Oikonomou
email: g.oikonomou@bristol.ac.uk

Theo Tryfonas
email: theo.tryfonas@bristol.ac.uk

Mengmeng Ge
email: gemengmeng.2011@my.bristol.ac.uk

Game Theoretic Approach for Cost-Benefit Analysis of Malware Proliferation Prevention

Many existing research efforts in the field of malware proliferation aim at modelling and analysing its spread dynamics. Many malware dissemination models are based on the characteristics of biological disease spread in human populations. In this work, we utilise game theory in order to extend two very commonly used malware spread models (SIS and SIR) by incorporating defence strategies against malware proliferation. We consider three different security mechanisms, "patch", "removal" and "patch and removal" on which our model is based. We also propose a cost-benefit model that describes optimal strategies the defender could follow when cost is taken into account. Lastly, as a way of illustration, we apply our models on the well studied Code-Red worm.

Introduction

With the ever growing importance of networked computing, malicious software, known as malware, has been a considerable threat to the realm of interconnected computers. Often built by cyber criminals, malware aims to compromise target computers with the ultimate goal of stealing sensitive data or gaining access to private systems. Malware includes a variety of malicious software such as computer viruses, worms, trojan horses, key-loggers and many others.

Defence mechanisms such as firewalls and anti-viruses have been developed in order to defend against malicious software. Those mechanisms investigate the problem of malware at micro level by utilising experimental and heuristic findings, such as virus signatures, in order to prevent or detect and cure a computer's infection. Nevertheless, malware spread in a network of computers underlines the need for a network-level solution. The increasing number of malware which bases its function on new techniques that are difficult to detect and mitigate renders conventional defence mechanisms unsuitable. In light of these challenges epidemiological models which can describe the dynamics of malware proliferation over a computer network have been proposed.

Additionally, Game Theory has been introduced in a number of occasions across the fields of computer and network security (e.g. [START_REF] Roy | A survey of game theory as applied to network security[END_REF][START_REF] Dijk | FlipIt: the game of "Stealthy takeover[END_REF]) to describe the interactions between attacker and defender and the ways they may affect each other. As malware acts based on inscribed behaviour coded by cyber criminals, approaching it as a threat agent on its own right under the premise of game theory becomes a reasonable assumption.

Our work aims at combining well-known epidemiology models with a game theoretic framework that can describe the state of the system when the defender uses various strategies against the proliferation of a random-scanning worm. We develop a game between defender and malware, taking into account the spread dynamics, so that defenders manage to compute their optimal strategy by minimising the cost of security, on a cost-benefit basis.

The rest of the paper is structured as follows. In Section 2 we give a basic background on epidemiology models and game theory as it is applied in malware analysis. Special emphasis is given to the "FLIPIT" game. In Section 3 we present the models and methods that we have developed. In Section 4 we present an application of our approach to the well studied case of the Code-Red worm computing an optimal strategy for the defender. Finally, Section 5 discusses the conclusions drawn form this work.

Background

General Description of Epidemiology Models

The way that viruses and worms spread in a computer network shares common characteristics with the dissemination of biological diseases in human populations, in a way that the analysis of malware can benefit from investigating the behaviour of biological diseases. There are two kinds of models for analysing malware proliferation in epidemiology: stochastic models and deterministic models. Stochastic models are used to analyse small-scale networks, while deterministic models are used to analyse largescale networks [START_REF] Andersson | Stochastic Epidemic Models and Their Statistical Analysis[END_REF]. In our work, in order to study the effect of mass action, we consider malware spread in a large computer network, thus we utilise deterministic models.

In general, individuals in the epidemic population have several states, including susceptible, infected, recovered. A large fraction of the models used, rely on the transitions between those states. Among these, two models have been widely used, the Susceptible-Infected-Recovered (SIR) created by Kermack and McKendrick in 1927 [4-6] and a modified version of it, known as the Susceptible-Infected-Susceptible (SIS) model. Both models assume that all individuals within a closed population (i.e. no births and deaths) are susceptible to the malware in the initial phase and an individual may go through each state sequentially.

In the SIS model, the state transitions of an individual form a circulation. The individual may recover from the infection, but there is still a chance to be reinfected. In other words, an individual becomes again susceptible to the malware after its recovery. In the SIR model, the final state is described as the recovered state. An infected individual can recover from the infection and become immune to the malware. An immunised individual cannot be reinfected by the same malware.

Typically, disease spread depends on common shared characteristics of the individuals in a population. In a network of computers, malware exploits certain vulnerabilities in the system in order to infect a host. Common practice of malware is to exploit vulnerabilities in software that the victim-host has installed. Thus, in order for a host to be considered as susceptible to a certain malware, it has to have installed the specific software version that bears the vulnerability that the malware can exploit. In case it doesn't then it cannot be infected and thus cannot be considered as susceptible. In the real world, not every host in a network carries the same vulnerabilities. However, in our work, in order to simplify our model, we have made the assumption that our network is homogeneous. In other words, in a single network architecture an individual can infect every other individual. Furthermore, the network is assumed to be symmetric so that no preferential direction of the malware proliferation exists.

Mathematical Specification of Standard Epidemiology Models

In this section we present the mathematical specification of two commonly used epidemiology models, SIR and SIS. In general, such models are formulated over a fixedsize network. Nodes represent individuals. Links or edges between nodes represent contacts between individuals. The infection spreads along direct links between nodes.

The SIR Model In the SIR model [START_REF] Kermack | A contribution to the mathematical theory of epidemics[END_REF][START_REF] Kermack | Contributions to the mathematical theory of epidemics. II. the problem of endemicity[END_REF][START_REF] Kermack | Contributions to the mathematical theory of epidemics. III. further studies of the problem of endemicity[END_REF], the total population is divided into three parts: i) susceptible nodes (denoted by S), ii) infected nodes (denoted by I) and iii) recovered nodes (denoted by R). The differential equations 1, 2 and 3 depict the rate of change of the susceptible nodes, infected nodes and recovered nodes respectively over time [START_REF] Capasso | A generalization of the kermack-McKendrick deterministic epidemic model[END_REF]. Here β denotes the probability of a susceptible node to be infected by another infected node when they come in contact in each time unit, also regarded as the infection rate; γ denotes the probability of an infected node to recover from an infection and become immune to the malware in each time unit, also regarded as the recovery rate. In our research a contact is considered as a network link between two nodes, and since all nodes are connected with each other, either directly or through a number of hops depending on the network's topology, they are always in contact with each other.

dS dt = -βIS (1)
dI dt = βIS -γI (2)
dR dt = γI (3)
The SIS Model In the SIS model, the total population is divided into two parts, susceptible nodes (denoted by S) and infected nodes (denoted by I). Equations 4 and 5 model the rate of change of susceptible nodes and infected nodes respectively over time [START_REF] Van Der Molen | Math on malware[END_REF]. Again, β denotes the probability of a susceptible node to get infected by an infected node when they come in contact in each time unit, also regarded as the infection rate; γ denotes the probability of an infected node to recover from an infection and become susceptible again to the malware in each time unit, also regarded as the recovery rate. Even though the term "recovery rate" is used in both the SIR and the SIS model, it is used for different purposes.

dS dt = -βIS + γI (4)
dI dt = βIS -γI (5)

Brief Introduction to Game Theory

Game theory provides us with a set of analytical tools designed to describe and analyse the phenomena observed when two or more decision makers interact [START_REF] Osborne | A course in game theory[END_REF]. Decision makers are identified as unique players and the formal description of the interaction between them is denoted as a game [START_REF] Turocy | Texas a&m university[END_REF]. The basic assumption of game theory is that every player acts rationally, aiming at the best possible outcome, and take into account other players' decisions. Solution to a game is the description of the strategies that each player has to follow in order to achieve the best possible outcome. Nash equilibrium is the solution of the game that describes a steady state, where each player gets the best possible payoff. A deviation from the Nash Equilibrium strategy always leads in lesser payoff. Games are divided in various categories based on the nature of their parts:

• Cooperative -Non-cooperative Games. In general, games are divided into cooperative and non-cooperative games based on the way that players interact with each other. In cooperative games, all players try to maximise the overall payoff, while in non-cooperative games, each player cares only about his own gain and cost. In the field of network security, the research falls under the category of non-cooperative games since there is no cooperation between the attacker and the defender [START_REF] Roy | A survey of game theory as applied to network security[END_REF]. • Static -Dynamic Games. Under the category of non-cooperation, games are divided into static and dynamic games. In static games, all players make their decisions simultaneously not knowing other player strategies. They are one-shot games where each player has a pre-computed move list, each move denoted as a strategy, from which he has to choose the best move in order to maximise his personal benefit. Benefit, also known as payoff, refers to a player's net gain when he choses to play a strategy, and is described by Equation 6. In dynamic games, a player can alter his move during the game. The game is played into stages in each of which each player has to choose his move. A strategy in such games is defined as the combination of sequential moves chosen by the player in order to maximise his total benefit. Each stage of a dynamic game can be considered as a static game leading to a structure of sequential static games.

Benef it = Gain -Cost (6)
• Perfect -Imperfect Games. A game where the players choose their strategies simultaneously, without knowing the choices of the other players, is an imperfect information game. Contrary, in perfect information games every player knows exactly the strategies that other players have followed before his turn. Thus only games where players play sequentially can be considered as perfect information games.

• Complete -Incomplete Information Games. Complete information games indicate that players know the available strategies and payoffs of the other players but do not necessarily know the strategies that have been played by other players. On the contrary, in incomplete information games players may not have access to other players' available strategies and payoffs during the game. • Pure -Mixed Strategies. Pure strategies refer to deterministic actions taken by a player in the game for every possible situation that s/he can face (for every other players' actions). In mixed strategies a player's move is not based on a deterministic action-decision, but involves a probabilistic combination of the available pure strategies [START_REF] Osborne | A course in game theory[END_REF].

Game Theory in Security and Malware Analysis

Traditional network security mechanisms such as Intrusion Detection and/or Prevention Systems (IDS/IPS) analyse malware at a level of specific technical detail. They focus on collecting, dissecting and recording its structure and behaviour. This allows them to respond to attacks that are based on well known techniques. For instance, IDS algorithms apply malware-signature identification or make use of heuristic algorithms to detect suspicious system behaviours that indicate possible infection. Nevertheless, since they mostly rely on such experimental findings, they are proved to be insufficient against sophisticated attacks which may utilise unknown techniques (e.g. zero-day attacks).

A shortcoming of the traditional network security solutions is that they lack a macrolevel quantitative decision framework [START_REF] Roy | A survey of game theory as applied to network security[END_REF] and various researchers have focused their work on utilising game theory in order to provide a holistic solution [START_REF] Lin | A game theoretic approach to decision and analysis in strategies of attack and defense[END_REF][START_REF] Wu | On modeling and simulation of game theorybased defense mechanisms against DoS and DDoS attacks[END_REF][START_REF] Khouzani | A dynamic game solution to malware attack[END_REF][START_REF] Dijk | FlipIt: the game of "Stealthy takeover[END_REF]. The relationship between attacker and defender can be modelled as the interaction between two competing parts in a game theoretic scenario. The malware's goal is to spread widely, whereas the defender aims at protecting the network against the attack (minimising spread) whilst keeping costs as low as possible. Game theory can be used for studying decision making problems in multiplayer scenarios, to examine and evaluate all possible scenarios given the outcomes of each player's strategy and return the best one.

The "FLIPIT" Game: In order to develop our game we first devised a cost-benefit model to help us compute the gain of each strategy. The cost-benefit model was originally based on another game theoretic model known as "The FLIPIT Game" [START_REF] Dijk | FlipIt: the game of "Stealthy takeover[END_REF]. In FLIPIT, the authors have developed a model that describes the situation in which an attacker periodically takes over a system and is not immediately detected by the defender. There are two players, the attacker and the defender, and a shared resource. The two opponents compete to control the shared resource. The attacker tries to put the resource into a bad state, while the defender puts the resource into a good state. The objective of each player is to control the resource for the largest possible fraction of time and minimise at the same time their total cost. Players do not know the current situation of the game when other players make a move; they learn that only when they make a move. Making a move incurs cost and taking over control gains benefit. Each player loses some points per move and gains some points per second when he is in control.

The mathematical description of the game is provided below. Here we assume that the defender is player 0 and the attacker is player 1. Player i pays k i points per move and gains one point per second when the source is under his control.

γ i (t) = G i (t) t (7)
The total period of time t is the time the resource is controlled by the defender plus the time controlled by the attacker as shown in Equation 8.

G 0 (t) + G 1 (t) = t (8)
Thus, for each player, the gain rate γ i (t) is equal to the fraction of time that player i has the shared resource under control, as shown in Equation 9.

γ 0 (t) + γ 1 (t) = 1 (9)
Equation 10 calculates the benefit of a strategy, which is denoted as the gain minus the total cost. The aim of each player in the game is to maximise the value of benefit.

B i (t) = G i (t) -k i • N i (t) (10)
The generic description of a shared resource taken under control by an attacker is suitable to describe the situation of a computer network under attack from a worm. In our work we view the network as the shared resource which both attacker and defender try to take under control. However, in this context the shared resource cannot be instantly fully taken over, since a worm spreads in a fraction of the total population in each time step rather the whole population. Hence, only a fraction of the shared resource can be taken over by the attacker.

Models and Methods

Game Theoretical Models of Malware Proliferation

Worms have the ability to self-replicate and spread without human intervention in a network [START_REF] Saudi | Edowa worm classification[END_REF], resembling human viruses. In the human virus spread example this could mean that all individuals are always in contact with each other. In a network it means that an infected node can infect every other node in the network since all nodes are linked with one anoother. Random-scan worms have the ability to spread without topology constrains since they rely on random IP scans. On the other hand, worms spreading via emails have specific routes according to the email list of each infected computer. In our work we model random-scan worms.

There are three security mitigation practices against worm dissemination: i) Remove, ii) Patch and iii) both Patch and Remove. Under the SIR and SIS models, a susceptible node can either be patched against the certain worm and become immune to it or stay in the susceptible state. If a susceptible node is infected then it can either stay infected and consequently spread the worm or it can use the removal tool (e.g. an antivirus) in order to remove the worm. However, the removal tool does not encompass immunisation functionality. Thus, when an infected node removes the worm it returns back to the susceptible state where it can subsequently be reinfected. However, if an infected node uses both the remove tool and the patch against the worm then it moves to recovery state where it is immune against the specific worm. For each of the three security strategies we set up differential mathematical expressions, as in SIR and SIS models, which describe the dynamics of the system.

Patch Strategy: When the Patch Strategy is used, susceptible nodes can become immune to the worm, but infected nodes cannot recover from the infection. In this case, the worm and the defender seem to take part in a race. If the worm spreads very fast, it will infect most computers in a short time before defenders notice it; if people in the network can patch their computers much faster than the worm proliferation, the wide-range infection can be avoided. The model is depicted in Figure 1. The mathematical specification of the Patch Strategy is given in Equations 11,12 and 13, where S is the susceptible population, I is the infected population and R is the immune population. β is the probability that a susceptible node gets infected in each time unit, also regarded as infection fraction, and γ is the immunisation rate.

dS dt = -βIS -γS (11)
dI dt = βIS (12)
dR dt = γS (13)
Removal Strategy: When Removal Strategy is used, infected nodes can recover from the infection when the worm is detected and removed. However, nodes that have recovered from an infection are still susceptible to the specific worm since no immunisation against it is included. In this case the model is transformed into a SIS model where the system reaches an equilibrium where the number of infected nodes and the number of susceptible nodes stay almost constant. The model is depicted in Figure 2.

The mathematical specification of Removal Strategy is given in Equations 14 and 15. Again, S refers to the susceptible population and I refers to the infected population. β is the probability that a susceptible node gets infected and r is the removal or recovery rate. As seen, no recovered population is found in the system.

dS dt = -βIS + γI ( 14
)
dI dt = βIS -γI (15)
Patch and Removal Strategy: The last strategy devised is the Patch and Removal. In this strategy both moves of patch and removal are available. A susceptible node can become immune to the worm when patch is used. Furthermore, an infected node can recover from the infection if the worm is removed and then become immune to the worm by using the patch. This is the most efficient, yet costly, way to eliminate malware spread. Eventually, all nodes in the network will be immune against the specific worm. The strategy model is shown in Figure 3. The differential equations that describe the dynamics of the model are shown in Equations 16, 17, 18 and 19. S refers to the susceptible population, I refers to the infected population, R is used for the recovered and immunised population and Q for the that becomes immune to the malware. As before, β is the probability that a susceptible node gets infected, γ refers to the immunisation rate when a susceptible node uses the specific patch and λ is the "removal and patch" rate.

dS dt = -βIS -γS (16
)

dI dt = βIS -λI (17) dR dt = λI (18) dQ dt = γS + λI (19)

Cost-Benefit Analysis

In "FLIPIT", two opponents compete in order to gain full control of a shared resource and gain is defined by the time the resource is under one's control. In our epidemiology model, the shared resource is the node population of the network. In each time unit the two opponents (attacker and defender) perform actions to take under their control a part of the population (a number of neighbouring nodes). For instance, the attacker takes under control I nodes in each time unit, and I changes according to the equations presented above. Hence, gain is defined by the average fraction of node population under one's control. Therefore, by considering player 0 as defender and player 1 as attacker we define G i (t) the gain of player i and calculate it as shown in Equation 20, where P i (t) is the fraction of population under control by player i over time and t k is the total time for which our model is running.

G i (t) = 1 t k t k 0 P i (t)dt (20)
Since there are only two fractions of populations, one under the control of the defender and one under the control of the attacker, we can say that P 0 (t) = 1 -P 1 (t). Hence:

G 0 (t) + G 1 (t) = 1.
For player 0, we define cost (C 0 (t)) as the total number of moves made by player 0 (number of times that has used the security tool) (n 0 (t)), multiplied by each move's cost (k 0 ) (Equation 21).

C 0 (t) = n 0 (t)k 0 (21)
We define as cost for player 1 the perceived complexity of the algorithm that their malware implements. Each player's benefit is equal to the player's total gain minus the cost (Eq. 22).

B i (t) = G i (t) -C i (t) (22)
In order to compute cost, we utilise quantitative tables of operational complexity. A strategy by either player (e.g. Patch Strategy for the defender or Code-Red worm for the attacker) may encompass several actions, with each action characterised by a complexity level. We set up empirically three levels of perceived complexity, low, medium and high, and assign a score to each of them, 1, 2 and 3 respectively. Therefore, the cost of a move for player 0 or the total cost of player 1 is equal to the sum of the costs of the actions it involves.

Application of the Model to Code-Red's Parameters

In this section we apply our game theoretic model to a real case of malware proliferation, the well known Code-Red worm. Albeit old, we chose Code-Red because it is a random-scanning worm with no topology constraints and so its characteristics fit well into the generic nature of our abstraction. It is self-activated by exploiting a vulnerability which exists in the host operating system. Other worms, such as Conficker, utilise various spread methods, e.g. through email, which would warrant specialisation of the differential equations describing the proliferation and mitigation strategies.

Code-Red

Code-Red was discovered in July 2001. It exploits a buffer-overflow vulnerability in Microsoft IIS Web Server [START_REF] Moore | Code-Red: a case study on the spread and victims of an Internet worm[END_REF]. It produces a list of random IP addresses and launches 99 threads to search each computer in the list in order to infect as many vulnerable computers as possible. It has two versions: Code-Red v1 and Code-Red v2. Code-Red v1 spreads slowly because it generates an identical list by using a static seed. Code-Red v2 is the variant of Code-Red v1. It can infect new nodes by using a random seed for its pseudo-random generator. Therefore, the latter version has a higher spread speed.

The greatest damage caused by the worm is that it could launch a massive DoS attack.

Finally, this worm is memory resident. Thus a reboot can clear the worm from a host node. However, in order to prevent the infection or reinfection, nodes have to use a specific patch [START_REF] Moore | Code-Red: a case study on the spread and victims of an Internet worm[END_REF]. We make the assumption of a network with a population of 10,000 susceptible nodes and 1,000 nodes immune to the worm due to not every node running susceptible software -a Microsoft ISS Web Server. We set the maximum time of worm spread period at one week or t k = 168 hours. According to [START_REF] Vojnovic | On the race of worms, alerts, and patches[END_REF], an infected node infects other nodes with rate 1.62 nodes per hour. Thus the probability of a susceptible node in our network to get infected by an infected node in each second is equal to 1.62/N where N is the total population. Hence, β = 1.47 • 10 -4 . The costs for each player are shown in Tables 1 and2. Since the attacker uses a specific malware that is not able of changing behaviour, there is no reason in computing an optimal strategy since this has already been chosen (the Code-Red worm's inscribed behaviour). We analytically compute the defender's gain according to Equation 20, however the calculations are not shown in this paper due to space restrictions. Using Vensim as a simulation environment, we set up three simulations for the three security strategies (Patch, Remove, Patch and Remove) respectively, according to our models, in order to find the best strategy that the defender can follow. In our case study we assume that the defender has already chosen a patch rate or a removal rate so that he has only to chose the security strategy that he will follow. In an alternative scenario, the defender could also utilise our model in order to find both the security strategy and the rates that could give him the best possible benefit.

Patch Strategy

For the Patch Strategy the state of our system is shown in Figure 4. It can be seen that at the early hours of the worm spread the number of infected nodes increases sharply while on the other hand the number of susceptible nodes decreases. Since infected nodes cannot recover and patched nodes cannot be infected, the system reaches soon its equilibrium where the Infected population is near 8,351 nodes and the immunised population around 1,649 nodes. If the defender used a much larger patch rate then they might be able to compete with the attacker in this race. However, this would not be easy since the number of infected nodes increases exponentially, whereas the number of immunised nodes linearly. Furthermore, an increase in the security rate would increase the total security cost. Based on the results of the simulation, the cost (using the Table 2) and the gain (the average number of non-infected nodes) of the Patch strategy are given by Equation 23. Thus the Benefit for the Patch strategy is equal to

B 0 = G 0 -C 0 = -4, 652. C 0 = number of patches × cost of patch = 1649 • 4 = 6596 G 0 = 1944 (23)

Removal Strategy

For the Removal Strategy the state of our system is shown in Figure 5. It can be seen that after a period of time the system reaches an equilibrium where the populations of susceptible and infected nodes remain constant, 222 and 9,778 respectively. Again, the results would be better if the defender used a larger recovery rate. Based on simulation results, the cost and the gain (the average number of non-infected nodes) of the Removal strategy are given in Equation 24. Thus the Benefit for the Removal strategy is equal to

B 0 = G 0 -C 0 -156, 152. C 0 = number of removals × cost of removal = 52236 • 3 = 156708 G 0 = 556 (24)

Patch and Removal Strategy

The state of our system in time is shown in Figure 6. At the early hours of dissemination, the worm spreads exponentially into the network. However, as time passes more and more infected nodes recover form the infection and get immunised against the worm. Furthermore, the number of susceptible that are patched also increases. Thus, eventually every node in the network will be patched and thereby immunised against the worm.

Discussion

Our analysis indicates that the Patch and Removal strategy is the most efficient defence against worm dissemination, since this is the only one eventually leading to zero infected nodes left in the network. However, it is not the one that would give the best benefit to the defender. That is because, when the Patch strategy is used, infected nodes cannot be patched. Thus the total number of patches is significantly smaller and hence the total cost for the defender is lower. Therefore, although Patch and Removal is the most efficient strategy, when there are cost restrictions Patch strategy can also be used. An interesting approach could be the usage of a mixed strategy based on them.

Conclusions

In this paper we have integrated premises of game theory with malware proliferation models and developed a cost-benefit game-theoretic approach to evaluate defence strategies that mitigate malware proliferation. We applied our approach to a case study where defender could choose between three strategies i) "patch", ii) "removal" and iii) "patch and removal" and discuss our results for both the spread and the cost-benefit strategy selection. Our model can be extended by introducing more options for the defender, such as the ability to change the security (i.e. patch and removal) rate; as well as the attacker, e.g. change the manifested behaviour of the worm deployed.

As mentioned, we kept the rates of patch, removal and the infection rate constant in our simulations. However, defender can vary their security rates in order to achieve better results. Furthermore, an even more complicated approach would be to give the attacker the option to choose among strategies, in other words vary the malware's behaviour (e.g. metamorphic viruses). Thereby, we could establish a game where both players try to find their optimal strategies.

Fig. 2 .

 2 Fig. 1. Patch Strategy Model

Fig. 3 .

 3 Fig. 3. Patch and Removal Strategy Model

Fig. 4 .

 4 Fig. 4. Patch Strategy

Fig. 5 .

 5 Fig. 5. Removal Strategy

  The dissemination slowly fades. Based on the results of the simulation, the cost and the gain of the Patch and Removal strategy are given in Equation 25. Hence, the Benefit for the defender's Patch and Removal strategy is equal to B 0 = G 0 -C 0 -39, 519. C 0 = number of patches × cost of patch + number of removals × cost of patch and removal = 1771 • 4 + 8188 • 5 = 48024 G 0 = 8505 (25)

Fig. 6 .

 6 Fig. 6. Patch and Removal Strategy

Table 1 .

 1 The cost for Code-Red worm.

	Actions	Complexity Low:1 Medium:2 High:3	Total
	Exploit the buffer vulnerability	2
	Generate random IP addresses	1	4
	Launch 99 threads with generated IP addresses 1

Table 2 .

 2 The cost of each move for the defender

	Actions		Complexity Low:1 Medium:2 High:3	Total
	Patch	Detection Patch		2 2	4
	Removal	Detection Reboot	1	2	3
		Detection		2
	Patch and Removal	Reboot	1		5
		Patch		2

This work has been kindly supported by the Faculty of Engineering's Systems Centre and its Industrial Partners.