A Game-Theoretic Model of Monkeypox to Assess Vaccination Strategies
Sri Vibhaav Bankaru, Depts. of Biomedical Engineering, Math, & Chemistry, Samuel Kossol, William Hou, & Parsa Mahmoudi, with Dr. Jan Rychtar, Dept. of Mathematics
Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Central and West African countries. In this paper, we analyze a compartmental model of MPX dynamics. Our goal is to see whether MPX can be controlled and eradicated by voluntary vaccinations. We show that there are three equilibria - disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be eradicated in a semi-endemic equilibrium by vaccination. However, in a fully endemic equilibrium, MPX cannot be eradicated by vaccination alone.https://scholarscompass.vcu.edu/uresposters/1296/thumbnail.jp

Bankuru, Sri Vibhaav

Hou, William

Kossol, Samuel

Mahmoudi, Parsa

Rychtář, Jan

Taylor, Dewey

English

Bankaru, Sri Vibhaav

Rychtar, Jan

VCU Scholars Compass

Virginia Commonwealth University VCU Scholars Compass Undergraduate Research Posters Undergraduate Research Opportunities Program 2020 A Game-Theoretic Model of Monkeypox to Assess Vaccination Strategies Sri Vibhaav Bankaru Samuel Kossol William Hou Parsa Mahmoudi Jan Rychtar Follow this and additional works at: https://scholarscompass.vcu.edu/uresposters  © The Author(s) Downloaded from Bankaru, Sri Vibhaav; Kossol, Samuel; Hou, William; Mahmoudi, Parsa; and Rychtar, Jan, "A Game-Theoretic Model of Monkeypox to Assess Vaccination Strategies" (2020). Undergraduate Research Posters. Poster 279. https://scholarscompass.vcu.edu/uresposters/279 This Book is brought to you for free and open access by the Undergraduate Research Opportunities Program at VCU Scholars Compass. It has been accepted for inclusion in Undergraduate Research Posters by an authorized administrator of VCU Scholars Compass. For more information, please contact libcompass@vcu.edu. A GAME-THEORETIC MODEL OF MONKEYPOX TO ASSESS POTENTIAL VACCINATION STRATEGIESSri Vibhaav Bankuru1, Samuel Kossol1, William Hou2, Parsa Mahmoudi31Department of Biomedical Engineering, 2Department of Kinesiology and Health Sciences, 3Department of BiologyVirginia Commonwealth University, Richmond, VA 23284, USAIntroduction• Monkeypox virus (MPXV) is a rare zoonotic Orthopoxvirus that causes mon-keypox (MPX), a disease endemic to the Democratic Republic of the Congoand Nigeria.• There are likely less than 10000 cases of MPX worldwide. In 2003, 47cases were reported in USA and in 2018, 3 cases reported in UK.• Common symptoms of MPX, though relatively milder than smallpox, includefever, severe headaches, skin lesions, and myalgia• The case fatality rate is over 10%.• A smallpox vaccine is an effective prevention against MPX.• Administration of the smallpox vaccine has ceased since the disease’s erad-ication in 1980, resulting in lowered immunity against Orthopoxviruses ingeneral.• We analyze a game theoretical model of MPX with voluntary smallpox vac-cinations.MPX overview• MPX is capable of being transmitted amongst rodents, and primates.• The majority of reported human cases originate from an interaction with aninfected animal.• We consider only a moribund rope squirrel (Funisciurus anerythrus) andsun squirrels (Heliosciurus rufobrachium) .• The incubation period for the virus ranges from 5 to 21 days.• MPX infection is split into 2 distinct phases: the invasion period (fever, lym-phadenopathy, intense asthenia, severe headaches, and myalgia) and theskin eruption period (rash and fluid filled blisters, the number vary from afew to thousands across the body).Scheme of MPX transmissionShVhEh Ih RhSs Es Is RsβshIsNs+ βhhIhNhµhαhµhµhνh ρhµh + dh µhΛhΛsµsβssIsNsµsνsµs + dsρsµsFig. 1: Susceptible (S), Exposed (E), Infectious (I), Recovered (R), Vaccinated (V ). h - humans, s - squirrels. The ratesare explained in Table 1.ODE model of MPX transmissiondSsdt= Λs −(µs + βssIsNs)SsdEsdt= βssIsNsSs − (µs + νs)EsdIsdt= νsEs − (µs + ds + ρs)IsdRsdt= ρsIs − µsRsdShdt= Λh −(µh +βshIsNs+βhhIhNh+ αh)ShdVhdt= αhSh − µhVhdEhdt=(βshIsNs+ βhhIhNh)Sh − (µh + νh)EhdIhdt= νhEh − (µh + dh + ρh)IhdRhdt= ρhIh − µhRhSymbol Meaning ValueΛh Human birth rate 0.0328Λs Squirrel birth rate 2µh Human natural death rate 1/60µs Squirrel natural death rate 0.5dh Human MPX related death rate 3.12ds Squirrel MPX related death rate 17.5ρh Human recovery rate 28.08ρs Squirrel recovery rate 12νh Human infection rate 30.42νs Squirrel infection rate 120αh Vaccination rate variableβss Squirrel-to-squirrel trans. rate 40βsh Squirrel-to-human trans. rate 0.05βhh Human-to-human trans. rate 32.85CV Cost of vaccine $4CMPX Cost of MPX infection $100Table 1: Parameters of MPX dynamics. See preprint for the references.Results - Equilibria of the dynamicsDisease-free (0) Fully Endemic (∗) Semi-endemic (†)NsΛsµsΛs·(µs+ds+ρsνs+1+ρsµs)µs·(µs+ds+ρsνs)+ds·(1− 1R0ss)+µs+ρsΛsµsSs N0s N∗s · 1R0ss N†sEs 0(µs+ds+ρsνs)· I∗s 0Is 0Λs−µsN∗sds0Rs 0ρsµs· I∗s 0NhΛhµhsee preprint see preprintShµhµh+αh·N0h(µh+νh)(µh+dh+ρhνh)·I∗hβsh(I∗sN∗s)+βhh(I∗hN∗h) (µh+dh+ρh)(µh+νh)νhβhh·N†hVhαhµh+αh·N0hαhµh· S∗h(αhµh)· S†hEh 0(µh+dh+ρhνh)· I∗h(µh+dh+ρhνh)· I†hIh 0Λh−µhN∗hdhΛh−µhN†hdhRh 0ρhµh· I∗hρhµh· I†hTable 2: Three different equilibria of the MPX dynamics. The formulas for N ∗h and N†h are too long for the table and are given in manuscript.The disease-free equilibrium is (locally asymptotically) stable when R0ss < 1 and R0hh < 1. The endemic equilibrium is stable when R0ss > 1.The semi-endemic equilibrium is stable when R0ss < 1 and R0hh > 1.R0ss = βss ·1µs + ds + ρs· νsµs + νsR0hh = βhh ·µhαh + µh· 1µh + dh + ρh· νhµh + νh.Results - Herd immunity and Nash equilibrium• Herd immunity is impossible to achieve in the fully endemic equilibrium(due to the reservoir of MPX in squirrel population).• In the semi-endemic equilibrium, one can achieve herd immunity when thevaccination rate reachesαHI = max{0,νhβhhµh(µh + dh + ρh)(µh + νh)− µh}.• The Nash equilibrium vaccination rate αNE is given as a solution ofCnotV (αh) = CV whereCnotV (αh) = CMPX ·(βshIsNs+ βhhIhNh)(βshIsNs+ βhhIhNh)+ µh · ( νhνh + µh).0 0.02 0.04 0.06 0.08 0.1Vaccination rate h0102030405060CostFully endemic equilibrium0 0.02 0.04 0.06 0.08 0.1Vaccination rate h0102030405060CostSemi-endemic equilibriumFig. 2: Costs vs. Vaccination rate when the effective human-to-human transmission rate is high, βhh = 60. Left: the fullyendemic state. Right: the semi-endemic state (βss = 30 < (µs + ds + ρs) · µs+νsνs ). The same scenario occurs whenβsh = 0 and βss is arbitrary. All other parameters as in Table 1.Conclusions• We provided closed form formulas of the equilibrium states of the dynam-ics.• We showed a potential existence of the previously neglected semi-endemic equilibrium, in which there is no infection in the animal populationand the disease still persists in the human population.• We demonstrated that αNE is about 10 times more sensitive to parametersrelated to vectors than to a corresponding parameter related to humans. Itis therefore important to provide accurate estimates.• As cases of MPX become increasingly reported among humans, we hopethat the models like ours may serve as a predictive tool to better study thespread of MPX.AcknowledgementsThe work on this project was done as part of the course MATH/BIOL 380 - In-troduction to mathematical biology. We acknowledge the help and support of ourclassmates, the instructor Dr. Rychtárˇ, and Dr. Taylor.

A Game-Theoretic Model of Monkeypox to Assess Vaccination Strategies

Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Central and West African countries. In this paper, we analyze a compartmental model of MPX dynamics. Our goal is to see whether MPX can be controlled and eradicated by voluntary vaccinations. We show that there are three equilibria—disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be eradicated in a semi-endemic equilibrium by vaccination. However, in a fully endemic equilibrium, MPX cannot be eradicated by vaccination alone

A game-theoretic model of Monkeypox to assess vaccination strategies

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A Game-Theoretic Model of Monkeypox to Assess Vaccination Strategies

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