# How to bring a system of differential equations to zero with DifferentialEquations?

**URL:** <https://discourse.julialang.org/t/how-to-bring-a-system-of-differential-equations-to-zero-with-differentialequations/53659>\
**Category:** Biology, Health, and Medicine\
**Tags:** diffeq\
**Created:** [January 20, 2021, 8:48am UTC](https://discourse.julialang.org/t/how-to-bring-a-system-of-differential-equations-to-zero-with-differentialequations/53659 "2021-01-20T08:48:59Z")\
**Posts on this page:** 1\
**Showing post:** 15

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**Author:** ![Luigi\_Marongiu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/luigi_marongiu/32/7909_2.png) [@Luigi\_Marongiu](https://discourse.julialang.org/u/Luigi_Marongiu)\
**Post date:** [January 24, 2021, 6:55am UTC](https://discourse.julialang.org/t/how-to-bring-a-system-of-differential-equations-to-zero-with-differentialequations/53659/15 "2021-01-24T06:55:35Z")

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Sorry, I am no mathematician, so most of this is beyond my comprehension. But I have found this – the problem is to distinguish the bacteria (blue line) by adding phage (green line) in what is known as phage therapy. I calculated the minimal amount to add by the formula `Vp = (mu/phi) * (1-(r_s0/kappa)) - omega/phi` (which I obtained by equating dN/dt to zero end solving for P). I don’t get extinction using such amount, but If I add more and more phage, I achieve extinction:

 ![treatment_8](https://global.discourse-cdn.com/julialang/original/3X/f/7/f7e33612790aa68d2f698091ef2f7775093ed1cb.png)  
 ![treatment_1](https://global.discourse-cdn.com/julialang/original/3X/9/4/946d090432b7c276b8ef55e36dfc4d6febbcb58e.png)  
 ![treatment_half9](https://global.discourse-cdn.com/julialang/original/3X/f/d/fdfa64e17c78b1901d934e4b39a592cb137b6a1a.png)  
So my question is: how do I calculate the right amount of phage and its time of delivery?  
Perhaps the boundary problem is not the right tool, but the fact that if I add one billion phages I do reach extinction tells me there must be a solution. But Instead of randomly selecting values I would like to find the right amount straight away, even because the conditions will change in future models. Same thing for the time of administration. Here I subjectively chosen t=1000, but there should be a better time (specifically when bacteria are at a certain density, even if in this context the density is constant).  
I tried the suggestions but:

```julia
julia> function bc!(residual, u, p, t)
           residual[1] = (u[1][1] - N0)/N0 # keep the starting value constant
           residual[2] = (u[1][2] - I0)/I0 # keep the starting value constant
           residual[3] = u[end][1]/N0 # u[end][1] should be zero at the final timestep
       end
bc! (generic function with 1 method)
julia> N0 = 1.0*10^7
1.0e7
julia> I0 = 1E5
100000.0
julia> P0 = 2*10^8
200000000
julia> pGuess = 1.0
1.0
julia> tspan = (0.0,1500.0)
(0.0, 1500.0)
julia> bvp0 = BVProblem(modelFun!, bc!, [N0,I0,P0], tspan)
BVProblem with uType Array{Float64,1} and tType Float64. In-place: true
timespan: (0.0, 1500.0)
u0: [1.0e7, 100000.0, 2.0e8]
julia> sol0 = solve(bvp0, Shooting(Vern7()), dtmin=0.01, dt=0.5)
ERROR: MethodError: no method matching iterate(::DiffEqBase.NullParameters)
Closest candidates are:
  iterate(::Base.EnvDict) at env.jl:119
  iterate(::Base.EnvDict, ::Any) at env.jl:119
  iterate(::JuliaInterpreter.ExprSplitter) at /home/gigiux/.julia/packages/JuliaInterpreter/LBEYA/src/construct.jl:535
  ...
Stacktrace:
 [1] indexed_iterate(::DiffEqBase.NullParameters, ::Int64) at ./tuple.jl:84
 [2] 𝔅!(::Array{Float64,1}, ::Array{Float64,1}, ::DiffEqBase.NullParameters, ::Float64) at /home/gigiux/Documents/model/util/funCollect.jl:7
 [3] (::ODEFunction{true,typeof(modelFun!),LinearAlgebra.UniformScaling{Bool},Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing,Nothing})(::Array{Float64,1}, ::Vararg{Any,N} where N) at /home/gigiux/.julia/packages/DiffEqBase/3iigH/src/diffeqfunction.jl:248
{etc}

```

Can I reach such a solution with boundary problems?  
Thank you

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